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POINTS TO REMEMBER IN CLASS

XII MATHEMATICS

By

Balraj Khurana

INDEX

1 Relations and functions - Pg 2

2 Inverse trigonometric functions - Pg 5

3 Calculus identities - Pg 6

4 Continuity - Pg 7

5 Differentiation - Pg 8

6 Application of derivative - Pg 9

7 Indefinite integral - Pg 11

8 Definite integral - Pg 14

9 Matrices - Pg 16

10 Determinants - Pg 19

11 Solution of system of linear equations - Pg 21



RELATIONS AND FUNCTIONS

I RELATION

i Let A and B be two sets A relation between A and B is a collection of ordered pairs (a b) such that a

A and b B

ii If 119877 119860 rarr 119861 is a relation from A to B then 119877 sube 119860 times 119861

iii If n(A) = m n(B) = n then total number of relations from A to B is 2mn

iv Domain of R = 119886 (119886 119887) isin 119877

v Range of R = 119887 (119886 119887) isin 119877

vi Co-domain of R = 119861

II Equivalence Relation

Let S be a set and R a relation between S and itself We call R an equivalence relation on S if R has the following three properties

Reflexivity Every element of S is related to itself ⟹ (119886 119886) isin 119877 forall 119886 isin 119878 Symmetry If a is related to b then b is related to a (119886 119887) isin 119877 ⟹ (119887 119886) isin 119877 forall 119886 119887 isin 119878 Transitivity If a is related to b and b is related to c then a is related to c (119886 119887) isin 119877 (119887 119888) isin 119877 ⟹

(119886 119888) isin 119877 forall 119886 119887 119888 isin 119878

Antisymmetric - A relation is antisymmetric if a R b and b R a⟹ a=b for all values a and b

III FUNCTIONS

Definition - Any relation on A x B in which

i No two second elements have a common first element and

ii Every first element has a corresponding second element is called a function It is also called mapping

A function is said to map an element x in its domain to an element y in its range 119891 119860 rarr119861 119900119903 119891 119909 rarr 119891(119909) 119905ℎ119890119899 119891(119909) = 119910 where y is a function of x

DOMAIN - The set of all the first elements of the ordered pairs of a function is called the domain

RANGE - The set of all the second elements of the ordered pairs of a function is called the range

CODOMAIN - If (a b) is an ordered pair of the function 119891 119860 rarr 119861 then the set B is called the Co-Domain The

range is a subset of the co-domain

IV Some important facts about a function from A to B



Every element in A is in the domain of the function that is every element of A is mapped to some

element in the range (If some element in S has no mapping (arrow) then the relation is not a function)

No element in the domain maps to more than one element in the range

The mapping is not necessarily onto some elements of T may not be in the range

The mapping is not necessarily one-one some elements of T may have more than one element of S

mapped to them

S and T need not be disjoint

V Types of functions

Injections A function f from A to B is called one to one (or one- one) if whenever 119943(119961120783) = 119943(119961120784) ⟹

119961120783 = 119961120784 119873119900119905119890 119905ℎ119886119905 ℎ119890119903119890 119899(119860) le 119899(119861)

Surjections A function f from A to B is called onto if for all b in B there is an a in A such that f(a) = b

⟹ forall119910 isin 119861 exist119909 isin 119860 ∶ 119891(119909) = 119910 119873119900119905119890 119905ℎ119886119905 ℎ119890119903119890 119899(119860) ge 119899(119861) Range = Co-domain

Bijections are functions that are injective and surjective ie a function f from A to B is called a bijection

if it is one to one and onto119873119900119905119890 119905ℎ119886119905 ℎ119890119903119890 119899(119860) = 119899(119861)

VI Some special functions with their domain range and nature

1 Polynomial function p(x) = a0 + a1x+a2x2+hellip+anx

n domain = R range = R continuous

2 Constant Function f(x) = k domain = r range = k continuous

3 Identity function I(x) = x domain = R range = R continuous

4 Exponential function f(x) = ex or ax domain = R domain = (0 prop) continuous

5 Logarithmic function f(x) = logx or In x domain = (0 prop) range = R continuous

6 Square root function f(x) = radic119909 domain = (0 prop) range = (0 prop) continuous

7 Sine function - sin Rrarr [minus11] 119888119900119899119905119894119899119906119900119906119904

8 Cosine function - cos Rrarr [minus11] 119888119900119899119905119894119899119906119900119906119904

9 Tangent function - tan Rminus119909 119909 =(2119899+1)120587

2 rarr 119877 continuous in its domain

10 Secant function - sec Rminus119909 119909 =(2119899+1)120587

2 rarr 119877 minus (minus11) continuous in its domain

11 Cosecant function - cosec Rminus119909 119909 = 119899120587 119899 isin 119885 rarr 119877 minus (minus11) continuous in its domain

12 Cotangent function - cot Rminus119909 119909 = 119899120587 119899 isin 119885 rarr 119877 continuous in its domain

13 119865119897119900119900119903 119891119906119899119888119905119894119900119899 x = Greatest integer that is less than or equal to x domain= R range = Z

discontinuous



14 Ceiling function x = Least integer that is greater than or equal to xdomain= R range = Z

discontinuous

15 Reciprocal function f(x) = 1

119909 domain = R - orange = R - o continuous in R+ and R-

16 Modulus function f(x) = |119909| = 119909 119894119891 119909 ge 0minus119909 119894119891 119909 lt 0

Domain = R Range = R + continuous

17 Signum function f(x) = |119909|

119909 forall119909 ne 0

0 119909 = 0=

1 119909 gt 00 119909 = 0minus1 119909 lt 0

domain = R range = -1 0 1 discontinuous

VII COMPOSITION OF FUNCTIONS - function composition is the application of one function to the

results of another For instance the functions f X rarr Y and g Y rarr Z can be composed by computing

the output of g when it has an input of f(x) instead of x A function g ∘ f X rarr Z defined by (g ∘ f)(x) =

g(f(x)) for all x in X

The composition of functions is always associative That is if f g and h are three functions with suitably

chosen domains and codomains then f ∘ (g ∘ h) = (f ∘ g) ∘ h

The functions g and f are said to commute with each other if g ∘ f = f ∘ g

VIII INVERSE OF A FUNCTION - Let ƒ be a bijective function whose domain is the set X and whose

range is the set Y Then if it exists the inverse of ƒ is the function ƒndash1 with domain Y and range X

defined by the following rule

A function with a codomain is invertible if and only if it is both one-to-one and onto or a bijection and

has the property that every element y isin Y corresponds to exactly one element x isin X

Domain (f) = range(f-1) and range (f) = domain (f-1)

Inverses and composition - If ƒ is an invertible function with domain X and range Y then

There is a symmetry between a function and its inverse Specifically if the inverse of ƒ is ƒndash1 then the

inverse of ƒndash1 is the original function ƒ ie If 119891minus1 ∘ 119891(119909) = 119868119883 then 119891 ∘ 119891minus1(119910) = 119868119884

Only one-to-one functions have a unique inverse

If the function is not one-to-one the domain of the function must be restricted so that a portion of the

graph is one-to-one You can find a unique inverse over that portion of the restricted domain

The domain of the function is equal to the range of the inverse The range of the function is equal to the

domain of the inverse



IX Inverse of a composition

The inverse of g o ƒ is ƒndash1 o gndash1

The inverse of a composition of functions is given by the formula

X BINARY OPERATION on a set ndash Let A be a non-empty setA binary operation on the set A is a

function lowast 119860 times 119860 rarr 119860 such that ab isin 119860forall (119886 119887) isin 119860 times 119860

Commutative property - A binary operation on the set A is said to be commutative if ab = ba

forall 119886 119887 isin 119860

Associative property - A binary operation on the set A is said to be associative if a(bc) = (a b)c

forall 119886 119887 119888 isin 119860

Identity element of a binary operation ndash Given a binary operation lowast 119860 times 119860 rarr 119860 a unique element e isin 119860

if it exists is called the identity element for if ae = a = ea forall 119886 isin 119860

Inverse of an element - Given a binary operation lowast 119860 times 119860 rarr 119860 the identity element e isin 119860 an element a

is called invertible wrt if exist119887 isin 119860 119904119906119888ℎ 119905ℎ119886119905 119834 lowast 119835 = 119838 = 119835 lowast 119834 Then b is called the inverse of a

and is denoted by a-1 ie a a-1= e = a-1 a

INVERSE TRIGONOMETRIC FUNCTIONS

INVERSE TRIGONOMETRIC FUNCTIONS or cyclometric functions - are the so-called inverse

functions of the trigonometric functions when their domain are restricted to principal value branch to make

the trigonometric functions bijectiveThe principal inverses are listed in the following table

Name Usual notation Definition Domain of x for

real result

Range of usual

principal value

(radians)

Range of usual

principal value

(degrees)

arcsine y = sin-1 x x = sin y minus1 le x le 1 minusπ2 le y le π2 minus90deg le y le 90deg

arccosine y = cos-1 x x = cos y minus1 le x le 1 0 le y le π 0deg le y le 180deg



arctangent y = tan-1 x x = tan y all real numbers minusπ2 lt y lt π2 minus90deg le y le 90deg

arccotangent y = cot-1 x x = cot y all real numbers 0 lt y lt π 0deg lt y lt 180deg

arcsecant y = sec-1 x x = sec y x le minus1 or 1 le x 0 le y lt π2 or π2 lt

y le π

0deg le y lt 90deg or 90deg lt y le

180deg

arccosecant y = csc-1 x x = csc y x le minus1 or 1 le x minusπ2 le y lt 0 or 0 lt

y le π2

minus90deg le y le 0deg or 0deg lt y le

90deg

Properties of the inverse trigonometric functions

I COMPLEMENTARY ANGLES

119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2

II NEGATIVE ARGUMENTS

119904119894119899minus1(minus119909) = minus119904119894119899minus1119909

119888119900119904minus1(minus119909) = 120587 minus 119888119900119904minus1119909





III RECIPROCAL ARGUMENTS

119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587

2minus 119905119886119899minus1119909 = 119888119900119905minus1119909119894119891 119909 gt 0

119905119886119899minus1 (1

119909) = minus

120587

2minus 119905119886119899minus1119909 = minus120587 + 119888119900119905minus1119909119894119891 119909 lt 0

119888119900119905minus1 (1

119909) =

120587

2minus 119888119900119905minus1119909 = 119905119886119899minus1119909 119894119891 119909 gt 0

119888119900119905minus1 (1

119909) =

3120587

2minus 119888119900119905minus1119909 = 120587 + 119905119886119899minus1119909 119894119891 119909 lt 0

IV CONVERTION FORMULA

Use Pythagoras formula in a right triangle to get the 3rd side

p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA

119904119894119899minus1119909 plusmn 119904119894119899minus1119910 = 119904119894119899minus1[119909radic1 minus 1199102 plusmn 119910radic1 minus 1199092]

119888119900119904minus1119909 plusmn 119888119900119904minus1119910 = 119888119900119904minus1[119909119910 ∓ radic1 minus 1199092radic1 minus 1199102]

119905119886119899minus1119909 plusmn 119905119886119899minus1119910 = 119905119886119899minus1 (119909plusmn119910

1∓119909119910)

119905119886119899minus1119909 + 119905119886119899minus1119910 + 119905119886119899minus1119911 = 119905119886119899minus1 (119909+119910+119911minus119909119910119911

1minus119909119910minus119910119911minus119911119909)

VI MULTIPLE FORMULA

2119904119894119899minus1119909 = 119904119894119899minus1[2119909radic1 minus 1199092]

2119888119900119904minus1119909 = 119888119900119904minus1[21199092 minus 1]

2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092

3119904119894119899minus1119909 = 119904119894119899minus1[3119909 minus 41199093] 3119888119900119904minus1119909 = 119888119900119904minus1[41199093 minus 3119909]

3119905119886119899minus1119909 = 119905119886119899minus1 3119909minus1199093

1minus31199092

CALCULUS I ALGEBRAIC AND TRIGONOMETRICIDENTITIES

1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1

4 1 2 tan x sec2 x 5 1 2 cot x csc2 x

6 Sin (uplusmn119907) = sin cos cos sinu v u v

7 cos (uplusmn119907) = cos cos sin sinu v u v

8 tan(uplusmn119907) =

tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906

10 cos2u = cos2u ndash sin2u = 2 cos2u ndash 1 = 1 ndash 2sin2u = 1minus1199051198861198992119906

1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u

12 Sin3u= 3sinu ndash 4sin3u

13 Cos3u = 4cos3u ndash 3cosu

14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c

Csin sin sin law of cosines

2 2 2 2 cosc a b ab C

29 area of triangle using trig 1

Area sin2

ac B

II CONIC SECTION FORMULA

1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21

4 Hyperbola formula x

a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a

6 parameterization of ellipse 2 2

2 21 becomes cos sin

x yx a t y b t

a b

III FORMULAS OF LIMITS

a Change of base rule for logs logln

lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1

IV CONTINUITY DEFINITION - Continuity of a function(x) at a point ndash A function f(x) is said to be continuous at the

point x = a if lim119909rarr119886

119891(119909) = 119891(119886)

Continuity of a function f(x) at x = a means

i f(x) is defined at a ie the point a lies in the domain of f

ii lim119909rarr119886

119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)



iii lim119909rarr119886

119891(119909) = 119891(119886)

Discontinuity at a point- A function f(x) fails to be continuous at the point x = a if

i f(x) is not defined at a ie the point a does not lie in the domain of f


119891(119909) 119889119900119890119904 119899119900119905 119890119909119894119904119905 ie either any of LHL or RHL do not exist or if they exist they are

not equal

iii Limit exists but lim119909rarr119886

119891(119909) ne 119891(119886)

Left continuity at a point ndash A function is said to be left continuous at x = a if lim119909rarr119886minus

119891(119909) = 119891(119886)

Right continuity at a point ndash A function is said to be right continuous at x = a if lim119909rarr119886+

119891(119909) = 119891(119886)

Removable discontinuity ndash if x = a is a point such that Limit exists but lim119909rarr119886

119891(119909) ne 119891(119886)

Then f is said to have removable discontinuity at x = a

If f(x) and g(x) are continuous at x = a then so are f+g f - g kf fg 119891

119892 (provided g(x)ne 0)

Composition of two continuous functions is continuous

V DIFFERENTIATION

I Definition of derivative If y = f(x) then y1 = 119889119891(119909)

119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0

A function f of x is differentiable if it is continuous

Left hand derivative ndash LHD = Lfrsquo(a) = lim119909rarr119886minus

119943(119938minus119945)minus119943(119938)

119945

Right hand derivative ndash RHD = R frsquo(a) = lim119909rarr119886+

119891(119886minusℎ)minus119891(119886)

ℎ

When LHD amp RHD both exist and are equal then f(x) is said to be derivable or differentiable

II FORMULAS OF DERIVATIVES

1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)



III RULES OF DIFFERENTIATION

Chain rule if y = f(u) and u = g(x) then 119889119910

119889119909=

119889119891

119889119906119889119906

119889119909

Product rule If u and v are two functions of x then 119889(119906119907)

119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907

Quotient rule If u and v are two functions of x then 119889

119889119909(119906

119907) =

119907119906primeminus119906119907prime

1199072

Parametric differentiation if y =f(t) x= g(t) then dy

dx

dy

dtdx

dt

Derivative formula for inverses df

dx df

dxx f a

x a

1 1

( )

Logarithmic differentiation If y = f(x)g(x) then take log on both the sides

Write logy = g(x) log[f(x)] Differentiate by applying suitable rule for differentiation

If y is sum of two different exponential function u and v ie y = u + v Find 119889119906

119889119909119886119899119889

119889119907

119889119909 by

logarithmic differentiation separately then evaluate 119889119910

119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909

Intermediate Value Theorem If a function is continuous between a and b then it takes on every value

between f a( ) and f b( )

Extreme Value TheoremIf f is continuous over a closed interval then f has a maximum and

minimum value over that interval

Mean Value Theorem(for derivatives) If f x( ) is a continuous function over a b and f(x) is

differentiable in (ab)then at some point c between a and b f b f a

b af c

( ) ( )( )

(the tangent at x = c is

parallel to the chord joining (a f(a)) and (b f(b)) )

Rollersquos Theorem If (i) f x( ) is a continuous function over a b (ii) f(x) is differentiable in (ab) (iii) f(a)

= f(b)then there exists some point c between a and b such that frsquo(c) = 0 ( the tangent at x = c is parallel

to x axis )

VI APPLICATION OF DERIVATIVE I APPROXIMATIONS DIFFERENTIALS AND ERRORS

Absolute error - The increment ∆119909 in x is called the absolute error in x

Relative error - If ∆119909 is an error in x then Δ119909

x is called the relative error in x

Percentage error - If ∆119909 is an error in x then Δ119909

xtimes 100 is called the percentage error in x

Approximation -

1 Take the quantity given in the question as y + ∆119910= f(x + ∆119909)

2 Take a suitable value of x nearest to the given value Calculate ∆119961

3 Calculate y= f(x) at the assumed value of x]



4 calculate 119889119910

119889119909 at the assumed value of x

5 Using differential calculate ∆119910 =119889119910

119889119909times ∆119909

6 find the approximate value of the quantity asked in the question as y + ∆119910 from the values of y and ∆119910

evaluated in step 3 and 5

II Tangents and normals ndash

Slope of the tangent to the curve y = f(x) at the point (x0y0) is given by 119889119910

119889119909(11990901199100)

Equation of the tangent to the curve y = f(x) at the point (x0y0) is (y - y0) = 119889119910

119889119909(11990901199100)

(x minus x0)

Slope of the normal to the curve y = f(x) at the point (x0y0) is given by minus119889119909

119889119910(11990901199100)

Equation of the normal to the curve y = f(x) at the point (x0y0) is (y - y0) = minus119889119909

119889119910(11990901199100)

(x minus x0)

To curves y = f(x) and y = g(x) are orthogonal means their tangents are perpendicular to each other at the

point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1

III IncreasingDecreasing Functions

Definition of an increasing function A function f(x) is increasing at a point x0 if and only if there

exists some interval I containing x0 such that f(x0) gt f(x) for all x in I to the left of x0 and f(x0) lt f(x) for

all x in I to the right of x0

Definition of a decreasing function A function f(x) is decreasing at a point x0 if and only if there

exists some interval I containing x0 such that f(x0) lt f(x) for all x in I to the left of x0 and f(x0) gt f(x) for


To find the intervals in which a given function is increasing or decreasing 1 Differentiate the given function y = f(x) to get frsquo(x)

2 Solve frsquo(x) = 0 to find the critical points

3 Consider all the subintervals of R formed by the critical points( no of subintervals will be one

more than the no of critical points )

4 Find the value of frsquo(x) in each subinterval

5 frsquo(x) gt 0 implies f(x) is increasing and frsquo(x) lt 0 implies f(x) is decreasing

VII CONCAVITY

Definition of a concave up curve f(x) is concave up at x0 if and only if f (x) is increasing at x0 which

means frdquo(x)gt 0 at x0 ie it is a minima

Definition of a concave down curve f(x) is concave down at x0 if and only if f (x) is decreasing at x0

which means frdquo(x) lt 0 at x0 ie it is a maxima

The first derivative test If f (x0) exists and is positive then f(x) is increasing at x0 If f (x) exists and is

negative then f(x) is decreasing at x0 If f (x0) does not exist or is zero then the test fails



The second derivative test If f (x) exists at x0 and is positive then f (x) is concave up or has minima at

x0 If f (x0) exists and is negative then f(x) is concave down or has maxima at x0 If f (x) does not exist

or is zero then the test fails

VIII Critical Points

Definition of a critical point a critical point on f(x) occurs at x0 if and only if either f (x0) is zero or the

derivative doesnt exist

Definition of an inflection point An inflection point occurs on f(x) at x0 if and only if f(x) has a

tangent line at x0 and there exists and interval I containing x0 such that f(x) is concave up on one side of

x0 and concave down on the other side

IX Extrema (Maxima and Minima)

Definition of a local maxima A function f(x) has a local maximum at x0 if and only if there exists some

interval I containing x0 such that f(x0) gef(x) for all x in I

Definition of a local minima A function f(x) has a local minimum at x0 if and only if there exists some

interval I containing x0 such that f(x0) lef(x) for all x in I

Occurrence of local extrema All local extrema occur at critical points but not all critical points occur

at local extrema

The first derivative test for local extrema If f(x) is increasing (f (x) gt 0) for all x in some interval (a

x0] and f(x) is decreasing (f (x) lt 0) for all x in some interval [x0 b) then f(x) has a local maximum at

x0 If f(x) is decreasing (f (x) lt 0) for all x in some interval (a x0] and f(x) is increasing (f (x) gt 0) for all

x in some interval [x0 b) then f(x) has a local minimum at x0

The second derivative test for local extrema If f (x0) = 0 and f (x0) gt 0 then f(x) has a local

minimum at x0 If f (x0) = 0 and f (x0) lt 0 then f(x) has a local maximum at x0

To solve word problems of maxima and minima 1 Draw the figure and list down the facts given in the question

2 From the given function convert one variable in term of the other

3 Write down the function to be optimized and convert it into a function of one variable by using

the result of step 2

4 Then proceed to find maxima or minima by applying second derivative test

5 Evaluate all components of the question

X Absolute Extrema

Definition of absolute maxima y0 is the absolute maximum of f(x) on I if and only if y0 ge f(x) for all

x on I

Definition of absolute minima y0 is the absolute minimum of f(x) on I if and only if y0 le f(x) for all

x on I

The extreme value theorem If f(x) is continuous in a closed interval I then f(x) has at least one

absolute maximum and one absolute minimum in I

Occurrence of absolute maxima If f(x) is continuous in a closed interval I then the absolute maximum

of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I

Occurrence of absolute minima If f(x) is continuous in a closed interval I then the absolute minimum

of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I



Alternate method of finding extrema If f(x) is continuous in a closed interval I then the absolute

extrema of f(x) in I occur at the critical points andor at the endpoints of I

VII INDEFINITE INTEGRALS

Definition - if the derivative of F(x) is f(x) then ANTIDERIVATIVE or INTEGRAL of f(x) is F(x) it is

denoted byint119891(119909)119889119909 = 119865(119909) + 119862 where C is any constant of integration The process of finding the

antiderivative or integral is called INTEGRATION

Theorem 1 If two functions differ by a constant they have the same derivative

Theorem 2 If two functions have the same derivative their difference is a constant I FORMULA OF INTEGRATION

1 int[119891(119909) plusmn 119892(119909)]119889119909 = int119891(119909) 119889119909 plusmn int119892(119909)119889119909

2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905

4 int 119891(119909) 119892(119909)119889119909 = 119865(119909) 119892(119909) minus int119865(119909)119892prime(119909)119889119909

5

6

7

8

9 where u is a variable a is any constant

and e is a defined constant

II INTEGRAL OF TRIGONOMETRIC FUNCTIONS

1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961

radic120783minus119961120784= 119956119946119951minus120783119961 + 119914 = minus119940119952119956minus120783119961 + 119914 |119961| le 120783

12 int119941119961

119935radic119961120784minus120783= 119956119942119940minus120783119961 = minus119940119952119956119942119940minus120783119961 119961 ge 120783

13 int119941119961

120783+119961120784 = 119957119938119951minus120783119961 + 119914 = minus119940119952119957minus120783 119961 + C

III INTEGRAL OF POWERS OF TRIGONOMETRIC FUNCTIONS The integrals of powers of trigonometric functions

will be limited to those which may by substitution be written in the form int119906119899119889119906

1 Techniques of Integration Integrating Powers and Product of Sines and Cosinesint 119904119894119899119898119909119888119900119904119899119909119889119909

We have two cases both m and n are even or at least one of them is odd

2 Case I m or n odd Suppose n is odd - then substitute sinx = t Indeed we have cosxdx = dt and hence

int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957

3 Case II m and n are even Use the trigonometric identities sin2 u 1 2

2

cos u

cos2 u 1 2

2

cos u

IV INTEGRALS OF MULTIPLES OF SIN AND COS for integrals

int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961

int 119940119952119956(119950119961) 119940119952119956(119951119961)119941119961 use the transformation formula



1 Sin(mx)sin(nx) = 1

2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]

2 Sin(mx)cos (nx) = 1

2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]

3 cos(mx)cos(nx) = 1

2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]

V REDUCTION FORMULA In integrals of the formint 119957119938119951119951 119961119941119961 int 119940119952119957119951 119961119941119961 int 119956119942119940119951 119961119941119961 int 119940119952119956119942119940119951 119961119941119961

Use

1 For int 119957119938119951119951 119961119941119961 substitute tannx = tann-2x tan2x = tann - 2x(sec2x - 1) then put tanx = t

2 For int119940119952119957119951 119961119941119961 substitute cotnx = cotn-2x cot2x = cot n - 2x(cosec2x - 1) then put cotx = t

3 For int 119956119942119940119951 119961119941119961 substitute secnx = secn-2x sec2x = secn - 2x(tan2x + 1) then put secx = t

4 For int119940119952119956119942119940119951 119961119941119961 substitute cosecnx = cosecn-2x cosec2x = cosecn - 2x(cot2x + 1) then put cosecx = t

VI INTEGRALS INVOLVING radic119938120784 plusmn 119961120784119912119925119915 radic119961120784 plusmn 119938120784 ----Trigonometric substitutions may be used to eliminate

radicals from integrals

1 for radic1198862 minus 1199092 119904119906119887119904119905119894119905119906119905119890 119909 = 119886 119904119894119899119905 then dx = a cost dt

2 for radic1198862 + 1199092 119904119906119887119904119905119894119905119906119905119890 119909 = 119886 119905119886119899119905 then dx = a sec2t dt

3 for radic1199092 minus 1198862 119904119906119887119904119905119894119905119906119905119890 119909 = 119886 119904119890119888119905 then dx = a sect tant dt

VII Standard formula

1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1

radic1198862minus1199092 dx = 119904119894119899minus1 119909

119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1

radic1199092minus1198862 dx = 119897119900119892|119909 + radic1199092 minus 1198862| + C

7 intradic1198862 minus 1199092dx = 119909

2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C


2radic1199092 minus 1198862 minus

119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C

VIII Integrals of the form int120783

119938119961120784+119939119961+119940119941119961 or int

120783

radic119938119961120784+119939119961+119940119941119961 Apply completion of square method to convert

ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2

] and use suitable standard formula

IX Integrals of the formint119961120784+120783

119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929

Divide numerator and denominator by x2

Express denominator as (119909 plusmn1

119909)2plusmn 1198962 ( choose the sign between x and

1

119909 as opposite of that in

numerator

Substitute x + 1

119909 = t or x -

1

119909 = t as the case may be

Reduce the integral to standard form and apply suitable formula



X Integrals of the form int119953119961+119954

119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583

Evaluate 120582 and 120583 by equating the coefficients of like powers of x from both sides

Express given integral as 120582 int2119886119909+119887

119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961

Use completion of square method for the second integral to convert it into standard form

Then use suitable integral formula

XI Integrals of the form int(119953119961 + 119954)radic119938119961120784 + 119939119961 + 119940 119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583


Express given integral as 120582 int(2119886119909 + 119887)radic119938119961120784 + 119939119961 + 119940 119889 119909 + 120583 intradic119938119961120784 + 119939119961 + 119940 119889119909

Use the formula int(119891(119909))119899119891prime(119909)119889119909 =

(119891(119909))119899+1

119899+1 to evaluate the first integral and use completion of square

method for the second integral to convert it into standard form


XII Integrals of the form int120783

119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961

Divide numerator and denominator by cos2x

Express sec2x if any in the denominator as 1+tan2x

Put tanx = t so that sec2xdx = dt

XIII RATIONAL EXPRESSIONS OF SIN AND COS int119941119961

119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute

Then use completion of square method

XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961

write numerator = λ( derivative of denominator) + μ(denominator) ie

asinx + bcosx = λ( acosx minus bsinx) + μ(csinx + dcosx )

obtain the values of λ and μ by equating the coefficients of sinx and cosx from both the sides

Express the given integral as 120582 int119888119888119900119904119909minus119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate

XV THE METHOD OF PARTIAL FRACTIONS to integrate the rational function f(x) = 119875(119909)

119876(119909)

1 If degree(P) ge 119941119942119944119955119942119942 (119928) perform polynomial long-division Otherwise go to step 2

2 Factor the denominator Q(x) into irreducible polynomials linear and irreducible quadratic

polynomials

3 Find the partial fraction decomposition by usingthe following table

Form of rational function Form of partial function 119901119909 + 119902

(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862

1198881199092+119889119909+119890 where cx2+dx+e can not be

further factorised

A B C are real numbers to be determined by taking LCM and comparing the coefficients of like

terms from the numerator

4 Integrate the result of step 3

XVI To evaluate int119941119961

119961(119961119951+119948) 119899 isin 119873 119899 ge 2

Multiply numerator and denominator by xn-1

Then substitute xn = t so that n x n-1 dx = dt

Then apply partial fraction

XVII If a rational function contains only even powers of x in both numerator and denominator

Put x2 = y t in the given rational function

Resolve the rational function obtained in step 1 into partial fraction

Replace back y = x2 Then integrate

XVIII Integration by Parts ndash If u and g are two functions of x then the integral of product of two functions =

1st function times 119957119945119942 119946119951119957119942119944119955119938119949 119952119943 119957119945119942 120784119951119941119943119958119951119940119957119946119952119951 - integral of the product of the derivative of 1st

function and the integral of the 2nd function

Write the given integralint119906(119909) 119907(119909) 119889119909 where you identify the two functions u(x) and v(x) as the 1st and 2nd

function by the order

I ndash inverse trigonometric function

L ndash Logarithmic function

A ndash Algebraic function

T ndash Trigonometric function

E ndash Exponential function

Note that if you are given only one function then set the second one to be the constant function g(x)=1

integrate the given function by using the formula

int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909

XIX Integrals of the form int119942119961[119943(119961) + 119943prime(119961)] dx

Express the integral as sum of two integrals one containing f(x) and other containing frsquo(x)ie

int119942119961[119943(119961) + 119943prime(119961)] dx = int119942119961119943(119961)119837119857 +int 119942119961119943prime(119961)119837119857

Evaluate the first integral by integration by parts by taking ex as 2nd function

2nd integral on RHS will get cancelled by the 2nd term obtained by evaluating the 1st integral

We get int119942119961[119943(119961) + 119943prime(119961)] dx = ex f(x) + C

XX Integrals of the type int119942119938119961 119956119946119951119939119961119941119961 orint119942119938119961 119940119952119956119939119961119941119961

Apply integration by parts twice by taking eax as the first function



XXI INTEGRATION OF SOME SPECIAL IRRATIONAL ALGEBRAIC FUNCTIONS integrals of the

formint120593(119909)

119875radic119876119889119909

int1

(119886119909+119887)radic119888119909+119889119889119909 P and Q are both linear functions of x put Q = t2ie cx + d = t2

int1

(1198861199092+119887119909+119888)radic119901119909+119902119889119909 P is a quadratic expression and Q is linear expression of x put Q = t2

ie put px + q = t2

int1

(119886119909+119887)radic1199011199092+119902119909+119903119889119909 P is a linear expression and Q is quadratic expression of x put P =

1

119905

ie ax+ b = 1

119905

int1

(1198861199092+119887)radic1198881199092+119889dx P and Q are pure quadratic expressions put x=

1

119905to obtain int

minus119905dt

(119886+1198871199052)radic119888+1198891199052 then put c+dt2

= u2

int119901119909+119902

(1198861199092+119887)radic1198881199092+119889dx P and Q are pure quadratic expressions and 120593(119909) 119894119904 119897119894119899119890119886119903 put x = t2

VIII DEFINITE INTEGRAL

1 The Fundamental Theorem of Calculus Let f (x) be continuous on [a b] If F(x) is any antiderivative of f (x)

then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887

119886 where b the upper limit and a the lower limit are given valuesNotice that

the constant of integration does not appear in the final expression of equation

2 Areas above and below a curveIf the graph of y = f(x) between x = a and x = b has portions above and

portions below the X axis then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887

119886 is the sum of the absolute values of the positive

areas above the X axis and the negative areas below the X axis the value of b is the upper limit and the

value of a is the lower limit

3 Mean Value Theorem(for definite integrals) If f is continuous on a b then at some

point c in a b 1 b

af c f x dx

b a

4 Definite integral as the limit of a sum of all the strips between a and b having areas of

119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0

sum [119891(119909 + (119896 minus 1)ℎ)] times ℎ 119896=119899119896=1

119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]

Steps - 1 Find nh = b ndash a



2Evaluate f(a) f(a+h) f(a+ 2h) hellip fa + (n ndash 1)h and set pattern in terms of h h2 h3 etc

3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]

4After combining the terms of constant h h2 h3 together apply the summation formulas

1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1

Sina +sin(a+h) +sin(a+2h)+ hellip +sina+(n - 1) h = sin119886+(

119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)

cosa +cos(a+h) +cos(a+2h)+ hellip +cosa+(n - 1) h = cos119886+(

119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)

5 Properties of the Definite Integral

If f (x) and g(x) are defined and continuous on [a b] except maybe at a finite number of points then we have the

following linearity principle for the integral

(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888

P0 The value of the integral do not change if variable of integration is changed

int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886

P1 The integral changes its sign if limit of integration is interchanged

int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886

P2 The integral can be expressed as sum of sub-integrals

int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887

119886 where a lt c lt b

P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0

P5 int 119891(119909)119889119909 = int 119891(119909)119889119909 + int 119891(2119886 minus 119909)119889119909119886

0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886



IX AREA UNDER THE BOUNDED REGION

Area of the region bounded by the curve y = f(x) the x axis and ordinates x = a and x = b is int 119910119889119909119887

119886=

int 119891(119909)119889119909119887

119886

Area of the region bounded by the curve x = f(y) the y axis and ordinates y = a and y= b is int 119909119889119910119887

119886=

int 119891(119910)119889119910119887

119886

If y = f1(x) and y = f2(x) are two curves intersecting at the points (a b) and (c d) then the area enclosed

between the curves is given by int (119910119906119901119901119890119903 119888119906119903119907119890 minus 119910119897119900119908119890119903 119888119906119903119907119890)119888

119886119889119909

If x = f1(y) and x = f2(y) are two curves intersecting at the points (a b) and (c d) then the area enclosed


119886119889119910

WORKING RULE-

I Trace the graph of the curves and write about them in brief

II Find the points of intersection of the curves

III Express y in term of x befrom the equation of the curve if you are integrating wrt x ( or x

in term of y if you wish to integrate wrt y ) as the case may be

IV Consider the area under the bounded region as definite integral by using the concept

discussed above

V Evaluate the definite integral

VI Write the answer in sq units

MATRICES AND DETERMINANTS

DEFINITION A matrix A = [119938119946119947]119950times119951 is defined as an ordered rectangular array of numbers in

m rows and n columns 119912 = [

119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]

1 ROW MATRIX A matrix can have a single row A = [119938119946119947]120783times119951 = [ a11 a12 a13 hellip a1n]

2 COLUMN MATRIX - A matrix can have a single column A = [119938119946119947]119950times120783=[

119938120783120783

119938120784120783

119938119950120783

]

3 ZERO or NULL MATRIX ndash A matrix is called the zero or null matrix if all the entries are 0

4 SQUARE MATRIX - A matrix for which horizontal and vertical dimensions are the same (ie an

matrix)

5 DIAGONAL MATRIX - A square matrix A = [119938119946119947]119951times119951 is called diagonal matrix if aij = 0 for 119946 ne119947

6 SCALAR MATRIX - A diagonal matrix A = [119938119946119947]119951times119951 is called the scalar matrix if all its diagonal

elements are equal

7 IDENTITY MATRIX ndash A diagonal matrix A = [119938119946119947]119951times119951 is called the identity matrix if aij = 1 for i

= j it is denoted by In

8 UPPER TRIANGULAR MATRIX - A square matrix A = [119938119946119947]119951times119951 is called upper triangular

matrix if aij = 0 for 119946 gt 119947



9 LOWER TRIANGULAR MATRIX - A square matrix A = [119938119946119947]119951times119951 is called lower triangular

matrix if aij = 0 for 119946 lt 119947

MATRIX OPERATIONS

1 DEFINITION Two matrices A and B can be added or subtracted if and only if their

dimensions are the same (ie both matrices have the identical amount of rows and

columns

2 Addition

If A = [119938119946119947]119950times119951 and B = [119939119946119947]119950times119951 are matrices of the same type then the sum is a matrixC = [119914119946119947]119950times119951

obtained by adding the corresponding elements aij+bij ie A+B = C if aij+bij =cij

3 Matrix addition is commutative associative and distributive over multiplication -

A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction

If A = [119938119946119947]119950times119951 and B = [119939119946119947]119950times119951 are matrices of the same type then the difference is a

matrix D = [119941119946119947]119950times119951obtained by subtracting the corresponding elements aij - bij ie A -

B = C if aij - bij =dij

5 Equal matrices ndashTwo matrices are said to be equal if they have the same order and their

corresponding elements are also equal ie A = [119938119946119947]119950times119951 = B = [119939119946119947]119950times119951 if aij = bij for all I j

6 Scalar multiplication- If A = [119938119946119947]119950times119951 and B = [119939119946119947]119950times119951 are matrices of the same order and k

m are scalars then scalar multiplication is defined as kA=[kaij]

K(A+B) = Ka + Kb (m+n) A = mA+ nA (mk)A = m(kA) =k(mA)

7 Matrix Multiplication

DEFINITION When the number of columns of the first matrix is the same as the number of

rows in the second matrix then matrix multiplication can be performed

Let A = [119938119946119947]119950times119951 and B = [119939119946119947]119951times119953 Then the product of A and B is the matrix C which has

dimensions mxp The ijth element of matrix C is found by multiplying the entries of the ith row

of A with the corresponding entries in the jth column of B and summing the n terms The

elements of C are



Note That AxB is not the same as BxA

8 Properties of matrix multiplication

AB ne BA

A(BC) = (AB)C

AIn = A = InA

AB = 0 ⇏ 119860 = 0 119900119903 119861 = 0 If AB = AC then not necessarily B = C

If A is a square matrix of order n then A satisfies any given matrix polynomial

f(A) = amAm+am-1Am-1 + hellip + a2A2+a1A+a0I

9 Transpose of Matrices The transpose of a matrix is found by exchanging rows for

columns ie Matrix A = (aij) and the transpose of A isAT=(aji) where j is the column

number and i is the row number of matrix A (119860119879)119879 = 119860

(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879

Symmetric matrix For a symmetric matrix A = AT ie aij = aji

Skew -symmetric matrix For a skew-symmetric matrix AT = - A ie aji = - aij

Note that the diagonal elements of the skew symmetric matrix are 0

A + AT is a symmetric matrix

A - AT is a skew - symmetric matrix

Every square matrix A can be expressed as a sum of a symmetric P and skew symmetric Q

matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2

10 Elementary transformation - Following elementary row or column transformations

can be applied to a matrix

Interchange of any two rows or columns 119877119894 harr 119877119895 or 119862119894 harr 119862119895

Multiplication of any row or column by any non-zero number(scalar) 119877119894 harr 119896119877119894 or 119862119894 harr 119896119862119894



The addition to the elements of any row or column the scalar multiples of any other row or

column 119877119894 harr 119877119894 + 119896119877119895 or 119862119894 harr 119862119894 + 119896119862119895 k can be any number positive or negative

Working rule to find A-1 by elementary transformations

a) Write A = InA apply elementary row transformations to both the matrices A on LHS

and In on RHS till you get In= BA Then B is the required A-1

b) Write A = AIn apply elementary column transformations to both the matrices A on LHS

and In on RHS till you get In= AB Then B is the required A-1

NOTE Apply only one kind of transformations (row or column ) in all the steps in one

particular answer

11 The Determinant of a Matrix

DEFINITION Determinants play an important role in finding the inverse of a matrix

and also in solving systems of linear equations The determinant of a square matrix A is a

number associated with every square matrix and is denoted by det(A) or |A|

MINOR The minor of the element aij of |A| is given by Mij where Mij is the determinant of the (n-1)

x (n-1) matrix that is obtained by deleting row i and column j (where the element aij lies) of the

determinant of A

COFACTOR - The cofactor of the element aij given by Aij = (-1)i+j Mij

Sign convention for the cofactors [+ minusminus +

] [+ minus +minus + minus+ minus +

] etc

ADJOINT (OR ADJUGATE) OF A MATRIX - the transpose of the matrix of cofactors adj(A)

=[Cij]T=[Cji]

If A is a square matrix of order n then A adj A = adj A A = |119860|119868119899

If A is a square matrix of order n then|119886119889119895119860| = |119860|119899minus1

Determinant of a 2x2 matrix If A = [119938119946119947]120784times120784 = [119938120783120783 119938120783120784

119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784

Determinant of a 3x3 matrixIf 119912 = [119938119946119947]120785times120785= [

119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|

Determinant of a nxn matrix If A = [119938119946119947]119951times119951 then det A= |119912| = 119938120783120783119912120783120783 + 119938120783120784119912120783120784 + ⋯+119938120783119951119912120783119951 where Aij is the cofactor of the element aij given by Aij = (-1)i+j Mij

Singular matrix ndash A square matrix is said to be singular if |119912| = 120782



Non- Singular matrix ndash A square matrix is said to be non-singular if |119912| ne 120782

If A and B are non-singular matrices of the same order then AB and BA are also non-

singular matrices of the same order

12 THE PROPERTIES OF DETERMINANTS

P1 The value of the determinant remains unchanged if its rows and columns are interchanged

P2 119877119894 ⟷ 119862119894 ⟹ ∆= ∆primerArr |119860| = |119860119879| P3 If two rows or columns of a determinant is interchanged the sign of the determinant changes 119877119894 ⟷

119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆

P4 If any two rows or columns of a determinant is identical then its value is 0

P5 If each element of any row or column is multiplied by a scalar then the value of the determinant gets

multiplied by that scalar119877119894 ⟶ 119896119877119894119900119903 119862119894 ⟶ 119896119862119894 ⟹ ∆= 119896∆

P6 If each element of a row or column can be expressed as sum of two or more terms then the

determinant can be expressed as sum of two or more determinants of the same orders

P7 If any row (or column) of a determinant is proportional to any other row (or column) then the value

of the determinant is 0 119894 119890 119877119894 = 119896119877119895119900119903 119862119894 = 119896119862119895 ⟹ ∆= 0

P8 If to each element of any row or column is added the equimultiples of the corresponding elements of

one or more rows or columns the value of the determinant remains unchanged 119877119894 ⟶ 119877119894 +119896119877119895119900119903 119862119894 ⟶ 119862119894+119896119862119895 ⟹ ∆1= ∆

P9 If a determinant can be regarded as a polynomial function in x and if it becomes 0 by putting x = a

then (x ndash a) is a factor of the determinant

P10 If the elements of any row or column is multiplied by its corresponding cofactors and summed up

then the result is the determinant itself

P11 If the elements of any row or column is multiplied by the cofactors of any other row orcolumn and

summed up then the result is 0

P12 The determinant of the product of two square matrices of the same order is equal to the product of

their determinants ie |119860119861| = |119860||119861| P13 If each element of a particular row or column is 0 then the value of the determinant is 0

P14 If A is a square matrix of order n then |119896119860| = 119896119899|119860|

13 APPLICATION OF DETERMINANT

Area of the triangle whose vertices are (x1y1) (x2 y2) (x3 y3) is given by ∆= |1

2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||

Condition of collinearity of the points (x1y1) (x2 y2) (x3 y3) is given by |

1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0



Equation of a line passing through the points(x y) (x1y1) (x2 y2) is given by |

119909 119910 11199091 1199101 11199092 1199102 1

| = 0

14 The Inverse of a Matrix

DEFINITION If A = [119938119946119947]119951times119951is non-singular ( ie det(A) does not equal zero ) then there

exists an nxn matrix A-1 which is called the inverse of A such that 119860minus1 =119860119889119895 119860

|119860|

AA-1= A-1A = I where I is the identity matrix

If A and B are two invertible matrices of the same order then (AB) -1= B-1A-1

If A B and C are three invertible matrices of the same order then (ABC) -1= C-1B-1A-1

If A is an invertible matrix then AT is also invertible and (AT)-1 = (A-1) T

15 SOLVING SYSTEMS OF EQUATIONS USING INVERSE MATRIX METHOD

DEFINITION A system of linear equations is a set of equations with n equations and n

unknowns is of the form of

The unknowns are denoted by x1x2xn and the

coefficients (as and bs above) are assumed to be given In matrix form the system of equations

above can be written as

which can be expressed in matrix equation as AX =B

By pre-multiplying both sides of this equation by A-1 gives

A-1 (AX) = A-1 B ⟹ (119860minus1119860)119883 = 119860minus1119861 rArr 119868119883 = 119860minus1119861 rArr 119883 = 119860minus1119861

STEPS

i Evaluate |119860| Refer the note given below

ii Evaluate the cofactors of elements of A

iii Form the adjoint of A as the matrix of cofactors



iv Calculate 119860minus1 =119860119889119895 119860

|119860|

NOTE - From the above it is clear that the existence of a solution depends on the value of the

determinant of A There are three cases

1 If the |119860| ne 0 then the system is consistent with unique solution given by 119883 = 119860minus1119861

2 If |119860| = 0 (A is singular) and adjA B ne 0 then the solution does not exist The system is

inconsistent

3 If |119860| = 0 (A is singular) and adjA B = 0 then the system is consistent with infinitely

many solutionsto find these solutions put z = k in two of the equations and solve them by matrix

method

For homogeneous system of linear equations AX = 0 (B = 0)

1 If the |119860| ne 0 then the system is consistent with trivial solution x = 0 y = 0 z = 0


inconsistent


many solutions to find these solutions put z = k in two of the equations and solve them by matrix

method

______________________________________________________________________________________

DIFFERENTIAL EQUATIONS

An equation containing an independent variable a dependent variable and differential coefficients of

the dependent variable with respect to the independent variable is called a differential equation DE

The ORDER of a differential equation is the highest derivative that appears in the equation

The DEGREE of a differential equation is the power or exponent of the highest derivative that

appears in the equation

Unlike algebraic equations the solutions of differential equations are functions and not just numbers

IVP or an initial value problem is one in which some initial conditions are given to solve a DE

To form a DE from a given equation in x and y containing arbitrary constants (parameters) ndash

1 Differentiate the given equation as many times as the number of arbitrary constants involved

in it

2 Eliminate the arbitrary constant from the equations of y yrsquo yrsquorsquo etc

A first order linear differential equation has the following form



The general solution is given by where

called the integrating factor If an initial condition is given use it to find the constant C

Here are some practical steps to follow

1 If the differential equation is given as rewrite it in the form

where

2 Find the integrating factor

3 Evaluate the integral

4 Write down the general solution

5 If you are given an IVP use the initial condition to find the constant C

VARIABLE SEPARABLE FORM The differential equation of the form 119889119910

119889119909= 119891(119909 119910) is called

separable if f(xy) = h(x) g(y) that is 119889119910

119889119909= ℎ(119909)119892(119910) - - - (I)

In order to solve it perform the following STEPS

1 Solve the equation g(y) = 0 which gives the constant solutions of (I)

2 Rewrite the equation (I) as 119889119910

119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1

119892(119910)119889119910 = intℎ(119909)119889119909 to obtain G(y) = H(x) + C

4 Write down all the solutions the constant ones obtained from (1) and the ones given in (3)

5 If you are given an IVP use the initial condition to find the particular solution Note that it may

happen that the particular solution is one of the constant solutions given in (1) This is why Step 4 is

important

Homogeneous Differential Equations

A function which satisfies 119891(120582119909 120582119910) = 120582119899119891(119909 119910) for a given n is called a homogeneous function of

order n

A differential equation is called a HOMOGENEOUS DIFFERENTIAL EQUATION if it can be written

in the form M(xy)dx + N(xy)dy = 0 where M and N are of the same degree

A first-order ordinary differential equation is said to be homogeneous if it can be written in the form 119889119910

119889119909= 119865(

119910

119909)

Such equations can be solved by the change of variables 119907 =119910

119909 which transforms the equation into the separable

equation 119889119909

119909=

119889119907

119865(119907)minus119907



Steps For Solving a Homogeneous Differential Equation

1 Rewrite the differential in homogeneous form 119889119910

119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)

2 Make the substitution y = vx (for first eq) or x = vy (for 2nd eq) where v is a variable

3 Then use the product rule to get 119889119910

119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910

4 Substitute to rewrite the differential equation in terms of v and x or v and y only

5 Divide by xd or yd where d is the degree of the polynomials M and N

6 Follow the steps for solving separable differential equations

7 Re-substitute v = yx or v = x y

VECTORS

A quantity that has magnitude as well as direction is called a vector It is denoted by directed line segment

119860119861 119900119903 119886 where A is the initial point and B is the terminal point The distance AB is called the magnitude

denoted by |119808119809| = 119834 and the vector is dirtected from A to B

If 119886 = 1198861119894 + 1198862119895 + 1198863 then coefficients of 119894 119895 119886119899119889 ie a 1 a 2 and a 3 are called the DIRECTION RATIOS

of 119886

Any UNIT VECTOR along 119886 = 1198861119894 + 1198862119895 + 1198863 is given by =

| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632

are called the DIRECTION COSINES of 119886

FIXED VECTOR is that vector whose initial point or tail is fixed It is also known as localised

vector For example the initial point of a position vector is fixed at the origin of the coordinate axes

So position vector is a fixed or localised vector

FREE VECTOR is that vector whose initial point or tail is not fixed It is also known as a non-

localised vector For example velocity vector of particle moving along a straight line is a free vector

CO-INITIAL VECTORS are those

vectors which have the same initial point

are four coinitial vectors

CO-TERMINUS VECTORS are those

vectors which have a common terminal

point are four coterminous

vectors



NEGATIVE VECTOR A vector is said to

be negative of a given vector if its

magnitude is the same as that of the given

vector but the direction is reversed

POSITION VECTOR gives the position of a point with reference to the origin of the coordinate system

Position vector of P(x y z) is given as 119874119875 = 119903 = 119909119894 + 119910119895 + 119911 where |119874119875| = 119903 = radic1199092 + 1199102 + 1199112

COLLINEAR VECTORS are those vectors that act either along the same line or along parallel

lines These vectors may act either in the same direction or in opposite directions if 119886 119886119899119889 119886119903119890

119888119900119897119897119894119899119890119886119903 119905ℎ119890119899 119886 = 120582 PARALLEL VECTORS are two

collinear vectors acting along the same

direction

ANTI-PARALLEL VECTORS are two

collinear vectors acting in the opposite

directions

EQUAL VECTORS Two vectors are said to be equal if they have the same magnitude and direction

Since the three vectors have the same direction

Moreover the three vectors are also equal in magnitude

THE ADDITION OF VECTORSif 119886 = 1198861119894 + 1198862119895 + 1198863and = 1198871119894 + 1198872119895 + 1198873 are two vectors then 119886 +

= (1198861 + 1198871)119894 + (1198862 + 1198872) + (1198863 + 1198873) 1 Triangle law of vectors for addition of two vectors If two vectors can be represented both in

magnitude and direction by the two sides of a triangle taken in the same order then the resultant is

represented completely both in magnitude and direction by the third side of the triangle taken in the

opposite order

Corollary 1) If three vectors are represented by the three sides of a triangle taken in order then their resultant is zero

2) If the resultant of three vectors is zero then these can be represented completely by the three sides of a

triangle taken in order



2 Parallelogram law of vectors for addition of two vectorsIf two vectors are completely

represented by the two sides OA and OB respectively of a parallelogram Then according to the law of parallelogram of

vectors the diagonal OC of the parallelogram will be resultant such that 3 Polygon law of vectors for addition of more than two vectors If a number of vectors can be

represented both in magnitude and direction by the sides of an open convex polygon taken in the

same order then the resultant is represented completely in magnitude and direction by the closing

side of the polygon taken in the opposite order

MULTIPLICATION OPERATIONS FOR VECTORS

1 MULTIPLICATION OF A VECTOR BY A SCALAR When a vector 119886 is multiplied by a real

number say m then we get another vector m119886 The magnitude of m119886 is m times the magnitude of 119886

If m is positive then the direction of m119886 is the same as that of 119886 If m is negative then the direction

of m119886 is opposite to that of 119886

2 SCALAR MULTIPLICATION OF TWO VECTORS to yield a scalar

119886 = |119886||119887|119888119900119904120579

where |a| and |b| denote the length of 119886 and and θ is the angle between them

It is important to remember that there are two different angles between a pair of vectors depending

on the direction of rotation However only the smaller of the two is taken in vector multiplication

Some Properties of the Dot Product

For unit vector triad 119894 119895 119886119899119889 119894 119894 = 119895 119895 = =1 and 119947 = 119947 = = 0

If 119886 = 1198861119894 + 1198862119895 + 1198863and = 1198871119894 + 1198872119895 + 1198873 are two vectors then 119886 = 11988611198871 + 11988621198872 + 11988631198873

For a vector a where |a| denotes the length (magnitude) of a

Thus given two vectors the angle between them can be found by rearranging the above formula

120579 = cosminus1 119887

|119886||119887|

The magnitude of the projection of in the direction of = 119887

|119887|

The dot product is commutative

The dot product is distributive over vector addition

The dot product is bilinear

When multiplied by a scalar value dot product satisfies

Two non-zero vectors a and b are perpendicular if and only if a bull b = 0

the dot product does not obey the cancellation law If a bull b = a bull c and a ne 0 then a bull (b minus c) = 0

If a is perpendicular to (b minus c) we can have (b minus c) ne 0 and therefore b ne c



1 VECTOR MULTIPLICATION OF TWO VECTORS to yield another vector

The cross product of 119886 and written 119886 x is defined by x = a b sin

where a and b are the magnitude of vectors 119886 and is the angle between the vectors and is the unit vector (vector with magnitude = 1) that is perpendicular (at 90 degrees to orthogonal to

normal to) both 119886 and But there are two vectors that this could be - one on either side of the

plane formed by the two vectors) so we choose to be the one which makes (119886 ) a right handed triad

Some Properties of the Cross Product

The cross product of any two parallel vectors is the null vector since sin 0 = 0

i times i = j times j = k times k = 0

x 119947 = 119947 x = x = 119947 119947 x = minus x 119947 = minus x = minus119947

x = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k = | 119947 119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

|

The cross product is anti-commutative 119886 x = - x 119886

The cross product of a vector with itself is the null vector in particular 119886 x 119886 = 0

A unit vector perpendicular to both 119886 119886119899119889 is given by times

| times |

The area of the parallelogram formed by 119886 and = | 119886 x | Since area of the triangle OAB is half of the area of the parallelogram the area of the triangle formed

by two vectors = 1

2|119886 x |

It does not obey the cancellation law If a times b = a times c and a ne 0 then (a times b) minus (a times c) = 0 and by the

distributive law above a times (b minus c) = 0 Now if a is parallel to (b minus c) then even if a ne 0 it is possible

that (b minus c) ne 0 and therefore that b ne c

However if both a middot b = a middot c and a times b = a times c then it can be concluded that b = c

SCALAR TRIPLE PRODUCT

It is defined for three vectors in that order as the scalar

( times )It denotes the volume of the parallelopiped formed

by taking a b c as the co-terminus edges

ie V = magnitude of times = | times |

The value of scalar triple product depends on the cyclic order of the vectors and is independent of the position of the

dot and cross These may be interchange at pleasure However and anti-cyclic permutation of the vectors changes the

value of triple product in sign but not a magnitude



Properties of Scalar Triple Product

If119886 = 1198861 + 1198862119895 + 1198863 = 1198871119894 + 1198872119895 + 1198873and 119888 = 1198881119894 + 1198882119895 + 1198883 then their scalar triple product is given by

[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|

119886 ( times 119888 ) = (119886 times ) 119888 ie position of dot and cross can be interchanged without altering the product

in that order form a right handed system if gt 0

[119886 119886 119888 ] = 0 ie scalar triple product is 0 if any two vectors are equal

[119896119886 119886 119888 ] = 0 ie scalar triple product is 0 if any two vectors are parallel

Show that

If three vectors are coplanar then

Four points A( ) B( ) C( ) and D( ) are coplanarif [( minus 119886 ) (119888 minus 119886 ) (119889 minus 119886 ) ] = 0 = 0

LINEAR PROGRAMMING

The mathematical models which tells to optimise (minimize or maximise) the objective function Z subject

to certain condition on the variables is called a Linear programming problem (LPP)

The standard form of the linear programming problem is used to develop the procedure for solving a general

programming problem

1 A general LPP is of the form

Max (or min) Z = c1x1 + c2x2 + hellip +cnxn

subject to the constraints

x1 x2 xn are called decision variable

c1 c2hellip Cn a11 a12hellip amn are all known constants

Z is called the objective function of the LPP of n variables which is to be maximized or minimized

OBJECTIVE FUNCTION The Objective Function is a linear function of variables which is to be

optimised ie maximised or minimised eg profit function cost function etc The objective function

may be expressed as a linear expression



CONSTRAINTS Limited time labour resources etc may be expressed as linear inequations or

equations and are called constraints

OPTIMISATION A decision which is considered the best one taking into consideration all the

circumstances is called an optimal decision The process of getting the best possible outcome is

called optimisation

SOLUTION OF A LPP A set of values of the variables x1 x2hellipxn which satisfy all the constraints

is called the solution of the LPP

FEASIBLE SOLUTION A set of values of the variables x1 x2 x3hellipxn which satisfy all the

constraints and also the non-negativity conditions is called the feasible solution of the LPP

OPTIMAL SOLUTION The feasible solution which optimises (ie maximizes or minimizes as

the case may be) the objective function is called the optimal solution

2 Mathematical Formulation of Linear Programming Problems

There are mainly four steps in the mathematical formulation of linear programming problem as a

mathematical model We will discuss formulation of those problems which involve only two variables

1 Identify the decision variables and assign symbols x and y to them These decision variables are those

quantities whose values we wish to determine

2 Identify the set of constraints and express them as linear equationsinequations in terms of the decision

variables These constraints are the given conditions

3 Identify the objective function and express it as a linear function of decision variables It might take the

form of maximizing profit or production or minimizing cost

4 Add the non-negativity restrictions on the decision variables as in the physical problems negative

values of decision variables have no valid interpretation

3 Graphical Method of Solution of a Linear Programming Problem

The graphical method is applicable to solve the LPP involving two decision variables x and y Suppose

the LPP is to

Optimize Z = ax + by subject to the constraints

To solve an LPP by the graphical method includes two major steps

The determination of the solution space that defines the feasible solution - To determine the

feasible solution of an LPP we have the following steps

Step 1Since the two decision variable x and y are non-negative consider only the first quadrant of xy- plane

Step 2 Draw the line ax + by = c (1)

For each constraint the line (1) divides the first quadrant in to two regions say R1 and R2suppose (x1 y1) is a

point in R1 If this point satisfies the in equation ax + by ge 119900119903 le c then shade the region R1 If (x1 y1) does

not satisfy the inequation shade the region R2 Usually we take the point (x1 y1) as (0 0) if the line is not

passing through the origin

Step 3 Corresponding to each constraint we obtain a shaded region The intersection of all these shaded

regions is the feasible region



The determination of the optimal solution from the feasible region

There are two techniques to find the optimal solution of an LPPCorner Point Method and ISO- PROFIT

(OR ISO-COST) of the LPP

1 Corner Point Method

The optimal solution to a LPP if it exists occurs at the corners of the feasible region

The method includes the following steps

Step 1Find the feasible region of the LLP

Step 2Find the co-ordinates of each vertex of the feasible region

These co-ordinates can be obtained from the graph or by solving the equation of the lines

Step 3At each vertex (corner point) compute the value of the objective function

Step 4Identify the corner point at which the value of the objective function is maximum (or minimum

depending on the LPP)

The co-ordinates of this vertex is the optimal solution and the value of Z is the optimal value

2 ISO- PROFIT (OR ISO-COST) This method of optimization involves the following method

Step 1Draw the half planes of all the constraints

Step 2Shade the intersection of all the half planes which is the feasible region

Step 3Since the objective function is Z = ax + by draw a dotted line for the equation ax + by = k where k is

any constant Sometimes it is convenient to take k as the LCM of a and b

Step 4To maximize Z draw a line parallel to ax + by = k and farthest from the origin This line should

contain at least one point of the feasible region Find the coordinates of this point by solving the equations of

the lines on which it lies

To minimize Z draw a line parallel to ax + by = k and nearest to the origin This line should contain at least

one point of the feasible region Find the co-ordinates of this point by solving the equation of the line on

which it lies

Step 5 If (x1 y1) is the point found in step 4 then x = x1 y = y1 is the optimal solution of the LPP and

Z = ax1 + by1 is the optimal value

We may come across LPP which may have no feasible (infeasible) solution If the intersection of the

constraints is empty The given problem has no feasible solution Therefore the given LPP has no

solution

We may come across LPP which may have unbounded solution If the feasible region is an unbounded

convex region- then M is the maximum value of z if the open half plane determined by z = ax + by gt M

has no point in common with the feasible region otherwise z has no maximum value Similarly m is the

minimum value of z if the open half plane determined by z = ax + by lt m has no point in common with

the feasible region otherwise z has no minimum value

THREE DIMENSIONAL GEOMETRY

Points are defined as ordered triples of real numbers and the distance between points P1 = (x1 y1 z1) and P2 = (x2 y2 z2) is defined by the formula

P1P2 = radic(x2 minus x1)2 + (y2 minus y1)2 + (z2 minus z1)2

Distance of the point P(xyz) from the origin is radic1199092 + 1199102 + 1199112



Section formula - the coordinate of a point R dividing the line segment PQ joining P (119886) (x1 y1 z1) and

Q( ) (x2 y2 z2) in the ratio m n is given by Internal division

Vector form 119903 =119898 +119899

119898+ 119899 Cartesian form x =

1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division

Vector form 119903 =119898 minus119899

119898minus 119899 Cartesian form x =

1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899

Midpoint formula Vector form 119903 = +

2 Cartesian form x =

1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2

Position vector of centroid of a triangle with vertices A (119886 ) 119861( ) 119886119899119889 119862( 119888 ) is given by + +119888

3

DIRECTION COSINES OF A line - if α β γ be the angles which a given directed line makes with the positive

directions of the co-ordinate axes then cosα cosβ cosγ are called the direction cosines of the given line and are

generally denoted by l m n respectively Thus l = cosα m = cosβ and n = cosγ

l2 + m2 + n2 = 1

Direction RatiosIf a b c are three numbers proportional to the direction cosine l m n of a straight line then a b

c are called its direction ratios They are also called direction numbers or direction components

Direction Cosine of a Line joining two given Points P (x1 y1 z1) and Q(x2 y2 z2)

Angle between two Lines

Let θ be the angle between two straight lines AB and AC whose direction cosines are l1 m1 n1 and l2 m2 then

cosθ = l1l2 + m1m2 + n1n2 If direction ratios of two lines are a1 b1 c1 and a2 b2 c2 then angle between two lines is given by

Condition of perpendicularity If the given lines are perpendicular then θ = 900 ie cos θ = 0

=gt l1l2 + m1m2 + n1n2 = 0 or a1a2 + b1b2 + c1c2 = 0

Condition of parallelism If the given lines are parallel then θ = 00 1198971

1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882

Projection of a line joining two point P (x1 y1 z1) and Q (x2 y2 z2) on another line whose direction cosines are

l m n is AB = l(x2 ndash x1) + m(y2 ndash y1) + n(z2 ndash z1)



Perpendicular Distance of a Point P from a Line Let AB is straight line passing through point A (a b c)

and having direction cosines l m n

AN = projection of line AP on straight line AB = l(x ndash a) + m(y ndash b) + n(z ndash c) and AP = radic(xndasha)2+(yndashb)2+(zndash

c)2

there4 perpendicular distance of point P from AB PN = radicAP2 ndash AN2

Equation of Straight Line in Different Forms

VECTOR EQUATION OF A STRAIGHT LINE

(i) Line passing through a given point A( ) and parallel to a vector

Where is the pv of any general point P on the line

and λ is any real number Symmetrical FormEquation of straight line

passing through point P (x1 y1 z1) and whose

direction cosines are l m n is 119909ndash1199091

119897=

119910ndash1199101

119898 =

119911ndash1199111

119899

The vector equation of a straight line passing through

the origin and parallel to a vector is = n

Line passing through two given points A( ) and B( ) = = + λ( ndash ) Equation of straight line passing

through two points P (x1 y1 z1) and Q (x2 y2 z2) is 119909ndash119909 1

1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111

Equation of the line of intersection of two planes a1x + b1y + c1z + d1 = 0 a2x + b2y + c2z + d2 = 0 is a1x + b1y

+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0

The general coordinates of a point on a line is given by (x1 + lr y1 + mr z1 + nr) where r is distance between point

(x1 y1 z1) and point whose coordinates is to be written

SHORTEST DISTANCE BETWEEN TWO LINES

Two lines in space can be parallel intersecting or neither (called skew lines) Le

be two lines

(i) They intersect if

(ii) They are parallel if 1 and 2 are collinear Parallel lines are of the form

Perpendicular distance between them is constant and is equal to

(iii) For skew lines shortest distance between them (along common perpendicular) is given by

Equation of Plane in Different Forms



EQUATION OF A PLANE IN VECTOR FORM

Following are the four useful ways of specifying a plane i Normal form - A plane at a perpendicular distance d from the

origin and normal to a given direction has the equation

( is a unit vector)

ii A plane passing through the point A( ) and normal to has the

equation

iii Parameteric equation of the plane passing through A( ) and parallel to the plane of vectors ( ) and ( ) is

given by

iv Parameteric equation of the plane passing through A( ) B( ) C( )(A B C non-collinear) is given by

=gt

In Cartesian form the equation of the plane assumes the form Ax + By + Cz = D The vector normal to this plane is

and the perpendicular distance of the plane from the origin is

i General equation of a plane is ax + by + cz + d = 0

ii Normal form - Equation of the plane is lx + my + nz = d where d is the length of the normal from the origin to

the plane and (l m n) be the direction cosines of the normal

iii The equation to the plane passing through P(x1 y1 z1) and having direction ratios (a b c) for its normal is

a(x ndash x1) + b(y ndash y1) + c (z ndash z1) = 0

iv The equation of the plane passing through three non-collinear points (x1 y1 z1)(x2 y2 z2) and (x3 y3 z3) is

= 0

v The equation of the plane whose intercepts are a b c on the x y z axes respectively is 119909

119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)

vi Equation of YZ plane is x = 0 equation of plane parallel to YZ plane is x = d

vii Equation of ZX plane is y = 0 equation of plane parallel to ZX plane is y = d

viii Equation of XY plane is z = 0 equation of plane parallel to XY plane is z = d

Four points namely A (x1 y1 z1) B (x2 y2 z2) C (x3 y3 z3) and D (x4 y4 z4) will be coplanar if one point lies

on the plane passing through other three points



Angle between the planes is defined as angle between normals of the planes drawn from any point to the

planes Angle between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is

Note If a1a2 +b1b2 +c1c2 = 0 then the planes are perpendicular to each other

If 1198861

1198862=

1198871

1198872=

1198881

1198882 then the planes are parallel to each other

Perpendicular Distance The length of the perpendicular from the point P(x1 y1 z1) to the plane ax + by + cz

+ d = 0 is

Intersection of a Line and a Plane If the equation of a plane is ax + by + cz + d = 0 then direction cosines

of the normal to this plane are a b c So angle between normal to the plane and a straight line having

direction cosines l m n is given by cos θ = al+bm+cnradica2+b2+c2Then angle between the plane and the

straight line is π2 ndashθ

Plane and straight line will be parallel if al + bm + cn = 0

Plane and straight line will be perpendicular if al = bm = cn

PROBABILITY THEORY An experiment is a situation involving chance or probability that leads to results called outcomes

An outcome is the result of a single trial of an experiment

An event is one or more outcomes of an experiment

The sample space of an experiment is the set of all possible outcomes of that experiment

Probability is the measure of how likely an event is

The probability of event A is the number of ways event A can occur divided by the total number of possible

outcomes

P(A) = The Number Of Ways Event A Can Occur The total number Of Possible Outcomes

If P(A) gt P(B) then event A is more likely to occur than event B

If P(A) = P(B) then events A and B are equally likely to occur

If event A is impossible then P(A) = 0

If event A is certain then P(A) = 1

The complement of event A is P( ) = 1 - P(A)

The probability of a sample point ranges from 0 to 1

The sum of the probabilities of the distinct outcomes within a sample space is 1

Two events are mutually exclusive if they cannot occur at the same time (ie they have no outcomes in common)

Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring

Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the

second so that the probability is changed



The conditional probability P(B|A) of an event B in relationship to an event A is the probability that event B

occurs given that event A has already occurred The formula for conditional probability is

For events A and B provided that

Addition Rule1 When two events A and B are mutually exclusive the probability that A or B will occur is the

sum of the probability of each event P(A or B) = P(A) + P(B)

Addition Rule2 When two events A and B are non-mutually exclusive P(A or B) = P(A) + P(B) - P(A and B)

Addition Rule3 When two events A and B are independent the probability that A or B will occur is P(A or B) =

P(A) + P(B) - P(A) middot P(B)

Multiplication Rule 1 When two events A and B are independent the probability of both occurring is P(A and

B) = P(A) middot P(B)

Multiplication Rule 2 When two events A and B are dependent the probability of both occurring is P(A and B)

= P(A) middot P(B|A)

Total Probabi l i ty Theorem Let A1 A2 An be a set of mutually exclusive events that together

form the sample space S Let B be any event from the same sample space such that P(B) gt 0 Then p(B) = p(A 1 ) middot p(B|A 1 ) + p(A 2 ) middot p(B|A 2 ) + + p(A n) middot p(B|A n )

Bayes theorem Let A1 A2 An be a set of mutually exclusive events that together form the sample

space S Let B be any event from the same sample space such that P(B) gt 0 Then

A discrete variable is a variable which can only take a countable number of values Let X be a random variable that takes the numerical values X1 X2 Xn with probablities p(X1) p(X2)

p(Xn) respectively A discrete probability distribution consists of the values of the random variable X and their

corresponding probabilities P(X)

The probabilities P(X) are such that sum P(X) = 1

Bernoulli Trials - An experiment in which a single action is repeated identically over and over The possible

results of the action are classified as success or failure The trials must all be independent The binomial

probability formula is used to find probabilities for Bernoulli trials

The Binomial Distribution - The probability of achieving exactly k successes in n trials is P(X= r) = nCr prqn ndash r

where n = number of trials

k = number of successes

n ndash k = number of failures

p = probability of success in one trial

q = 1 ndash p = probability of failure in one trial

Expectation and Variance If X ~ B(np) then the expectation and variance is given by

1 E(X) = np

2 Var(X) = npq





RELATIONS AND FUNCTIONS

I RELATION

i Let A and B be two sets A relation between A and B is a collection of ordered pairs (a b) such that a

A and b B

ii If 119877 119860 rarr 119861 is a relation from A to B then 119877 sube 119860 times 119861

iii If n(A) = m n(B) = n then total number of relations from A to B is 2mn

iv Domain of R = 119886 (119886 119887) isin 119877

v Range of R = 119887 (119886 119887) isin 119877

vi Co-domain of R = 119861

II Equivalence Relation

Let S be a set and R a relation between S and itself We call R an equivalence relation on S if R has the following three properties

Reflexivity Every element of S is related to itself ⟹ (119886 119886) isin 119877 forall 119886 isin 119878 Symmetry If a is related to b then b is related to a (119886 119887) isin 119877 ⟹ (119887 119886) isin 119877 forall 119886 119887 isin 119878 Transitivity If a is related to b and b is related to c then a is related to c (119886 119887) isin 119877 (119887 119888) isin 119877 ⟹

(119886 119888) isin 119877 forall 119886 119887 119888 isin 119878

Antisymmetric - A relation is antisymmetric if a R b and b R a⟹ a=b for all values a and b

III FUNCTIONS

Definition - Any relation on A x B in which

i No two second elements have a common first element and

ii Every first element has a corresponding second element is called a function It is also called mapping

A function is said to map an element x in its domain to an element y in its range 119891 119860 rarr119861 119900119903 119891 119909 rarr 119891(119909) 119905ℎ119890119899 119891(119909) = 119910 where y is a function of x

DOMAIN - The set of all the first elements of the ordered pairs of a function is called the domain

RANGE - The set of all the second elements of the ordered pairs of a function is called the range

CODOMAIN - If (a b) is an ordered pair of the function 119891 119860 rarr 119861 then the set B is called the Co-Domain The

range is a subset of the co-domain

IV Some important facts about a function from A to B








mapped to them




119961120783 = 119961120784 119873119900119905119890 119905ℎ119886119905 ℎ119890119903119890 119899(119860) le 119899(119861)






















discontinuous




discontinuous






119909 forall119909 ne 0

0 119909 = 0=

1 119909 gt 00 119909 = 0minus1 119909 lt 0































forall 119886 119887 isin 119860


forall 119886 119887 119888 isin 119860











real result

Range of usual

principal value

(radians)

Range of usual

principal value

(degrees)








y le π


180deg


y le π2


90deg



119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2









119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587


119905119886119899minus1 (1

119909) = minus

120587


119888119900119905minus1 (1

119909) =

120587


119888119900119905minus1 (1

119909) =

3120587




p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq










mapped to them




119961120783 = 119961120784 119873119900119905119890 119905ℎ119886119905 ℎ119890119903119890 119899(119860) le 119899(119861)






















discontinuous




discontinuous






119909 forall119909 ne 0

0 119909 = 0=

1 119909 gt 00 119909 = 0minus1 119909 lt 0































forall 119886 119887 isin 119860


forall 119886 119887 119888 isin 119860











real result

Range of usual

principal value

(radians)

Range of usual

principal value

(degrees)








y le π


180deg


y le π2


90deg



119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2









119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587


119905119886119899minus1 (1

119909) = minus

120587


119888119900119905minus1 (1

119909) =

120587


119888119900119905minus1 (1

119909) =

3120587




p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






discontinuous






119909 forall119909 ne 0

0 119909 = 0=

1 119909 gt 00 119909 = 0minus1 119909 lt 0































forall 119886 119887 isin 119860


forall 119886 119887 119888 isin 119860











real result

Range of usual

principal value

(radians)

Range of usual

principal value

(degrees)








y le π


180deg


y le π2


90deg



119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2









119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587


119905119886119899minus1 (1

119909) = minus

120587


119888119900119905minus1 (1

119909) =

120587


119888119900119905minus1 (1

119909) =

3120587




p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq











forall 119886 119887 isin 119860


forall 119886 119887 119888 isin 119860











real result

Range of usual

principal value

(radians)

Range of usual

principal value

(degrees)








y le π


180deg


y le π2


90deg



119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2









119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587


119905119886119899minus1 (1

119909) = minus

120587


119888119900119905minus1 (1

119909) =

120587


119888119900119905minus1 (1

119909) =

3120587




p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq








y le π


180deg


y le π2


90deg



119904119894119899minus1119909 + 119888119900119904minus1119909 =120587

2

119904119890119888minus1119909 + 119888119900119904119890119888minus1119909 =120587

2

119905119886119899minus1119909 + 119888119900119905minus1119909 =120587

2









119904119894119899minus1 (1

119909) = 119888119900119904119890119888minus1119909

119888119900119904119890119888minus1 (1

119909) = 119904119894119899minus1119909

119888119900119904minus1 (1

119909) = 119904119890119888minus1119909

119904119890119888minus1 (1

119909) = 119888119900119904minus1119909

119905119886119899minus1 (1

119909) =

120587


119905119886119899minus1 (1

119909) = minus

120587


119888119900119905minus1 (1

119909) =

120587


119888119900119905minus1 (1

119909) =

3120587




p h

b



Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq





Use 119904119894119899minus1 (119901

ℎ) = 119888119900119904minus1 (

119887

ℎ) = 119905119886119899minus1 (

119901

119887) = 119888119900119904119890119888minus1 (

ℎ

119901) = 119904119890119888minus1 (

ℎ

119887) = 119888119900119905minus1 (

119887

119901)

V SUM FORMULA




1∓119909119910)



VI MULTIPLE FORMULA



2119905119886119899minus1119909 = 119905119886119899minus1 2119909

1minus1199092 = 119904119894119899minus1 2119909

1+1199092 = 119888119900119904minus1 1minus1199092

1+1199092



1minus31199092


1 a3 + b3 = (a+b)(a2 ndash ab + b2)

2 a3 - b3 = (a - b)(a2 + ab + b2)

3 sin cos2 2x x 1





tan tan

tan tan

u v

u v

1

9 1199041198941198992119906 = 2119904119894119899119906119888119900119904119906 =2119905119886119899119906

1+1199051198861198992119906


1+1199051198861198992119906

11 tan( )2u 2

1 2

tan

tan

u

u



14 Tan3u = 3119905119886119899119906minus1199051198861198993119906

1minus31199051198861198992119906

15 sin2 u 1 2

2

cos u

16 cos2 u 1 2

2

cos u

17 tan2 u 1 2

1 2

cos

cos

u

u

18 Sin3u = 3119904119894119899119906minus1199041198941198993119906

4

19 cos3u = 3119888119900119904119906+1198881199001199043119906

4

20 sinusinv = 1

2[119888119900119904(119906 minus 119907) minus 119888119900119904(119906 + 119907)]



21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq





21 cosucosv = 1

2[119888119900119904(119906 + 119907) + 119888119900119904(119906 minus 119907)]

22 Sinucosv = 1

2[119904119894119899(119906 + 119907) + 119904119894119899(119906 minus 119907)]

23 cosusinv = 1

2[119904119894119899(119906 + 119907) minus 119888119900119904(119906 minus 119907)]

24 sinu + sinv = 2119904119894119899(119906+119907)

2119888119900119904

(119906minus119907)

2

25 sinu - sinv = 2119888119900119904(119906+119907)

2119904119894119899

(119906minus119907)

2

26 cosu + cosv = 2119888119900119904(119906+119907)

2119888119900119904

(119906minus119907)

2

27 cosu - cosv = 2119904119894119899(119906+119907)

2119888119900119904

(119907minus119906)

2

28 law of sinesa

A

b

B

c




Area sin2

ac B


1 Circle formula

2 2 2x h y k r

2 Parabola formula

24x h p y k

3 Ellipse formula

x

a

y

bc a b

2

2

2

2

2 21


a

y

bc a b

2

2

2

2

2 21

5 eccentricity e

c

a



x yx a t y b t

a b



lna x

x

a

b limsin

x

x

x0 = 1

c limsin

x

x

x = 0

d lim119909rarr119886

119909119899minus119886119899

119909minus119886= 119899119886119899minus1

e lim119909rarr0

119890119909minus1

119909= 1

f lim119909rarr0

119886119909minus1

119909= 119897119900119892119890119886

g lim119909rarr0

log (1+119909)

119909= 1



119891(119909) = 119891(119886)




119891(119909)119890119909119894119904119905119904 119894 119890 lim119909rarr119886minus

119891(119909) = lim119909rarr119886+

119891(119909)




119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






119891(119909) = 119891(119886)





not equal


119891(119909) ne 119891(119886)


119891(119909) = 119891(119886)


119891(119909) = 119891(119886)


119891(119909) ne 119891(119886)





V DIFFERENTIATION


119889119909=

f x

f x h f x

hh( ) lim

( ) ( )

0



119943(119938minus119945)minus119943(119938)

119945


119891(119886minusℎ)minus119891(119886)

ℎ



1 119889(119862)

119889119909= 0

2 119889(119909)

119889119909= 1

3 119889(119909119899)

119889119909= 119899119909119899minus1

4 119889(119890119909)

119889119909= 119890119909

5 119889(119890119886119909+119887)

119889119909= 119886119890119886119909+119887

6 119889(119886119909)

119889119909= 119886119909 119897119900119892119886

7 119889(119897119900119892119909)

119889119909=

1

119909

8 119889(119904119894119899119909)

119889119909= 119888119900119904119909

9 119889(119888119900119904119909)

119889119909= minus119904119894119899119909

10 119889(119905119886119899119909)

119889119909= 1199041198901198882119909

11 119889(119888119900119905119909)

119889119909= minus1198881199001199041198901198882119909

12 119889(119904119890119888119909)

119889119909= 119904119890119888119909119905119886119899119909

13 119889(119888119900119904119890119888119909)

119889119909= minus119888119900119904119890119888119909119888119900119905119909

14 119889(119904119894119899minus1119909)

119889119909=

1

radic1minus1199092

15 119889(119888119900119904minus1119909)

119889119909= minus

1

radic1minus1199092

16 119889(119905119886119899minus1119909)

119889119909=

1

1+1199092

17 119889(119888119900119905minus1119909)

119889119909= minus

1

1+1199092

18 119889(119904119890119888minus1119909)

119889119909=

1

|119909|radic1199092minus1

19 119889(119888119900119904119890119888minus1119909)

119889119909= minus

1

|119909|radic1199092minus1

20119889119891(119886119909+119887)

119889119909= 119886119891prime(119886119909 +b)





119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







119889119909=

119889119891

119889119906119889119906

119889119909


119889119909= 119906

119889119907

119889119909+ 119907

119889119906

119889119909= 119906119907prime + 119906prime119907


119889119909(119906

119907) =


1199072


dx

dy

dtdx

dt


dx df

dxx f a

x a

1 1

( )




119889119909119886119899119889

119889119907

119889119909 by


119889119909as

119889119910

119889119909=

119889119906

119889119909+

119889119907

119889119909







b af c

( ) ( )( )





to x axis )







Approximation -






4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq





4 calculate 119889119910



119889119909times ∆119909





119889119909(11990901199100)


119889119909(11990901199100)

(x minus x0)


119889119910(11990901199100)


119889119910(11990901199100)

(x minus x0)


point of contact

119905ℎ119890 119888119900119899119889119894119905119894119900119899 119900119891 119900119903119905ℎ119900119892119900119899119886119897119894119905119910 119900119891 119905119908119900 119888119906119903119907119890119904 1198881 119886119899119889 1198882 119894119904 119889119910

119889119909]1198881

times119889119910

119889119909]1198882

= minus1














VII CONCAVITY
























at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq




















at local extrema













X Absolute Extrema


x on I


x on I


















2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq














2 int 119896119891(119909)119889119909 = 119896 int 119891(119909)119889119909 + 119862

3 int 119891(119892(119909)) 119892prime(119909)119889119909 = int119891(119905)119889119905 119908ℎ119890119903119890 119892(119909) = 119905


5

6

7

8




1 int119956119946119951119961119941119961 = minus119940119952119956119961 + 119940

2 int119940119952119956119961119941119961 = 119956119946119951119961 + 119940

3 int119956119942119940119961119941119961 = 119949119952119944|119956119942119940119961 + 119957119938119951119961| + 119940

4 int119940119952119956119942119940119961119941119961 = 119949119952119944|119940119952119956119942119940119961 minus 119940119952119957119961| + 119940

5 int 119957119938119951119961119941119961 = 119949119952119944|119956119942119940119961| + 119940 = minus119949119952119944|119940119952119956119961| + 119940

6 int119940119952119957119961119941119961 = 119949119952119944|119956119946119951119961| + 119940

7 int119956119942119940120784119961119941119961 = 119957119938119951119961 + 119940

8 int119940119952119956119942119940120784119961119941119961 = minus119940119952119957119961 + 119940

9 int119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

10 int119940119952119956119942119940119961119957119938119951119961119941119961 = 119956119942119940119961 + 119940

11 int119941119961


12 int119941119961


13 int119941119961







int 119956119946119951119950119961119940119952119956119951 119961119941119961 = int 119957119950(120783 minus 119957120784)119951120784

119941119957


2

cos u

cos2 u 1 2

2

cos u


int 119956119946119951(119950119961) 119940119952119956(119951119961)119941119961 int 119956119946119951(119950119961) 119956119946119951(119951119961)119941119961





2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






2[119888119900119904(119898 minus 119899)119909 minus 119888119900119904(119898 + 119899)119909]


2[119904119894119899(119898 minus 119899)119909 + 119904119894119899(119898 + 119899)119909]


2[119888119900119904(119898 minus 119899)119909 + 119888119900119904(119898 + 119899)119909]


Use











1 int1

1198862+1199092 119889119909 = 1

119886 tanminus1 119909

119886 +

C

2 int1

1198862minus 1199092 119889119909 =

1

2119886 119897119900119892 |

119886+119909

119886minus119909| + C

3 int1

1199092minus 1198862 119889119909 =

1

2119886 119897119900119892 |

119909minus119886

119909+119886| + C

4 int1


119886 + C

5 int1

radic1198862+1199092 dx = 119897119900119892|119909 + radic1198862 + 1199092| + C

6 int1



2radic1198862 minus 1199092 +

1198862

2 119904119894119899minus1 119909

119886 + C

8 intradic1198862 + 1199092dx = 119909

2radic1198862 + 1199092 +

119886

2

2119897119900119892|119909 + radic1198862 + 1199092| + C



119886

2

2119897119900119892|119909 + radic1199092 minus 1198862| + C


119938119961120784+119939119961+119940119941119961 or int

120783


ax2+ bx + c = a [(119909 +119887

2119886)2+ (

radic4119886119888minus1198872

2119886)2



119961120786+120640119961120784+120783119941119961 int

119961120784minus120783

119961120786+120640119961120784+120783119941119961 int

120783

119961120786+120640119961120784+120783119941119961 119960119945119942119955119942 120640 isin 119929




1


numerator

Substitute x + 1

119909 = t or x -

1






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






119938119961120784+119939119961+119940119941119961 or int

119953119961+119954

radic119938119961120784+119939119961+119940119941119961

Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583



119938119961120784+119939119961+119940119889119909 + 120583 int

120783

119938119961120784+119939119961+119940119889119909 OR 120582 int

2119886119909+119887

radic119938119961120784+119939119961+119940119889119909 + 120583 int

120783

radic119938119961120784+119939119961+119940119941119961




Put px+q = 120582119889(1198861199092+119887119909+119888)

119889119909+ 120583 = 120582(2119886119909 + 119887) + 120583




(119891(119909))119899+1





119938+119939119956119946119951120784119961119941119961 int

120783

119938+119939119940119952119956120784119961119941119961 int

120783

119938119956119946119951120784119961+119939119940119952119956120784119961119941119961 int

120783

(119938119956119946119951119961 + 119939119940119952119956119961)120784119941119961

int120783

119938+119939119956119946119951120784119961+119940119940119952119956120784119961119941119961





119938119956119946119951119961 + 119939119940119952119956119961

put sinx = 2119905119886119899

119909

2

1+1199051198861198992 119909

2

and cosx = 1minus1199051198861198992

119909

2

1+1199051198861198992 119909

2

then substitute


XIV 119816119847119853119838119840119851119834119845119852 119848119839 119853119841119838 119839119848119851119846int119938119956119946119951119961+119939119940119952119956119961

119940119956119946119951119961 + 119941119940119952119956119961119941119961





119888119904119894119899119909+119889119888119900119904119909119889119909 + 120583 int

119888119888119900119904119909+119889119904119894119899119909

119888119904119894119899119909+119889119888119900119904119909dx And evaluate


119876(119909)



polynomials



(119886119909 + 119887)(119888119909 + 119889)

119860

119886119909 + 119887+

119861

119888119909 + 119889



1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq





1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(119888119909 + 119889)(119890119909 + 119891)

119860

119886119909 + 119887+

119861

119888119909 + 119889+

119862

119890119909 + 119891

119901119909 + 119902

(119886119909 + 119887)2

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)2(119888119909 + 119889)

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

119888119909 + 119889

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)3

119860

119886119909 + 119887+

119861

(119886119909 + 119887)2+

119862

(119886119909 + 119887)3

1199011199092 + 119902119909 + 119903

(119886119909 + 119887)(1198881199092 + 119889119909 + 119890)

119860

119886119909+119887+

119861119909+119862


further factorised





119961(119961119951+119948) 119899 isin 119873 119899 ge 2




















int119906(119909) 119907(119909)119889119909 = 119906(119909) int 119907(119909)119889119909 minus int [(119889

119889119909119906(119909)) (int 119907(119909)119889119909)] 119889119909












formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






formint120593(119909)

119875radic119876119889119909

int1


int1


ie put px + q = t2

int1


1

119905

ie ax+ b = 1

119905

int1


1

119905to obtain int

minus119905dt


= u2

int119901119909+119902




then int 119891(119909)119889119909 = 119865(119887) minus 119865(119886)119887









point c in a b 1 b

af c f x dx

b a


119891(119886 + 119896 minus 1 ℎ) ℎ that is

int 119891(119909)119889119909 = limℎrarr0


119887

119886

= limℎrarr0

ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]





3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






3Use int 119891(119909)119889119909119887

119886 = lim

ℎrarr0ℎ[119891(119886) + 119891(119886 + ℎ) + 119891(119886 + 2ℎ) + ⋯+ 119891(119886 + (119899 minus 1)ℎ)]


1 + 2 + 3 + ⋯+ (119899 minus 1) =(119899minus1)119899

2

12+22

+ 32+ ⋯+ (119899 minus 1)2 =

(119899minus1)119899(2119899minus1)

6

13+23

+ 33+ ⋯+ (119899 minus 1)3 =

(119899minus1)21198992

4

119886 + 119886119903 + 1198861199032 + ⋯+ 119886119903119899minus1 = 119886(119903119899minus1)

(119903minus1)|119903| gt1


119899minus1

2)ℎ sin(

119899ℎ

2)

sin(ℎ

2)


119899minus1

2)ℎsin(

119899ℎ

2)

sin(ℎ

2)




(i) int (119891(119909) plusmn g(119909))119887

119886119889119909 = int 119891(119909)119889119909 plusmn int 119892(119909)119889119909

119887

119886

119887

119886

(ii) int 120572119891(119909)119889119909119887

119886= 120572 int 119891(119909)119889119909

119887

119886

(iii) int 119891(119909)119889119909 = 0119888

119888


int 119891(119909)119889119909 = int 119891(119905)119889119905119887

119886

119887

119886


int 119891(119909)119889119909 = minusint 119891(119909)119889119909119886

119887

119887

119886


int 119891(119909)119889119909 = int 119891(119909)119889119909 +119888

119886 int 119891(119909)119889119909119887

119888

119887


P3 int 119891(119909)119889119909 = int 119891(119886 + 119887 minus 119909)119889119909119887

119886

119887

119886

P4 int 119891(119909)119889119909 = int 119891(119886 minus 119909)119889119909119886

0

119886

0


0

119886

0

2119886

0

P6 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(2119886 minus 119909) = 119891(119909)

0 119894119891 119891(2119886 minus 119909) = minus 119891(119909)

2119886

0

P7 int 119891(119909)119889119909 = 2int 119891(119909)119889119909

119886

0 119894119891 119891(minus119909) = 119891(119909)119894 119890 119894119891 119891 119894119904 119890119907119890119899 119891119906119899119888119905119894119900119899

0 119894119891 119891(minus119909) = minus 119891(119909) 119894 119890 119894119891 119891 119894119904 119900119889119889 119891119906119899119888119905119894119900119899

119886

minus119886





119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







119886=

int 119891(119909)119889119909119887

119886


119886=

int 119891(119910)119889119910119887

119886



119886119889119909



119886119889119910

WORKING RULE-






discussed above






119938120783120783119938120783120784 ⋯ 119938120783119951

⋮ ⋱ ⋮119938119950120783119938119950120784 ⋯ 119938119950119951

]



119938120783120783

119938120784120783

119938119950120783

]



matrix)



elements are equal









MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







MATRIX OPERATIONS



columns

2 Addition




A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC

4 Subtraction















elements of C are





AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







AB ne BA

A(BC) = (AB)C

AIn = A = InA







(119860 + 119861)119879 = 119860119879 + 119861119879

(119860 119861)119879 = 119861119879119860119879

(119896 119860)119879 = 119896 119860119879







matrices where P = (119860+119860119879)

2 and Q =

(119860minus 119860119879)

2















particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq













particular answer







determinant of A




] etc


=[Cij]T=[Cji]




119938120784120783 119938120784120784] then|119912| = 119938120783120783119938120784120784 minus 119938120784120783119938120783120784


119938120783120783 119938120783120784 119938120783120785

119938120784120783 119938120784120784 119938120784120785

119938120785120783 119938120785120784 119938120785120785

] then

det A= |119912| = 119938120783120783 |119938120784120784 119938120784120785

119938120785120784 119938120785120785| minus 119938120783120784 |

119938120784120783 119938120784120785

119938120785120783 119938120785120785| + 119938120783120785 |

119938120784120783 119938120784120784

119938120785120783 119938120785120784|











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq











119877119895 or 119862119894 ⟷ 119862119895 ⟹ ∆= minus∆





















2|

1199091 1199101 11199092 1199102 11199093 1199103 1

||


1199091 1199101 11199092 1199102 11199093 1199103 1

| = 0




119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






119909 119910 11199091 1199101 11199092 1199102 1

| = 0




|119860|














STEPS







|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq






|119860|





inconsistent



method




inconsistent



method

______________________________________________________________________________________











in it









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq









where






119889119909= 119891(119909 119910) is called


119889119909= ℎ(119909)119892(119910) - - - (I)




119892(119910)= ℎ(119909)119889119909 and then

3 integrate int1





important



order n




119889119909= 119865(

119910

119909)



equation 119889119909

119909=

119889119907

119865(119907)minus119907





119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







119889119909= 119865 (

119910

119909) or

119889119909

119889119910= 119865 (

119909

119910)



119889119909= 119907 + 119909

119889119907

119889119909 119900119903

119889119909

119889119910= 119907 + 119910

119889119907

119889119910





VECTORS





of 119886


| |=

119834120783

radic119834120783120784+119834120784

120784+119834120785120784 +

119938120784

radic119938120783120784+119938120784

120784+119938120785120784 +

119938120785

radic119938120783120784+119938120784

120784+119938120785120784 then 119897 =

119938120783

radic119938120783120784+119938120784

120784+119938120785120784119846 =

1198862

radic11988612+1198862

2+11988632and n =

1198863

radic11988612+1198862

2+11988632













vectors













direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq















direction



directions








opposite order


















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq

















119886 = |119886||119887|119888119900119904120579









120579 = cosminus1 119887

|119886||119887|


|119887|




















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq















119939120783 119939120784 119939120785

|




| times |


by two vectors = 1

2|119886 x |

















[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







[119886 119888 ] = 119886 ( times 119888 ) = |

119938120783 119938120784 119938120785

119939120783 119939120784 119939120785

1198881 1198882 1198883

|





Show that



LINEAR PROGRAMMING




programming problem
















called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq









called optimisation




















the LPP is to






































which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq




























which it lies





solution














Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq







Vector form 119903 =119898 +119899


1198981199092+1198991199091

119898+119899 y =

1198981199102+1198991199101

119898+119899 z =

1198981199112+1198991199111

119898+119899

External division



1198981199092minus1198991199091

119898minus119899 y =

1198981199102minus1198991199101

119898minus119899 z =

1198981199112minus1198991199111

119898minus119899



1199091+1199092

2 y =

1199101+1199102

2 z =

1199111 +1199112

2


3




l2 + m2 + n2 = 1










1198972=

1198981

1198982=

1198991

1198992 or

1198861

1198862=

1198871

1198872=

1198881

1198882








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq








c)2









119897=

119910ndash1199101

119898 =

119911ndash1199111

119899





1199092ndash1199091=

119910ndash1199101

1199102ndash1199101 =

119911ndash1199111

1199112ndash1199111


+ c1z + d1 + 120582(a2x + b2y + c2z + d2 ) = 0





be two lines











( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq








( is a unit vector)


equation


given by


=gt









= 0


119886+

119910

119887+

119911

119888= 1 (a b c

ne 0)











If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq








If 1198861

1198862=

1198871

1198872=

1198881



+ d = 0 is













outcomes













































1 E(X) = np

2 Var(X) = npq



































1 E(X) = np

2 Var(X) = npq



Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, … · 2020-04-01 · Vidyarthi Tutorials H10...

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Transcript of Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, … · 2020-04-01 · Vidyarthi Tutorials H10...