Free Mathematics eBook for JEE Main

8/12/2019 Free Mathematics eBook for JEE Main

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C l a s s

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Class

TABLE OF CONTENT

RELATIONS AND FUNCTIONS---------------------------------------------------- 1

TRIGONOMETRY FUNCTION------------------------------------------------------ 1 - 2

INVERSE TRIGONOMETRY FUNCTION---------------------------------------- 2 - 3

QUADRATIC EQUATIONS AND INEQUALITIES----------------------------- 3 - 4

COMPLEX NUMBERS---------------------------------------------------------------- 4

PERMUTATION AND COMBINATIONS------------------------------------------ 5

BINOMIAL THEOREM---------------------------------------------------------------- 5SEQUENCE AND SERIES----------------------------------------------------------- 6

STRAIGHT LINES---------------------------------------------------------------------- 6

CONIC SECTIONS--------------------------------------------------------------------- 7

THREE DIMENSIONAL GEOMETRY--------------------------------------------- 8

DIFFERENTIAL CALCULUS-------------------------------------------------------- 9

CONTINUITY OF FUNCTIONS----------------------------------------------------- 10

DIFFERENTIAL AND APPLICATION--------------------------------------------- 10 - 11 INTEGRAL CALCULUS-------------------------------------------------------------- 12 - 13

PROBABILITY-------------------------------------------------------------------------- 14

MATRICES------------------------------------------------------------------------------- 15 - 16

DETERMINANT------------------------------------------------------------------------- 16

VECTORS-------------------------------------------------------------------------------- 17 - 18

STATISTICS----------------------------------------------------------------------------- 18

DIFFERENTIAL EQUATIONS------------------------------------------------------ 19



Class

C l a s s

1

1 2 1 2 1 2( ) ( ) , . f x f x x x x x X = = "

( ) ( ( )) gof x g f x x A= "

X gof I = Y fog I =

sin( ) sin cos cos sina b a b a b=m m cos( ) cos cos sin sina b a b a b = m

tan tantan( )1 t an tan

a ba b a b= mm sin sin 2sin cos

2 2 A B A B A B + -+ =

sin sin 2cos sin

2 2 A B A B

A B + -- =

cos cos 2 cos cos

2 2 A B A B

A B + -+ =

cos cos 2sin sin

2 2 A B A B

A B + -- = -

2

sin sin sina b c

R A B C

= = =

2 2 2

cos 2

b c a

A bc

+ -=

tan cot2 2

B C b c A

b c

- - = +

( )( )sin

2 A s b s c

bc- - =

( )( )tan

2 ( ) A s b s c

s s a- - = -

1sin sin2 2

sin sin( ) sin( 2 ) ...... ; 2sin( / 2)

n n

tonterms n

ba ba a b a b b p

b

- + + + + + + =

RELASTIONS AND FUNCTIONS

TRIGONOMETRY FUNCTION

1. A function f :

2. A function f: is onto (or surjective) if given any suchthat

3. A function f: is one one and onto (or bijective), if f is both one- one and onto

4. The composition of functions f: and g is the functiongiven by

5. A function f: is invertible if such that and

6. A function f: is invertible if and only if f is one - one and onto.

1. 2.

3. 4.

5. 6.

7. 8. Sine Rule:

9. Cosine rule: 10.

11. 12.

13.

X

X Y , f x y

X Y

A B C : C

X Y $ g : Y X

X Y

Y

y Y $( ) =

: B gof A

is one-one (or injective) if

x X


4/22



Class

C l a s s

3

2 0ax bx c+ + = 2 4

2b b ac x

a- -=

0, 0

0, 0

a c

a c

> > < < 0, 0 0

0, 0, 0

a b c

a b c

> > > < < < 0, 0, 0

0, 0, 0

a b c

a b c

> < > < > <

QUADRATIC EQUATIONS AND INEQUALITIES

1. For the quadratic equation ,

(i) If b = 0(ii)

If c = 0 one root is zero other is b /a(iii) If b = c =0 both roots are zero.(iv) If a = c roots are reciprocal to each other (v) If Roots are of opposite signs.

(vi) If both roots are negative

(vii) If both roots are positive(viii) If signs of a = sign of b sign of c Greater root in magnitude is negative

roots are of equal magnitude but of opposite sign



Class

C l a s s

4

1i = -

1 2 ( ) ( ) z z a c i b d + = + + +

1 2 ( ) ( ) z z ac bd i ad bc= - + +

2 2 2 2

a bi

a b a b

-++ +

1

z

1 z -

( ) 2 2 2 2 1 0 1a ba ib i ia b a b

- + + = + = + + 4 4 1 4 2 4 31, , 1,k k k k i i i i i i+ + += = = - = -

1 2| | | | z z z z l- + - = 1 2| | z z l- <

1 2| | z z l- = ( ) ( )1 11, 1 3 , 1 3

2 2- + - - - -

2 21,cos sin

3 3i

p p +

4 4cos sin

3 3

ip p +

(cos sin ) cos sinni n i nq q q q+ = + cos sin z iq q= + 1

2 cos ;n n z n z q+ =

12 sinn n z i n z

q- =

2. If p +iq (p and q being real) is root of the quadratic equation, wherealso a root of the quadratic equation.

th3. Every equation of n degreethan n roots, it is an identity.

4. An inequality of the form log f (x) > b is equivalent to the following systems ofainequalities :

b(a) f(x) > 0, f (x) > a for a > 1.

b(b) F(x)>0, f(x) < a for a < 1.

1. Let and(a)

(b)

2. For any non-zero complex number z = a + ib , there exists the complexnumber , denoted by or called the multiplicative inverse of z such

that

3. For any integer k, 4. , represents an ellipse if , having the points and as its

foci. And if , then z lies on a line segment connecting and .

5. Cube root of unity The three cube root of unity are , which arethe same as and

6. De Moivre's Theorem: for all real values of n, If , using De Moivre's Theorem

, then p iq is

has exactly n roots and if the equation has more

Then

(n

z a ib c id 2

(a 0, b 0)

z z 1 2

1)

= +1

z

COMPLEX NUMBERS

z = +

1 2 z



Class

C l a s s

5

PERMUTATION AND COMBINATIONS

BINOMIAL THEOREM

1. The number of permutations of n different things, taken r at a time, where repetition isallowed, is denoted by and given by

2. The number of permutations of n objects taken all at a time. Where repetition isr allowed, is n .

3. The number of permutations of n objects where p objects are of first kind, p objects1 2thare of the second kind, ..., p objects are of the k kind and rest, if any, l are allk

different is

4. The number of combinations of n different things taken r at a time, denoted by

, is given by

5. Number of circular permutations of n things when p alike and the rest different takenall at a time distinguish clockwise and anticlockwise arrangement is

1. The expansion of a binomial for any positive integral n is given by Binomial Theorem.which is

The coefficients of the expansions are arranged in an array. This array is called Pascal'striangle. When the index is other than a positive integer such as negative integer or

n nfraction. The number of terms in the expansion of (1 + x) is innite and the symbol Ccannot be used to denote the coefficients of the general term.

2. The general term of an expansion isThe total number of terms in the expansion of is n +1.

3. In the expansion , if n is even, then the middle term is the term. If n is odd,then the middle terms are and terms.

, where

,

r p

n !

( )!r

p

nn

n r =

-

0 r n

1 2

!! !..... !k

n p p p

r c

n !

!( )!r cn

n r n r = - 0 r n

( 1)!!

n p-

0

1 2 2 11 2 1( ) ..... .

n n n n n nn na b nC a nC a nC a b nC a b nC b

- - --+ = + + + + +

( ) na b+ 1 .r n r r

r T nC a b-

+ = ( )na b+

( )na b+

12

thn + 1

2

thn +

1

12

thn + +


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Class 6

SEQUENCE AND SERIES

STRAIGHT LINES

th n1. The general term or the n term of the A.P is given by a = a + (n -1)d.

The sum S of the first n terms of an A.P is given byn

2. The sum S of the first n terms of G.P is given by or if r n

3. A series who's each term is formed, by multiplying corresponding terms of an A.P.and a G.P., is called ar Arithmetic-geometric series. Summation of n terms

4. Harmonical progression is dened as a series in which reciprocal "of its terms are in A.P. The standard from of a H.P. is

1. An acute angle (say and is given by

,

2. Equation of the line passing through the points (x , y ) and (x , y ) is given by1 1 2 2y - y = .1

3. Equation of a line making intersects a and b on the x- and y- axis, respectively, is

.

4. The perpendicular distance (d) of a line Ax + By +C =0 from a point (x , y )is given by1 1

5. Distance between the parallel lines Ax + By +C =0 and Ax + By +C =0, is given by1 2.

1

q ) between lines L m1 2 1 2 L mwith slopes and

[ ]2 ( 1) ( )

2 2nn n

S a n d a l = + - = + ( 1)

1

n

n

a r S

r -=

-

( 1)1

na r r

--

12

(1 ) [ ( 1) .1 (1 ) 1

nn

na dr r a n d S r

r r r

-- + -= + -- - -

1 1 1......

2a a d a d + + +

+ +

2 2

1 2

tan1m m

m mq -=

+ 1 21 0m m+

2 11

2 1

( ) y y x x x x

- --

1

x ya b

+ =

1 1

2 2

| | Ax By C d

A B

+ +=+

1 2

2 2

| |C C d

A B

-=+


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Class 7

CONIC SECTIONS

2 2 21. The equation of a circle with centre (h, k) and the radius r is (x h) + (y k) = r .

2. Equation of tangent: xx + yy +g(x + x ) +f(y + y ) +c =01 1 1 1

23. The equation of the parabola with focus at (a, 0) a > 0 and directrix x = - a is y =4ax.

4. The equation of the parabola is a line segment perpendicular to the axis of theparabola, through the focus and whose end points lie on the parabola.

25. Length of the latus rectum of the parabola y = 4ax is 4a.

26. The parametric equation of the parabola is x = at , y = 2at.

7. An ellipse is the set of all points in a plane, the sum of whose distances from twoxed points in the plane is a constant.

8. The equations of an ellipse with foci on the x-axis is ; its parametric equationis

9. Latus rectum of the ellipse is

10.The eccentricity of an ellipse is the ratio between the distances from the centre of the

ellipse to one of the foci and to one of the vertices of the ellipse.

11.A hyperbola is the set of all points in a plane, the difference of whose distances from twoxed points in the plane is a constant.

12. The equation of a hyperbola with foci on the x-axis is

Two asymptotes:

13.Latus rectum of the hyperbola : is

2 2

2 2 1 x ya b

+ = cos ; sin x a y bq q= =

2 22 2 1

x ya b

+ = 22b

a

2 22 2 1

x ya b

- = 2 2

2 2 0 x ya b

- = 2 2

2 2 1 x ya b

- = 22b

a


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Class 8

THREE DIMENSIONAL GEOMETRY

1. The coordinates of the points R which divides the line segment joining two pointsP(x , y , z ) and Q(x , y ,z ) internally and externally in the ratio m : n are given by1 1 1 2 2 2

and , respectively.

2. The coordinates of the centroid of the triangle, whose vertices are (x , y , z ), (x ,1 1 1 2y ,z ), (x , y ,z ) are2 2 3 3 3

3. If l, m ,n are the direction cosines and a, b, c are the direction ratios of a line then

4. If l , m ,n and l , m ,n are the direction cosine of two lines; and is the acute angle1 1 1 2 2 2between two lines; then

5. Equation of a line through a point (x , y , z ) and having direction cosine, m, n is1 1 1

6. Shortest distance between and is

7. The equation of plane through a point whose position vector is and perpendicular tothe vector is

8. Vector equation of a plane that passes through the line of intersection of planesand is where is any nonzero constant.

9. The distance of a point whose position vector is from the plane is

10. Skew line: Two straight line are said to be skew lines if they are parallel norintersecting.

Shortest distance:

11. Equation of a sphere: where centre is (a,b,c) and radiusRGeneral form:

Centre and

q

=0

2 1 2 1 2 1, ,

mx nx my ny mz nz

m n m n m n

+ + +

+ + +

2 1 2 1 2 1, ,

mx nx my ny mz nz

m n m n m n

- - -

+ - -

1 2 3 1 2 3 1 2 3, ,

3 3 3 x x x y y y z z z + + + + + +

2 2 2 2 2 2 2 2 2

, ,a b c

l m na b c a b c a b c

= = =+ + + + + +

1 2 1 2 1 2cos | |l l m m n nq = + +

1 1 1 x x y y z z

l m n- - -= =

1 1r a bl= +

r ur ur 2 2r a bm= +

r uur uur ( )( )1 2 2 1

1 2

.b b a a

b b

-

ur uur uur ur

ur uur

a

r

N uur ( ) .r a N -r r uur

1 1.r n d =

r ur 2 2.r n d =

r uur ( )1 2 1 2.r n n d d l l+ = +r ur uur l ar .r n d =

r r $.d a n-r

( )( )( )

2 1 1 2 2 1

1 2 2 1

x x m n m n

m n m n

- -

-

2 2 2 2( ) ( ) ( ) x a y b z c R- + - + - =

2 2 2( ) 2 2 2 0 x y z ux vy wz d + + + + + + =

, ,u v w

a a a- - -

2 2 22 2 2

u v w d a a a a

+ + -


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Class 9

DIFFERENTIAL CALCULUS

LIMIT

1. A function f has the limit L as X approaches a if the limit from the left exists and bothlimit are L that is, remember x a

Let and . If l and m are finite then:

(i)

(ii)

, provided(iii)

; where k is a constant. (iv)

, where k is a constant. (v)

If then(vi)

2. Limit in case of composite function: ; provided f is continuous at X

3.

(Where x is measured in radian)

4. ,

m 0

lim ( ) lim ( ) lim ( ) x a x a x a

f x f x f x

= = im x a

L it

im ( ) x a

L f x l

= im ( ) x a

L g x m

=

( )im ( ) ( ) x a

L f x g x l mf

=

im ( ). ( ) . x a

L f x g x l m

= ( )

im( ) x a

f x l L

g x m=

im ( ) im ( ) x a x a

L Kf x k L f x kl

= =

[ ]im ( ) im ( ) x a x a

L f x k L f x k

+ = +

( ) ( ), f x g x im ( ) im ( ) x a x a

L f x L g x

[ ] ( )im ( ) im ( ) x a x a L f g x f L g x = 1

10 0 0 0

sin sinim im im im 1sin sin x x x x

x x x x L L L L x x x x

--

= = = =

( ) 1/

0

1im 1 im 1

x x

x x L L x e

x + = + =

( ) /

0

1im 1 im 1 im 1

x axa x a

x x x

a L L x L e

x x + = + = + =


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Class 10

CONTINUITY OF FUNCTIONS

DIFFERENTIAL AND APPLICATION

1. All Polynomials, Trigonometrically, exponential and logarithmic function arecontinuous in their domain.

2. If f(x) is continuous and g(x) is discontinuous at x =0 then the product functionis not necessarily be discontinuous at x =a.

e.g. f(x) & g(x)

n3. For any positive integer n and any continuous function f, [if (x)] andcontinuous. When n is even, the inputs of f in are restricted to inputs x for which

4. If f(X) and (g(x) are continuous, then so are f(x) + g(x), f(x)- g(X), and f(x). g(x).5. If f(X) and g(x) are continuous, so is g(x) / f(x), so long as the inputs x do not yield

outputs f(x) =0.

1. Interpretation of the Derivative: If y = f(x) then,(a) m = f'(a) is the slope of the tangent line to y =f(x) at x =a and the equation of

the tangent line at x =a is given by y= f(a) +f'(a) (x-a).(b) f'(a) is the instantaneous rate of change of f(x) at x =a.(c) If f(x) is the position of an object at time x then f'(a) is the velocity of the object

at x =a.2. Basic Properties and Formulas : If f(x) and g(x) are differentiable functions (the

derivative exists), c and n are any real numbers1. (cf') = cf' (x)

2.

3.

4.

5.

6.

are

(fg)' = f'g +fg

f(x) 0

( ) ( ) ( ) x f x g xf = sin , 0

0, 0

x x

x

p

= ( )n f x

( )n f x

( ) ' '( ) '( ) f g f x g x =

'

2

' ' f f g fg g g

-=

1( )n nd

x nxdx

-=

( ( ( )) '( ( )) '( )d

f g x f g x g xdx

=


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Class 11

3. Increasing / Decreasing:(i) If f'(x) > 0 for all x in an interval I then f(x) is increasing on the interval I.(ii) If f'(x) 0 to the left of x =c and f'(x) < 0 to the right of x =c.2. A rel. max. of f(x) if f'(x) < 0 to the left of x =c and f'(x) > 0 to the right of x =c.3. Not a relative extrema of f(x) if f'(x) is the same sign on both sides of x=c.

nd2 Derivative Test: If x =c is a critical point of f(x) such that f' =0 then x =c.1. Is a relative maximum of f(x) if f''(c ) < 0.2. Is a relative maximum of f(x) if f''(c ) > 0.3. F(x) may have a relative maximum and minimum, or neither if f''(c ) =0.

6. Mean value theorem: If f(x) is continuous on the closed interval [a,b] anddifferentiable on the open interval (a,b) then there is a number a < c< b such that

7. Rolle's theorem: If a function f(x) is continuous on the closed interval [a,b] anddifferentiable in an interval (a,b) and also f(a) = f(b), then there exist at least onevalue of c of x in the interval (a,b) such that f' =0.

8. L' Hospital's Rule:

If f(a) =0 or

9. Length of Sub- tangent = ; sub- normal = .

Length of tangent =

Length of normal =

10. Any curve which cuts every member of a given family ofcurves at right angle, is called an orthogonal trajectory of the family.

,

Orthogonal trajectory:

=0 or

1( ) ( )'( )

nn f b f a f c nx

b a- -= = -

( ) '( ) ''( )lim

( ) '( ) ''( ) x a f x f a f a

x a af f f= =

'( )af

1 1 1( , )dx y x ydy

1 1 1( , )

dy y x y

dx

1 1

2

1

( , )

1 x y

dx ydy

+

1 1

2

1( , )

1 x y

dy y

dx

+


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Class 12

INTEGRAL CALCULUS

1. Fundamental theorem of calculus:Part I: If f(x) is continuous on [a,b] then is alsocontinuous on [a,b] and

Part II: f(x) is continuous on [a,b], F(x) is an anti- derivative of f(x) i.e.

then

2. Integration by Substitution: The substitution u = g(x) will convert

using du = g'(x)dx.

3. Integration by parts: and . Choose u and dv from integral

and compute du by differentiating u and compute v using

4. Integration by partial fraction: If integrating where the degree of P(x) is

smaller than the degree of Q(x). Factor denominator as completely as possible andfind the partial fraction decomposition of the rational expression. Integrate the partialfraction decomposition (P.F.D).

5. Some properties of definite integral:(a) If an interval [a, b] (a < b), the function f(x) and satisfy the

condition , then

(b) If m and M are the smallest and greatest values of a function f(x) on an interval[a,b] and then

(c) , where a < c < b

(d)

(e) , where a is the period of the function and

(f)

= - v =dv

a b

n I

udv uv vdu

( ) ( )

x

a

g x f t dt = '( ) ( ) ( )

x

a

d g x f t dt f xdx

= =

( ) ( ) F x f x dx=

( ) ( ) ( )b

a

f x dx F b F a= -

( )

( )

( ( )) '( ) ( ) g bb

a g a

f g x g x dx f u du= |b b

ba

a a

udv uv vdu= -

( )( )

P xdx

Q x

( ) xf ( ) ( ) f x xf

( ) ( )

a b

b a

f x dx x dxf

( ) ( ) ( )a

b

m b a f x dx M b a- -

( ) ( ) ( )b c b

a a c

f x dx f x dx f x= +

( )b

a

f x dx =

( )b

a

f a b x dx+ -

( )na

a

f x dx-

=

0

( )a

n f x dx

0

2 ( ) , if f(x) is even function, i.e. f(-x) =f(x)( )

0, if f(x) is odd function, i.e. f(-x) =f(x)

aa

a

f x dx f x dx

-

=


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Class 13

(g)

6. Leibnitz Rule:

7. If a series can be put in the form

or , then its limit as is

APPLICATION OF INTEGRALS

8. Net area: represents the area between the curve y =f(x), x axis and two co-

ordinates at x =a and x =b (b > a) with area above x axis positive and area below x axis negative.

9. Area between curve:

And

If the curves intersect then the area of each portion must be found individually.

10 The volumes of the solid generated by the revolution about the x-axis of the area

bounded by the curve y =f(x), the x- axis and the ordinates x =a, x =b is

n

2

00

2 ( ) , f(2a-x)=f(x)( )

0,if f(2a-x)=-f(x)

aa f x dx f x dx

=

( )

( )

( ) '( ) ( ( )) '( ) ( ( )) g x

f x

d f t dt g x F g x f x F f x

dx= -

1

0

1 r n

r

r f

n n

= -

=

1

1 r n

r

r f

n n

=

=

0

1

( ) f x dx

( )b

a

f x dx

( ) [upper function]-[lower function]dx

b

a

y f x A= =

( ) [right function]-[left function]dy

b

a x f y A= =

2dx

b

a

yp


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Class 14

PROBABILITY

1. Probability of an event: For a finite sample space with equally likely outcomesProbability of an event where n (A) = number of elements in the set A,

n(S) = number of elements in the set S.

2. If A and B are any two events, then P(A or B) = P(A) + P(B)- P(A and B)Equivalently,

3. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B)

4. If A is any event, then P (not A) = 1 P(A)

5. The conditional probability of an event E, given the occurrence of the event F is givenby

6. Theorem of total Probability: let {E , E , .,E } be an partition of a sample space1 2 nand suppose that each of E1, E2, ..,En has nonzero probability. Let a be any eventassociated with S then P(A) = P(E ) P(A |E )+P (E )P(A|E ) +..+P(E ) P(A|E )1 1 2 2 n n

7. Bayes' theorem: If E , E , .,E are events which constitute a partition of1 2 nsample space S, i.e. E , E , .,E are pair wise disjoint and and1 2 n

A be any event with non zero probability, then

8. Let X be a random variable whose possible values x , x , x ,.x occur with1 2 3 nprobabilities p , p , p ,..,p respectively. The mean of X, denoted by is the number1 2 3 n

The mean of a random variable X is also called the expectation of X, denoted byE(X).

9. Trails of a random experiment are called Bernoulli trails, if they satisfy the followingconditions:(a) There should be a finite number of trails.(b) The trails should be independent.(c) Each trail has exactly two outcomes: success or failure.(d) The probability of success remains the same in each trail.

For Binomial distributionn-xB (n, p), P (X = x) = n q , x = 0, 1, , n (q = 1- p)Cx

,( )( )( )

n A P A

n S

=

( ) ( ) ( ) ( ) p A B P A P B P A B= + -U I

( )( | ) , ( ) 0

( ) P E E

p E F P F P F

= I

1 2 .... n E E E S =U U U

1

( ) ( | ))

( | ) ( ) ( | )

i i

i n

j j j

P E P E A

P E A P E P A E =

=

1

n

i ii

x p=

m


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Class 15

MATRICES

1. Order of a matrix: A matrix which has in rows and n columns is called a matrix oforder .

J=1,2,nth thHere denotes the element of i row and j column.

2. Properties of scalar Multiplication : If A, B are Matrices of the same order and p, q areany two scalars then

i. p(A + B) = pA + pBii. (p + q) A = pA + qAiii. P(qA) = (pqA) = q(PA)iv. (-pA) = -(pA) = p(-A)v. tr(kA) = k tr(A)

3. Multiplication of matrices: If A and B be any two matrices, then their product AB willbe defined only when number of column in A is equal to the number of rows in B. If

and then their product , will be matrix of order ,

where

4. Positive Integral Powers of Matrix :For any positive integral m, n

m n m+ni. A A = Am n mn n mii. (A ) = A = (A )

n miii. I = I, I = Iiv. A =I where A is a square of order n.0 n

5. Transpose of a matrix: The matrix obtained from a given matrix A by changing itsrows into columns or columns into rows is called transpose of matrix A and is denoted

Tby A or A'. From the definition it is obvious that if order of A is.Properties of Transpose

T Ti. (A ) = A

ii. T T Tiii. (AB) = B AT Tiv. (kA) = k(A) K is scalar

Tv. I = ITvi. tr (A) = tr (A)

T T T T T Tvii.(A A A .A A ) = A A A A A1 2 3 n-1 n n n-1 3 2 1

m

a

m

n m

n

ij

where i=1,2,..mij m n A a =

ij m n

A a

=

ij n p B b

=

ij AB C c = =

1

( )n

ij ij ir rjr

AB c a b=

= =

( )T T T A B A B =


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Class 16

6. Inverse of matrix: If A and B are two matrices such that AB = I = BA then B is called-1 -1the inverse of A and it is denoted by A , thus A = B

To find inverse matrix of a given matrix A we use following formula

Thus exists

1. To find the value of a third order determinant

Let

Be a third order determinant. To find its value we expand it by any row or column asthe sum of three determinants of order 2. If we expand it by first row then

2. Minor: The Determinant that is left by cancelling the row and column intersecting ata particular element is called the minor of that element.

i+j3. Cofactor: The cofactor of an element a is denoted by F and is equal to (-1) Mij ij ijwhere M is a minor of element a . ij

4. Multiplication of two determinants: Multiplication of two third determinants isdefined as follows:

5. System of linear equation in three unknowns : Using Crammer's rule of determinantwe get i.e.

Case-I: If andI. If at least one ofii. is not zero then the system of equation is inconsistent i.e. has no solution.ii. If d = d =d = 0 or1 2 3iv. are all zero then the system is consistent and has infinitely many solutions.

| |

D0D ,D D3

D1,D2,D3

AB = I = BA

A 0

1 2,

DETERMINANT

1 .

| |adj A

A A

- =

1 A-

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

D =

22 23 21 23 21 22

11 12 1332 33 31 33 31 32

a a a a a aa a a

a a a a a aD = - +

1 1 1 1 1 1 1 1 1 2 1 3 1 1 1 2 1 3 1 1 1 2 1 3

2 2 2 2 2 2 2 1 2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

3 3 3 3 3 3 3 1 3 2 3 3 3 1 3 2 3 3 3 1 3 2 3 3

a b c l m n a l b l c l a m b m c m a n b n c n



+ + + + + + = + + + + + +

+ + + + + +

1 2 3

1 x y z = = =D D D D

31 2, , x y z

DD D= = =D D D



Class 17

VECTORS

1. Triangle law of vector addition of two vector Magnitude of

2. Components of vector: Consider a vectorThe quantities A and A are called x- and y- components of the vector is itselfx ynot a vector but is a vector and so is and

3. Scalar Product :

4. Vector product: =

5. Given vectors , , where are non coplanarvectors, will be coplanar if and only if

6. Scalar triple product :(a). If k, and then

(b). [a b c] = volume of the parallelepiped whose coterminous edges are formed by

(c). are coplanar if and only if [ ] = 0

(d). Four points A, B, C, D with position vectors respectively are coplanar ifand only if =0 i.e. if and only if

(e) Volume of a tetrahedron with three coterminous edges

(f) Volume of prism on a triangle base with three coterminous edges

7. Lagrange's identity:

8. Reciprocal system of vectors:If be any three non coplanar vector so that then the three vectors

defined by the equations are called the reciprocal

system of vectors to the given vector

:

that lies in xy plane as shown

(here is the angle between the vectors)

Rur 2 2 2 cos R A B AB q= + +

Aur

1 2 A A A= +

ur uur uur

$1 2, x y A A i A A j= =

uur uur$ $ x y A A i A j = +ur $

x A A=ur

x A i

$ $. cos y x A j A A q= sin y A A q=

. cos A B AB q=ur ur

q C A B= ur ur ur $cos AB nq

1 1 1 2 2 2, x a y b z c x a y b z c+ + + +r r r r r r

3 3 3 x a y b z c+ +r r r , ,a b c

r r r

1 1 1

2 2 2

3 3 3

0

x y z

x y z

x y z =

$1 2 3a a i a j a= + +

r ur uur uur$ $

1 2 3b b i b j b k = + +r ur uur ur

$ $1 2 3c c i c j c k = + +r ur uur ur

$

( ) 1 2 3

1 2 3

1 2 3

. [ ]

a a a

a b c abc b b b

c c c

= =r r r r rr

, ,a b cr r r

, , ,a b c d r r r ur

, ,a b cr r r , ,a b c

r r r

[ ] ABAC ADuuuruuuruuur [ ] 0b ac ad a- - - =

r rr rur r

1, , | [ ] |

6a b c abc=r r r r r r

1, , |[ ] |

2a b c abc=r r r r rr

. .

( ).( ) ( . )( . ) ( . )( . ). .

a c a d a b c d a c b d a d b c

b c b d = = -

r r r urr r r ur r r r ur r ur r r

r r r ur

' , ' '

[ ] [ ] [ ]

b c c a a ba b c

abc abc abc

= = =r r r r r r

r ur rr rr r r r r r r

abcr rr [ ] 0abc

r r r

' ' 'a b cr rur

, ,a b cr r r

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Class 18

9. Application of vector in geometry:

(a) Vector equation of a straight line passing through two points and is

(b) Vector equation of a plane passing through the points is

or

( c ) Vector equation of a plane passing through the point and perpendicular tois

(d) Perpendicular distance of a point P(r) from a line passing through and parallelto is given by

PM = =

(e) Perpendicular distance of a point P from a plane passing through the points

and is given by PM =

1. Mean deviation for ungrouped data2. Mean deviation for grouped data , where

3. Variance and standard deviation for ungrouped data ,

4. Variance and standard deviation of a distance frequency distribution

,

5. Variance and standard deviation of a continuous frequency distribution ,

6. Coefficient of variation (C.V) = ,

For series with equal means, the series with lesser standard deviation is moreconsistent or less scattered.

N =

STATISTICS

f i

ar b

r ( r a t b = + -r r r r

, ,a b cr r r

(1 )r s t a sb tc= - - + +r r r r .( ) [ ]r b c c a a b abc + + =r r r r r r r r r r

ar n

r

. .r n a n=r r r r

ar

br

| ( ) |

| |

r a b

b

- r r r

uur

1/222 ( ).( )

| |

r a br a

b

- - -

r r rr r

uur

( ).( )r a b c c a a b

b c c a a b

- + + + +

r r r r r r r r

r r r r r r br

cr

| |. ( ) i

x x M D x

n

-=

| |. ( ) i

x M M D M

n

-=

| |. ( ) i i

f x x M D x

N

-=

22 1 ( )i x xns = -

21 ( )i x x

ns = -

22 1 ( )i i f x xns = -

21 ( )i i f x x

ns = -

2 1 ( )i i f x xn

s = - 2

21 i ii i

f x f x

n N s

= -

100 xs 0 x


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Class 19

DIFFERENTIAL EQUATIONS

1. A differential equation of the form

functions of x only is called a first order linear differential equation.(a)Differential equation of the form

(b)

(c)

(d)Differential equation of homogeneous type: An equation in x and y is said to be

homogeneous if it can be put in the form where and are both

homogeneous functions of the same degree in x & y.

(e)Differential Equation reducible to homogeneous form: A differential equation of the form , where can be reduced tohomogeneous form by usingX = X + h, y = Y + k so that

2. Linear differential equations: =

is called the integrating factor for this equation.

, where P and Q are constants or

f x y g x y

( , ) ( , )

dy Py Q

dx+ =

( ) : ( )

dy f x y f x dx c

dx= = +

( ) ( ) ( )

( )dy dy

f x g y f x dx cdx g y

= = +

( ) :( )

dy dv f ax by c dx

dx a bf v= + + =

+

( , )( , )

dy f x ydx g x y=

( )dx dv

c x f v v

= +-

1 1 1

2 2 2

a x b x cdydx a x b x c

+ +=+ +

1 1

2 2

a ba b

dY dydX dx

=

;

dy Py Q

dx+ = pdx ye

pdxQe c +

pdxe


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Class

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