Revision Notes Maths XII

Relations and Functions

RelationLet A be a non-empty set and R A A . Then, R is called arelation on A.

If ( , )a b R , then we say that a is related to b and we writeaRb.

If ( , )a b R , then we write b R a/ .

Types of Relation

1. Empty Relation

A relation R on a set A is called empty relation, if noelement of A is related to any element of A i.e.,R A A= is the empty relation.

2. Universal Relation

A relation R on a set A is called universal relation, if eachelement of A is related to every element of A i.e.,R A A= .

Both the empty relation and the universal relation aresometimes called trivial relations.

3. Reflexive Relation

A relation R defined on set A is said to be reflexive, if( , ) ,x x R x A .

4. Symmetric Relation

A relation R defined on set A is said to be symmetric, if( , )x y R ( , )y x R , x y A,

5. Transitive Relation

A relation R defined on set A is said to be transitive, if( , )x y R and ( , )y z R ( , ) , , ,x z R x y z A

6. Equivalence Relation

A relation R on a set A is called an equivalence relation, ifit is reflexive, symmetric and transitive.

7. Equivalence Classes

Given, an arbitrary equivalence relation R on anarbitrary set X, R divides X into mutually disjoint subsetsAi called partitions satisfying A A Xi j = andA A i ji j i are called equivalenceclass and it is denoted by [ ]a .

Function

Every function is a relation but every relation is not afunction.

Types of Function

1. One-one (Injective) Function

A function f A B: is said to be one-one, if f x f x( ) ( )1 2= x x1 2= or x x1 2 f x f x( ) ( )1 2where, x x A1 2,

2. Many-one Function

A function f A B: is said to be many-one, if two ormore than two elements in A have the same image in B,

i e. ., if x x1 2 , then f x f x( ) ( )1 2= .

3. Onto (Surjective) Function

A function f A B: is said to be onto, if every element inB has its pre-image in A, i.e., if for each y B , there existsan element x A , such that f x y( ) = .

1

FAST TRACK

Revision NotesBRUSH UP your concept in Fast Track Mode

= , . The subset A

Let A and B be two non-empty sets. Then, a relation f fromA to B which associates to each element x A, a uniqueelement of f(x) B is called a function from A to B and wewrite f : A B. Here, A is called the domain of f ,i.e., dom(f ) = A, B is called the codomain of f .Also, { f(x) : x A} B is called the range of f .

2

4. Into Function

A function f A B: is said to be into, if atleast oneelement of B do not have a pre-image.

5. One-one and Onto (Bijective) Function

A function f X Y: is said to be one-one and onto, if f isboth one-one and onto.

6. Composite Function

Let f A B: and g B C: , thengof : A C , such that (gof ) ( ) { ( )},x g f x x A=

(i) In generally, gof fog .(ii) In generally, if gof is one-one, then f is one-one. And if

gof is onto, then g is onto.

7. Invertible Function

A function f X Y: is defined to be invertible, if thereexists a function g Y X: , such that gof = IX andfog = IY. The function g is called the inverse of f and it isdenoted by f 1. Thus, f is invertible, then f must beone-one and onto and vice-versa.

(i) If f X Y: , g Y Z: and h Z S: are functions, thenho(gof ) = (hog)of.

(ii) Let f X Y: and g Y Z: be two invertible functions.Then, gof is also invertible with ( )gof 1 = f og 1 1.

Binary OperationLet S be a non-empty set and be an operation on Ssuch that

a S, b S a b S , a b S,

Then, is called a binary operation on S.

GroupAn algebraic structure ( )S consisting of a non-void set Sand a binary operation defined on S is called a group, ifit satisfies following axioms.

(i) Closure property We say that on S satisfies theclosure property, if

a S, b S a b S , a b S, (ii) Commutative law Operation on S is said to be

commutative, if a b b a a b S = , , .(iii) Associative law Operation on S is said to be

associative, if ( ) ( );a b c a b c = a b c S, , .(iv) Identity law An element e S is said to be the

identity element of a binary operation on set S, ifa e e = a = a a S,

(v) Invertible law or inverse law

An element a S is said to be invertible, if thereexists an element b S , such that

a b b a e == , b S.Element b is called inverse of element a.

Zero is identity for the addition operation on R but it is notidentity for addition operation on N.

Fast Track Revision Notes Mathematics-XII

Inverse Trigonometric FunctionsTrigonometric functions are not one-one and onto on their natural domains and ranges, so their inverse does not exist in all

values but their inverse may exists in some interval of their domains and ranges. Thus, we can say that, inverse of

trigonometric functions are defined within restricted domains of corresponding trigonometric functions. Inverse of f is

denoted by f 1.

Let y f x x= =( ) sin be a function. Its inverse is x y= sin ,1 i.e., sin sinx x Inverse 1 .

sin (sin ) 1 1x x sin sin

1 1 1xx

(sin )sin

xx

1 1

Domain, Range and Principal Values of Inverse Trigonometric Functions

Function Domain Range Principal value branch

sin1 x [1, 1]

2 2

, 2 2

y , where y x= sin 1

cos1 x [1, 1] [ , ]0 0 y , where y x= cos 1

tan1 x R

2 2

, < < 2 2

y , where y x= tan 1

cosec1x ( , ] [ , ) 1 1

2 2

0, { } 2 2

y , y 0, where y x= cosec 1

sec 1 x ( , ] [ , ) 1 1 [ , ]02

0 y , y

2

, where y x= sec 1

cot1 x R ( , )0 0 <

3

T-ratios of Some Standard Angles

Angle ()0 0 = 30

6 = 45

4 = 60

3 = 90

2 =

Ratio

sin 0 12

1

2

3

2

1

cos 1 32

1

2

1

2

0

tan 0 13

1 3

Transformation of one Inverse Trigonometric Function to another Inverse Trigonometric Functions

Function Transformation of a Function

(i) sin1 x cos ( ) 1 21 xtan

1

21

x

xcot

121 x

xsec

1

2

1

1 x

cosec 1

1

x

(ii) cos 1 x sin ( ) 1 21 xtan

121 x

xcot

1

21

x

x

sec

1 1

x cosec1

1

1 2x

(iii) tan1 xsin

+

1

21

x

xcos

+

1

2

1

1 x

cot

1 1

xsec +1 21 x

cosec 1+

1 2x

x


Properties of Inverse Trigonometric Functions

Property I

(i) sin (sin ) =1 ,

2 2

,

(ii) cos (cos ) =1 , [ , ]0

(iii) tan (tan ) =1 , 2 2

,

(iv) cot (cot ) , ( , ) = 1 0

(v) cos (cos )ec ec =1 ,

2 2

0, { }

(vi) sec (sec ) =1 ,

[ , ]02

(vii) sin (sin ) =1 x x, x [ 1, 1]

(viii) cos (cos ) =1 x x, x [ 1, 1]

(ix) tan (tan ) =1 x x, x R

(x) cot (cot ) =1 x x, x R

(xi) cos (cos )ec ec 1 x = x, x , ( , ] [ )1 1

(xii) sec (sec ) =1 x x, x ( , ] [ )1 1,

Property II

(i) sin ( ) sin , = 1 1x x x , [ ]1 1

(ii) cos ( ) = 1 x cos1 x, x , [ ]1 1

(iii) tan ( ) tan = 1 1x x, x R

(iv) cot ( ) =1 x cot ,1 x x R

(v) cos ( )ec 1 x = cosec 1x, x 1

(vi) sec ( ) 1 x = sec 1 x, x 1

Property III

(i) sin ( / ) cos =1 11 x xec , x 1 or x 1

(ii) cos ( / ) sec =1 11 x x, x 1 or x 1

(iii) tan =1 1x

cot ,

cot ,

>

+

(ii) sin sin sin [ ], = 1 1 1 2 21 1x y x y y x

if 1 1x y, and x y2 2 1+ or

if xy > 0 and x y2 2 1+ >

(iii) cos cos cos [ ], + = 1 1 1 2 21 1x y xy x y

if 1 1x y, and x y+ 0

(iv) cos cos cos [ ], = + 1 1 1 2 21 1x y xy x y

if 1 1x y, and x y

(v) tan tan tan , + = +

1 1 11

1x yx y

xyxy

Property VI

(i) 2 2 11 1 2sin sin ( ) = x x x , 12

1

2x

(ii) 2 2 11 1 2cos cos ( ) = x x , 0 1 x

(iii) 22

1

1 12

tan sin , =+

x

x

x| |x 1 or 1 1x

(iv) 21

1

1 12

2tan cos , =

+

x

x

xx 0

(v) 22

1

1 12

tan tan , =

x

x

x < 1 1x

Some Useful Trigonometric Formulae

(i) sin( ) sin cos cos sinA B A B A B+ = +(ii) sin( ) sin cos cos sinA B A B A B =

(iii) cos( ) cos cos sin sinA B A B A B+ =

(iv) cos( ) cos cos sin sinA B A B A B = +

(v) tan( )tan tan

tan tanA B

A B

A B+ = +

1

(vi) tan ( )tan tan

tan tanA B

A B

A B =

+1

(vii) cot ( )cot cot

cot cotA B

A B

B A+ =

+1

(viii) cot ( )cot cot

cot cotA B

A B

B A = +

1

(ix) 2 sin cos sin ( ) sin ( )A B A B A B= + + (x) 2 cos cos sin ( ) sin ( )A B A B A B= +

(xi) 2 cos cos cos ( ) cos ( )A B A B A B= + +

(xii) 2 sin sin cos ( ) cos ( )A B A B A B= +

(xiii) sin sin sin cosC DC D C D+ = +

2 2 2

(xiv) sin sin cos sinC DC D C D = +

2 2 2

(xv) cos cos cos cosC DC D C D+ = +

2 2 2

(xvi) cos cos sin sinC DC D C D = +

2 2 2

= +

2 2 2

sin sinC D D C

(xvii) sin sin cos2 2x x x=(xviii) cos cos sin2 2 2x x x= = = 1 2 2 12 2sin cosx x

(xix) tantan

tan2

2

1 2x

x

x=

(xx) 1 2 2 2+ =cos cosx x; 1 2 2 2 =cos sinx x

(xxi) sintan

tan2

1 2x

x

x= 2

+; cos

tan

tan2

1

1

2

2x

x

x=

+

(xxii) sin sin sin3 3 4 3x x x= ;

cos cos cos3 4 33x x x=

(xxiii) tantan tan

tan3

3

1 3

3

2x

x x

x=

4


Matrices

MatrixA matrix is an ordered rectangular array of numbers orfunctions. The number or functions are called the elementsor the entries of the matrix.

Order of MatrixA matrix of order m n is of the form

A

a a a a

a a a a

a a a a

n

n

m m m mn

=

11 12 13 1

21 22 23 2

1 2 3

K

K

K K K K K

K

Its element in the ith row and jth column is aij.

If m n= , then matrix is a square matrix.A matrix is denoted by the symbol [ ].

i.e., [A] = [aij]m n

We shall consider only those matrices whose elements arereal number or function taking real values.

Types of Matrices(i) Row matrix A matrix having only one row and many

columns, is called a row matrix.

(ii) Column matrix A matrix having only one column andmany rows, is called a column matrix.

5

(iii) Zero matrix or null matrix If all the elements of a matrixare zero, then it is called a zero or null matrix. It is denotedby symbol O.

(iv) Square matrix A matrix in which number of rows andnumber of columns are equal, is called a square matrix.

(v) Diagonal matrix A square matrix is said to be a diagonalmatrix, if all the elements lying outside the diagonalelements are zero.

(vi) Scalar matrix A diagonal matrix in which all principaldiagonal elements are equal, is called a scalar matrix.

(vii) Unit matrix or identity matrix A square matrix having 1(one) on its principal diagonal and 0 (zero) elsewhere, iscalled an identity matrix. It is denoted by symbol I.

(viii) Equality of Matrix Two matrices are said to be equal, iftheir order is same and their corresponding elements arealso equal.

Addition of MatricesLet A and B be two matrices each of order m n . Then, thesum of matrices A B+ is defined, if matrices A and B are ofsame order.

If A aij m n= [ ] , B aij m n= [ ]Then, A B a bij ij m n+ = + [ ]

Properties of Addition of MatricesIf A, B and C are three matrices of same order m n , then

(i) Commutative law A B B A+ = +(ii) Associative law ( ) ( )A B C A B C+ + = + +(iii) Existence of additive identity A zero matrix (O) of order

m n (same as of A), is additive identity, ifA O A O A+ = = +

(iv) Existence of additive inverse If A is a square matrix,then the matrix ( )A , is additive inverse, ifA A O A A+ = = +( ) ( )

If A and B are not of same order, then A B+ is not defined.

Difference of MatricesIf A aij= [ ], B bij= [ ] are two matrices of the same order m n ,then difference, A B is defined as a matrix D d ij= [ ], whered a b i jij ij ij= , , .

Multiplication of a Matrix by a ScalarLet A aij m n= [ ] be a matrix and k be any scalar. Then, thematrix obtained by multiplying each element of A by k is calledthe scalar multiple of A by k, i.e., kA kaij m n= [ ] .

(i) k A B kA kB( )+ = + (ii) ( )k l A kA lA+ = +

Multiplication of MatricesLet A aij m n= [ ] and B bij n p= [ ] be two matrices such that thenumber of columns of A is equal to the number of rows of B,then multiplication of A and B is denoted by AB, is given by

c a bij ik kjk

n

==

1

where, c ij is the element of matrix C and C = AB.

Generally, it is not commutative AB BA .

Properties of Multiplication of Matrices

(i) Associative law ( ) ( )AB C A BC=(ii) Existence of multiplicative identity

A I A I A = = , I is called multiplicative identity.(iii) Distributive law A B C AB AC( )+ = +

Transpose of a MatrixThe matrix obtained by interchanging the rows andcolumns of a given matrix A, is callled transpose of amatrix. It is denoted by A or AT .

Properties of Transpose of Matrices

(i) ( )A B A B+ = + (ii) ( )kA kA = (iii) ( )AB B A = (iv) ( )A A =

Symmetric and Skew-Symmetric MatrixA square matirx A is callled symmetric, if A A = .A square matrix A is called skew-symmetric, if A A = .

Properties of Symmetric andSkew-symmetric Matrix

(i) For any square matrix A with real number entries,A A+ is a symmetric matrix and A A is askew-symmetric matrix.

(ii) Any square matrix can be expressed as the sum ofsymmetric and a skew-symmetric matrix.

i.e., A A A A A= + + 12

1

2( ) ( )

(iii) The principal diagonal elements of askew-symmetric matrix are always zero.

Elementary Operations of a MatrixThere are six operations (transformations) on a matrix,three of which are due to rows and three due tocolumns, which are known as elementary operations ortransformations.

(i) The interchange of any two rows or twocolumns Symbolically, the interchange of ith andjth rows is denoted by R Ri j and interchange ofith and jth columns is denoted by C Ci j .

(ii) The multiplication of the elements of any rowor column by a non-zero number Symbolically,the multiplication of each element of the ith rowby k, where k 0 is denoted by R kRi i . Thecorresponding column operation is denoted byC kCi i .

(iii) The addition to the elements of any row orcolumn, the corresponding elements of anyother row or column multiplied by anynon-zero number Symbolically, the addition tothe elements of ith row, the correspondingelements of jth row multiplied by k is denoted byR R kRi i j + .The corresponding column operation is denotedby C C kCi i j + .


6

Invertible MatricesIf A is a square matrix of order m and there exists anothersquare matrix B of the same order m, such that AB BA I= = ,then B is called the inverse matrix of A and it is denoted byA1.

A rectangular matrix does not posses inverse matrix.

If B is an inverse of A, then A is also the inverse of B.

Properties of Invertible MatricesLet A and B be two non-zero matrices of same order.

(i) Uniqueness of inverse If inverse of a square matrixexists, then it is unique.

(ii) AA A A I = =1 1

(iii) ( )AB B A =1 1 1

(iv) ( )A A =1 1

(v) ( ) ( )A A = 1 1 or ( ) ( )A AT T =1 1

where, A or AT is transpose of a matrix A.

Determinants

DeterminantEvery square matrix A is associated with a number, called itsdeterminant and it is denoted by det(A) or A.

Expansion of Determinant of Order (22)a a

a aa a a a

11 12

21 2211 22 12 21=

Expansion of Determinant of Order ( )3 3a a a

a a a

a a a

a a a a a

11 12 13

21 22 23

31 32 33

11 22 33 32 23= ( )

a a a a a12 21 33 31 23( ) + a a a a a13 21 32 31 22( )

Similarly, we can expand the above determinant correspondingto any row or column.

Properties of Determinants(i) If the rows and columns of a determinant are

interchanged, then the value of the determinant does notchange.

(ii) If any two rows (columns) of a determinant areinterchanged, then sign of determinant changes.

(iii) If any two rows (columns) of a determinant are identical,then the value of the determinant is zero.

(iv) If each element of a row (column) is multiplied by anon-zero number k, then the value of the determinant ismultiplied by k. By this property, we can take out anycommon factor from any one row or any one column of adeterminant.

(v) If A is a n n matrix, then| | | |kA k An= .(vi) If each element of any row (column) of a determinant is

added k times the corresponding element of another row(column), then the value of the determinant remainsunchanged.

(vii) If some or all elements of a row (column) of a determinantare expressed as sum of two (more) terms, then thedeterminant can be expressed as sum of two (more)determinants.

Minors and CofactorsMinors Minor of an element aij of a matrix is thedeterminant obtained by deleting i th row and jthcolumn. It is denoted by Mij.

If A

a a a

a a a

a a a

=11 12 13

21 22 23

31 32 33

, then

Minors of A are

Ma a

a a11

22 23

32 33

= ,

Ma a

a a12

21 23

31 33

= ,

Ma a

a a13

21 22

31 32

= , etc.

The minor of an element of a determinant of ordern n( ) 2 is a determinant of order n 1.

Cofactor If Mij is the minor of an element aij, then thecofactor of aij is denoted by Cij or Aij and defined asfollows

A C Mij iji j

ij= = +( )1

Cofactors of A are C Miji j

ij= +( )1

where, i = 1 2 3, , and j = 1 2 3, , .

If elements of a row (column) are multiplied withcofactors of any other row (column), then their sum iszero.

Area of TriangleLet A x y B x y( , ), ( , )1 1 2 2 and C x y( , )3 3 be the vertices of aABC. Then, its area is given by

= 12

1

1

1

1 1

2 2

3 3

x y

x y

x y

= + + 12

1 2 3 2 3 1 3 1 2x y y x y y x y y( ) ( ) ( )


(i) Since, area is positive quantity. So, we always take theabsolute value of the determinant.

(ii) If area is given, use both positive and negative valuesof the determinant for calculation.

Condition of CollinearityThree points A x y B x y( , ), ( , )1 1 2 2 and C x y( , )3 3 arecollinear, when = 0.

i e. .,

x y

x y

x y

1 1

2 2

3 3

1

1

1

0=

Adjoint of a MatrixThe adjoint of a square matrix A is defined as the transposeof the matrix formed by cofactors.

Let A aij= [ ] be a square matrix of order n, then adjoint of A,i e. ., adj A CT= , where C Cij= [ ] is the cofactor matrix of A.

Properties of Adjoint of Square MatrixIf A and B are square matrices of order n, then

(i) A A A( ) | |adj = I A An = ( )adj(ii) adj adj( ) ( )A AT T=

(iii) | | | |adj A A n= 1, if| |A 0

(iv) | [ ( )]| | |( )adj adj A A n= 12

' if| |A 0

(v) ( ) ( ) ( )adj adj adjAB B A=

Inverse of a MatrixA square matrix A has inverse, if and only if A is anon-singular (| | )A 0 matrix. The inverse of A is denotedby A1 i.e.,

AA

A =1 1| |

( )adj ,| |A 0

Properties of Inverse of a Square Matrix (A)

(i) ( )A A =1 1 (ii) ( )AB B A =1 1 1

(iii) ( ) ( )A AT T =1 1 (iv) ( )kA kA =1 1

(v) adj adj( ) ( )A A =1 1

A is said to be singular, if| |A = 0.

System of Linear EquationsLet the system of equations be

a x b y c z d1 1 1 1+ + = ,a x b y c z d2 2 2 2+ + =

and a x b y c z d3 3 3 3+ + =Then, this system of equations can be written as AX B=

where, A

a b c

a b c

a b c

=

1 1 1

2 2 2

3 3 3

, X

x

y

z

=

and B

d

d

d

=

1

2

3

A system of equations is consistent or inconsistentaccording as its solution exists or not.

(i) For a square matrix A in matrix equation AX B=(a) | |A 0, then system of equations is consistent and

has unique solution.

(b) | |A = 0 and (adj A) B O , then there exists nosolution, i e. ., inconsistent.

(c) | |A = 0 and ( )adj A B O= , then system of equationsis consistent and has an infinite number ofsolutions.

(ii) When B =

0

0

0

, in such cases, we have

(a) | |A 0 System has only trivial solutioni.e., x = 0, y = 0 and z = 0

(b) | |A = 0 System has infinitely many solutions.

7


Continuity and Differentiability

Continuous FunctionA real function f is said to be continuous, if it is continuousat every point in the domain f.

Continuity at a PointSuppose f is a real valued function on a subset of the realnumbers and let c be a point in the domain of f. Then, f iscontinuous at c, if lim ( ) ( )

x cf x f c

= .

i.e., if f c f x f xx c x c

( ) lim ( ) lim ( )= = +

, then f x( ) is continuous at

x c= . Otherwise, f x( ) is discontinuous at x c= .

Graphically, a function f x( ) is said to be continuous at a

point, if the graph of the function has no break point.

Some Basic Continuous Function(i) Every constant function is continuous.

(ii) Every identity function is continuous.

(iii) Every polynomial function is continuous.

Algebra of Continuous FunctionIf f and g are two continuous functions in domain D, then

(i) ( )f g+ is continuous.(ii) ( )f g is continuous.(iii) cf is continuous.

(iv) fg is continuous.

(v)f

gis continuous in domain except at the points,

where, g x( ) = 0.

(vi) If f is continuous, then f is also continuous.

(vii) Every rational function is continuous.

(viii) Suppose f and g are real valued functions such that(fog) is defined at c, if g is continuous at c and f iscontinuous at g c( ), then (fog) is continuous at c.

Differentiability or DerivabilityA function f is said to be derivable or differentiable at x c= ,if its left hand and right hand derivatives at c exist and areequal.

(i) Right Hand Derivative Rf af a h f a

hh =

+

( ) lim( ) ( )

0

(ii) Left Hand Derivative Lf af a h f a

hh =

( ) lim( ) ( )

0

f x( ) is differentiable at x a= , if Rf a L f a = ( ) ( ).Otherwise, f x( ) is not differentiable at x a= .

(i) Graphically, a function is not differentiable at a cornerpoint of a curve.

(ii) Every differentiable function is continuous. But acontinuous function need not be differentiable.

Differentiation

The process of finding derivative is called differentiation.

Derivatives of Standard Functions

(i)d

dxx nxn n( ) = 1

(ii)d

dx(constant) = 0

(iii)d

dxcx cn xn n( ) = 1

(iv)d

dxx x(sin ) cos=

(v)d

dxx x(cos ) sin=

(vi)d

dxx x(tan ) sec= 2

(vii)d

dxx x x(cos ) cos cotec ec=

(viii)d

dxx x x(sec ) sec tan=

(ix)d

dxx x(cot ) cos= ec 2

(x)d

dxe ex x( ) =

(xi)d

dxa a a ax x e( ) log ,= > 0

(xii)d

dxx

xe(log ) =

1, x > 0

(xiii)d

dxx

x aa

e

(log )log

= 1 , a > 0, a 1

(xiv)d

dxx

x(sin ) =

1

2

1

1, < 1

Derivative of Composite Function byChain RuleLet f be a real valued function which is a composite of twofunctions u and v, i.e., f vou= . Suppose t u x= ( ) and if bothdt

dxand

dv

dtexist, we have

df

dx

dv

dt

dt

dx= .

Derivatives of Two or More Functions

(i)d

dxu v

du

dx

dv

dx( ) =

(ii)d

dxu v w

du

dx

dv

dx

dw

dx( ) = K K

(iii)d

dxu v u

d

dxv v

d

dxu( ) ( ) ( ) = + [product rule]

(iv)d

dx

u

v

vd

dxu u

d

dxv

v

=

( ) ( )2

[quotient rule]

(v) y f x g x= [ ( )] ( )

dydx

= +

[ ( )]( )

( )( ) log ( ) ( )( )f x

g x

f xf x f x g xg x

(vi) If y f g x= [ ( )], then dydx

f g x g x= [ ( )] ( )

Useful Logarithmic Formulae(i) log log loga a amn m n= +

(ii) log log loga a am

nm n=

(iii) log logan

am n m=

(iv) loga a = 1; a > 0 and a 1(v) log log loga b am m b= + ; a b a b> > 0 0 1 1, , , and

m > 0(vi) log loga mm a =1 ; a m a> > 0 0 1, , and m 1

(vii) log logb bm

am

a= 1 ; a b> >0 0, and b 1

(viii) loglog

logb

m

m

aa

b= ; a b b m> > >0 0 1 0, , , and m 1

(ix) a mamlog = ; a m a> > 0 0 1, ,

8


> > >

For m n a

a

0 0 0

1

, ,

and

Rolles TheoremIf a function y f x= ( ) is defined in [ , ]a b and

(i) f x( ) is continuous in [ , ]a b .

(ii) f x( ) is differentiable in ( , )a b and

(iii) f a f b( ) ( )=Then, there will be atleast one value of c a b ( , ) suchthat f c =( ) 0.

Lagranges Mean Value TheoremIf a function f x( ) is said to be defined on [ , ]a b and

(i) continuous in [ , ]a b and

(ii) differentiable in ( , )a b , then there will be atleast one

value of c a b ( , ) such that f c f b f ab a

=

( )( ) ( )

.

Lagranges mean value theorem is valid irrespective ofwhether f a f b( ) ( )= or f a f b( ) ( ) .

9


Application of Derivatives

Rate of Change of QuantitiesIf y f x= ( ) is a function, where y is dependent variable and xis independent variable. Then,

dy

dx[or f x ( )] represents the

rate of change of y w.r.t. x anddy

dx x x

= 0

[or f x ( )0 ]

represents the rate of change of y w.r.t. x at x x= 0.

(i) Average rate of change of y w.r.t. xy

x=

(ii) Instantaneous rate of change of y w.r.t. xdy

dx=

(iii) Related rate of change = =dydx

dy dt

dx dt

/

/, if

dx

dt 0

Here,dy

dxis positive, if y increases as x increases and

dy

dxis

negative, if y decreases as x increases.

Marginal CostMarginal cost represents the instantaneous rate of changeof the total cost at any level of output. If C x( ) represents thecost function for x units produced, then marginal cost,

denoted by MC, is given by MC = ddx

C x{ ( )}.

Marginal Revenue

Marginal revenue represents the rate of change of total

revenue with respect to the number of items sold at an

instant. If R x( ) represents the revenue function for x units

sold, then marginal revenue, denoted by MR, is given by

MR = ddx

R x{ ( )}.

Total cost = Fixed cost + Variable costi.e., C x f c v x( ) ( ) ( )= +

Some Useful Results(i) Area of a square = x2 and perimeter = 4x

where, x is the side of the square.

(ii) Area of a rectangle = xy and perimeter = +2( )x ywhere, x and y are length and breadth of rectangle.

(iii) Area of a trapezium = 12

(Sum of parallel sides)

Perpendicular distance between them

(iv) Area of a circle = r 2

and circumference of a circle

= 2r, where r is the radius of circle.

(v) Volume of sphere = 43

3r and surface area = 4 2r

where, r is the radius of sphere.

(vi) Total surface area of a right circular cylinder

= +2 2 2 rh rCurved surface area of right circular cylinder

= 2rhand volume = r h2

where, r is the radius and h is the height of the cylinder.

(vii) Volume of a right circular cone = 13

2r h,

Curved surface area = rland total surface area = + r rl2

where, r is the radius, h is the height and l is the slantheight of the cone.

(viii) Volume of a parallelopiped = xyzand surface area = + +2( )xy yz zxwhere, x,y and z are the dimensions of parallelopiped.

(ix) Volume of a cube = x3 and surface area = 6 2xwhere, x is the side of the cube.

(x) Area of an equilateral triangle = 34

(Side)2.

Increasing and DecreasingFunctions

(i) Increasing functions Let I be an open intervalcontained in the domain of a real valued function f .Then, f is said to be

(a) increasing on I, if x x1 2 0 such that f is differentiable in theinterval ( ,c h c h + ).


11

First Derivative Test(i) If f x >( ) 0 to closely left of c and f x

12

Partial Fractions

Form of theRational Functions

Form of thePartial Fractions

1. px q

x a x b

( ) ( )

, a b Ax a

B

x b+

2. px q

x a

( )2

A

x a

B

x a( ) ( )+

2

3. px qx r

x a x b

2

2

( ) ( )

A

x a

B

x b

C

x b( ) ( ) ( )+

+

2

4. px qx r

x a x bx c

2

2

( ) ( )

A

x a

Bx C

x bx c( )+ +

2, where

x bx c2 cannot be factorisedfurther.

Integration by PartsLet u and v be two differentiable functions of a singlevariable x, then the integral of the product of twofunctions is

= uv dx u v dx

d

dxu v dx dx

I II

If two functions are of different types, then consider the1st function (i.e., u) which comes first in word ILATE,where

I : Inverse trigonometric function e.g., sin1 x

L : Logarithmic function e.g., log x

A : Algebraic function e.g., 1, x, x2

T : Trigonometic function e.g., sin , cosx x

E : Exponential function e.g., e x

Some Important Integrals

(i)dx

x a a

x a

x aC

( )log

2 2

1

2=

++

(ii)dx

a x a

a x

a xC

( )log

2 2

1

2= +

+

(iii)dx

x ax x a C

( )log| |

2 2

2 2

= + +

(iv)dx

x ax x a C

2 2

2 2

+= + + + log

(v) a x dxx

a xa x

aC2 2 2 2

21

2 2 = + + sin

(vi) x a dxx

x a2 2 2 2

2 =

+ +a x x a C2

2 2

2log

(vii) x a dxx

x a2 2 2 2

2+ = +

+ + + +a x x a C2

2 2

2log

Some Standard Substitutions

Expression Substitution

1. a x2 2 x a= sin or a cos

2. a x2 2+ x a= tan or a cot

3. x a2 2 x a= sec or a cosec

4. a x

a x

+

ora x

a x

+

x a= cos 2

5. x

x

or ( ) ( )x x x = + cos sin2 2

Integration of Irrational and TrigonometricFunctions

Integral Substitution

(i) x

(x a ) (x b )dx

2

2 2 2 2+ +Let x y2 = and proceed forpartial fraction of

y

y a y b( ) ( )+ +2 2

(ii)dx

a b x cos Put costan

tan

x

x

x=

+

12

12

2

2

, then

put tanx

t2

=

(iii)dx

a b x sin Put sintan

tan

x

x

x=

+

22

12

2

, then

put tanx

t2

=

(iv)dx

a x b xsin cos+Put a r= cos and b r= sin

(v)a x b x c

p x q xdx

sin cos

sin cos

+ ++

Put a x b xsin cos+ + c= +A d

dxp x q x( sin cos )

+ +B p x q x( sin cos )

(vi)a x b x

p x q x rdx

sin cos

sin cos

++ +

Put a x b xsin cos+= + +A d

dxp x q x r( sin cos )

+ + +B p x q x r( sin cos )

(vii)dx

ax b px q( )+ +Put px q t+ =

(viii)dx

ax bx c px q( )2 + + +Put px q t+ =

(ix) + + +

dx

px q ax bx c( )( )2Put px q

t+ = 1

(x) + +

dx

px q ax b( )2 2Put x

t= 1


Definite Integral

An integral is of the form of f x dxa

b( ) is known as definite

integral and is given by

f x dx g b g aa

b( ) ( ) ( ) =

where, a and b are lower and upper limits of an integral.

Definite Integral as a Limit of Sum

Let us define a continuous function f x( ) in [ , ]a b divide

interval into n equal sub-intervals, each of length h, so that

hb a

n=

Then,a

b

hf x dx h f a f a h = + +( ) lim [ ( ) ( )0 + +f a h( )2

+ + + f a n h{ ( ) }]1where, nh b a=

Some Standard Formulae

1. n n= + + + +1 2 3 = +n n( )12

2. n n2 2 2 2 21 2 3= + + + + = + +n n n( )( )1 2 16

3. n n3 3 3 3 31 2 3= + + + + = +n n2 21

4

( )

Fundamental Theorem of CalculusTheorem 1 Let f be a continuous function defined on theclosed interval [ , ]a b and A x( ) be the area of function.

[ . ., ( ) ( ) ].i e A x f x dxa

x= Then, =A x f x( ) ( ), for all x a b [ , ].

Theorem 2 Let f be a continuous function defined on theclosed interval [ , ]a b and F be an anti-derivative of f.

Then, f x dx F x F b F aab

a

b( ) [ ( )] ( ) ( )= =

Properties of Definite Integral

(i)a

b

a

bf x dx f t dt =( ) ( )

(ii)a

b

b

af x dx f x dx = ( ) ( )

(iii)a

af x dx =( ) 0

(iv)a

b

a

c

c

bf x dx f x dx f x dx = +( ) ( ) ( ) , where a c b< < .

(v)a

b

a

bf x dx f a b x dx = + ( ) ( )

(vi)0 0

a af x dx f a x dx = ( ) ( )

(vii)0

2

0 02

a a af x dx f x dx f a x dx = + ( ) ( ) ( )

(viii)0

2

02 2

0 2

aa

f x dxf x dx f a x f x

f a = =( ) ( ) , ( ) ( ),

, (

if even

if =

x f x) ( ), odd

(ix) aa

f x dx( ) = = =

2 if even

if odd0

0

af x dx f x f x

f x f x

( ) , ( ) ( ),

, ( ) ( ),

13


Application of Integrals1. The area enclosed by the curve y f x= ( ), the X-axis and

the ordinates at x a= and x b= , is given bya

by dx | | .

2. The area enclosed by the curve x f y= ( ), the Y-axis and

the abscissae at y c= and y d= , is given byc

dx dy | | .

.

3. If the curve y f x= ( ) lies below the X-axis, then areabounded by the curve y f x= ( ), X-axis and the ordinates

at x a= and x b= , is given bya

bydx .

4. Generally, it may happen that some position of the curveis above X-axis and some is below the X-axis which isshown in the figure. The area A bounded by the curvey f x= ( ), X-axis and the ordinates at x a= and x b= , isgiven by A A A= +| |2 1.

Y

Y

XX L M

x = bx = a

O

y =f x( ) B

Ay

dx

Y

X

x = f y( )

y = c

O

D

B Cy = d

xdy

A

Y

X

OX

Y

y f x= ( )

x = bx = a

Y

X

X

Y

x = bx = a

A2

A1

O

Y

X

14

5. The area enclosed between two curves, y f x1 = ( ) andy g x2 = ( ) and the ordinates at x a= and x b= , is givenby

a

by y dx 2 1 .

6. The area enclosed between two curves, x f y1 = ( ) andx g y2 = ( )and the abscissae at y c= and y d= , is given

byc

dx x dy 2 1 .

7. Area bounded by the two curves, y = f x( ) and y g x= ( )between the ordinates at x a= and x b= , is given by

a

c

c

bf x dx g x dx +( ) ( ) .

Curve SketchingThe points given below are of great help in curve sketching.

(i) If the equation of the curve contains only even powersof x, then it is symmetrical about Y-axis.

(ii) If the equation of the curve contains only even powersof y, then it is symmetrical about X-axis.

(iii) If the equation of the curve remains unchanged when xand y are interchanged, then it is symmetrical aboutthe line y x= .

(iv) If the equation of the curve remains unchangedwhen x and y are replaced by x and yrespectively, then the curve is symmetrical inopposite quadrants.


O

y = g x2 ( )

y = f x1 ( )

x = a x = b

Y

X

Y

X

Y

XO

yg x= (

)

Y

Xx b=x a=

yf x

=()

x c=

Differential Equations

Differential EquationAn equation containing an independent variable, dependentvariable and derivative of dependent variable with respect toindependent variable, is called a differential equation. Thederivates are denoted by the symbols

dy

dx

d y

dx

d y

dx

n

n, ,....,

2

2or y y y n , , ..., ... or y y yn1 2, ,...,

Order of a Differential EquationThe order of the highest derivative occurring in the differentialequation, is called order.

Degree of a Differential EquationThe power of the highest order derivative in the differentialequation, is called degree.

Solution of a Differential EquationA relation between the dependent and independent variableswhich, when substituted in the differential equation reduces itto an identity, is called a solution.

General Solution of aDifferential EquationA solution of a differential equation which contains as manyarbitrary constants as the order of the differential equation, iscalled the general solution or primitive solution of thedifferential equation.

Particular Solution of a DifferentialEquationThe solution obtained from the general solution givenparticular values to the arbitrary constants, is called aparticular solution of the differential equation.

Formation of a Differential EquationAn equation with independent, dependent variablesinvolving some arbitrary constants is given, then adifferential equation is obtained as follows

(i) Differentiate the given equation with respect to theindependent variable (say x) as many times as thenumber of arbitrary constants in it.

(ii) Eliminate the arbitrary constants.

(iii) The obtained equation is the required differentialequation.

The order of a differential equation representing a familyof curves is same as the number of arbitrary constantspresent in the equations corresponding to the family ofcurves.

Equation in Variable Separable FormIf the equation can be reduced into the form

f x( )dx + g y dy( ) = 0,we say that the variables have been separated.

Then, f x dx g y dy C( ) ( )+ =

O

x=

gy

2(

)

x=

fy

1(

)

y = c

y = d

Y

X

Y

X

15

Homogeneous Differential Equation

A differential equation of the formdy

dx

f x y

g x y= ( , )

( , )

where, f x y( , ) and g x y( , ) are homogeneousfunctions of x and y are of the same order.

If the homogeneous differential equation is in the formdx

dyF x y= ( , ), where F x y( , ) is a homogeneous function

of degree zero, then we make substitutionx

yv= , i.e.,

x vy= and we proceed further to find the generalsolution.

Linear Differential EquationA first order and first degree differential equation inwhich the degree of dependent variable and its

derivative is one and they do not get multiplied together, iscalled a linear differential equation.

There are two types of linear differential equations.

Type I Formdy

dxPy Q+ = , where P and Q are constants or

functions of x.

We find integrating factor (IF) = e P dx. Now, solution is y (IF) = [ ( )]Q dxIF + C.

Type II Formdx

dyPx Q+ = , where P and Q are constants or

functions of y, then IF = e P dy .

Its solution is x (IF) = + [ ( )]Q dy CIF .


Vector Algebra

VectorA quantity that has magnitude as well as direction, is calleda vector.

Since, the length is never negative, so the notation| |a

< 0has no meaning.

ScalarA quantity that has magnitude only, is called scalar.

Magnitude of a Vector

If a i j k

= + +a b c$ $ $, then | |a

= + +a b c2 2 2

Free VectorIf the initial point of a vector is not specified, then it is calleda free vector.

Type of Vectors

Zero or Null VectorA vector whose magnitude is zero i e. ., whose initial andfinal points coincide, is called a null vector or zero vector.

Unit VectorA vector whose magnitude is one unit. The unit vector in

the direction of a

is represented by $a. The unit vectors

along X-axis, Y-axis and Z-axis are represented by $ $ and $i j k, ,

respectively.

Coinitial VectorsTwo or more vectors having the same initial point are calledcoinitial vectors.

Collinear VectorsThe vectors which have same support are called collinearvectors.

Like VectorsThe vectors which have same direction are called likevectors.

Unlike VectorsThe vectors which have opposite directions are calledunlike vectors.

Equal VectorsTwo vectors are equal, if they have same magnitude anddirection.

Negative of a VectorA vector whose magnitude is same as that of given vectorbut the direction is opposite is called negative vector of the

given vector. e.g., Let AB

be a vector, then AB or BA

is a

negative vector.

Coplanar VectorsA system of vectors is said to be coplanar, if their supportsare parallel to same plane.

Addition of Vectors

If OA

=a

and AB b

= ,

then a b OA AB OB

+ = + =

O A

B

b

a

16

Triangle Law of Vector AdditionIf two vectors are represented along two sides of a triangletaken in order, then their resultant is represented by the thirdside taken in opposite direction i.e., from ABC, by trianglelaw of vector addition, we have

BC CA BA

+ =

Parallelogram Law of Vector AdditionIf two vectors are represented along the two adjacent sides ofa parallelogram, then their resultant is represented by thediagonal of the sides. If the sides OA and OC of parallelogram

OABC represents OA

and OC

respectively, then we get

OA OC OB

+ = or OA AB OB

+ = [QAB OC=

]

Thus, we may say that the two laws of vectors addition areequivalent to each other.

Properties of Vector Addition

(a) For any two vectors a

and b

,

a b b a

+ = + [commutative law]

(b) For any three vectors a b

, and c

,

a (b c ) (a b) c

+ + = + + [associative law]

(c) For any vector a

, we have a 0 0 a a

+ = + = .

The zero vector 0

is called the additive identity for thevector addition.

(d) For any vector a

, a a

+ =( ) 0

The vector a is additive inverse of a

.

(e) |a b| |a b|

+ + and|a b| |a| |b|

Difference of Vectors

If a

and b

are any two vectors, then their difference a b

is

defined as a ( b)

+ .

Section Formulae

Let A and B be two points with position vectors a

and b

respectively and P be a point which divides AB

internally in the ratio m n: . Then, position vector of P

= ++

m n

m n

b a.

If P divides AB externally in the ratio m:n. Then, position

vector of Pb a

=

m n

m n.

If R is the mid-point of AB, then ORa b

= +2

.

Component of a Vector

If a i j k = + +a a a1 2 3$ $ $, we say that the scalar components

of a

along X-axis, Y-axis and Z-axis are a a1 2, and a3,

respectively.

Important Results in Component Form

If a

and b

are any two vectors given in the component

form such that a i j k = + +a a a1 2 3$ $ $ and b i j k

= + +b b b1 2 3$ $ $

Then, (a a a b b b1 2 3 1 2 3, , ) ( , ,and ) are called direction

ratios of a

and b

, respectively.

(i) The sum (or resultant) of the vectors a

and b

isgiven by

a b i j k + = + + + + +( ) $ ( ) $ ( ) $a b a b a b1 1 2 2 3 3

(ii) The difference of the vectors a

and b

is given by

a b i j k = + + ( ) $ ( ) $ ( ) $a b a b a b1 1 2 2 3 3

(iii) The vectors a

and b

are equal, if and only if

a b a b1 1 2 2= =,and a b3 3=

(iv) The multiplication of vector a

by any scalar is

given by a i j k = + +( ) $ ( ) $ ( ) $a a a1 2 3

(v) Ifb

a

b

a

b

ak1

1

2

2

3

3

= = = (constant)

Then, vectors a

and b

will be collinear.

(vi) If it is given that l, m and n are direction cosines of avector, then

l m n$ $ $ $ cos $ $i j k i j k+ + = + +(cos ) ) (cos ) (is the unit vector in the direction of that vector, where , and are the angles which the vector makeswith X Y, and Z-axes, respectively.


C B

AO

a +b

a

b

b

O

a b

C

B A

Multiplication of a Vector by Scalar

Let a

be a given vector and be a scalar. Then, the

product of the vector a

by the scalar , denoted by a

, is

called the multiplication of vector a

by the scalar .

Let a

and b

be any two vectors and k and m be anyscalars. Then,

(i) k m k ma a a

+ = +( )

(ii) k m km( ) ( )a a

=

(iii) k k k( )a b a b

+ = +

Vector Joining Two PointsIf P x y z1 1 1 1( , , ) and P x y z2 2 2 2( , , ) are any two points, thenvector joining P1 and P2 is

P P1 2

= OP OP2 1

= ( $ $ $ ) ( $ $ $ )x +y z x +y z2 2 2 1 1 1i j k i j k+ +

= ( ) $ ( ) $ ( ) $x x + y y + z z2 1 2 1 2 1i j k

Dot Product or Scalar Product

(i) If is the angle between a and b

, then

a b a b =| || |cos

(ii) If is acute, then a b

> 0 and if is obtuse,

then a b

< 0

(iii) a b a b

= 0.

(iv) $ $ $ $ $ $i i j j k k = = =1 and $ $ $ $ $ $i j j k k i 0 = = =

(v) Projection of a

on b

=

a b

b| |

(vi) If a force F

displaces a particle from a point A to a

point B, then work done by the force = F . AB

(vii) Properties of scalar product

(a) a b b a = [commutative]

(b) a b c a b a c

+ = + ( )

(c) a a |a | a

= =2 2

where, a represents magnitude of vector a

.

(d) ( ) ( )a b a b

+ = a b2 2, where a and b represent

the magnitude of vectors a

and b

.

(e) ( ( )a b a b

= )

If , and are the direction angles of vectora i j k

= a + a + a1 2 3$ $ $, then its DCs is given as

cos

| |

, cos

| |

, cos

| |

= = a a a1 2 3

a a a

=

Cross Product or Vector Product

(i) If is the angle between the vectors a

and b

, then

a b a b n

=| || |sin $ , where, $n is a unit vector

perpendicular to the plane of a

and b

.

(ii) A unit vector perpendicular to both a

and b

is given by

$( )

| |

na b

a b

=

(iii) Area of a parallelogram with sides a

and b

=

| |a b

(iv) Area of a parallelogram with diagonals a

and b

= 1

2| |a b

(v) Area of a quadrilateral ABCD = 1

2( )AC BD

(vi) Area of a ABC = 1

2| |AB AC =

1

2| |BC BA

= 1

2| |CB CA

(vii) a b a b 0

=||

(viii) If a i j k

= + +a a a1 2 3$ $ $

and b i j k

= + +b b b1 2 3$ $ $, then a bi j k

=

$ $ $

a a a

b b b

1 2 3

1 2 3

(ix) a b b a

=

(x) a b c a b a c

+ = + ( )

(xi) a a 0

=

(xii) $ $ $,i j k = $ $ $,j k i = $ $ $k i j = ;$ $ $j i k = , $ $ $k j i = , $ $ $i k j =

(xiii) $ $ , $ $ ,i i 0 j j 0 = =

$ $k k 0 =

17


18

Three-Dimensional Geometry

Direction Cosines andDirection RatiosIf a line make angles , and with X-axis, Y-axis andZ-axis respectively, then l = cos , m = cos and n = cos are called the direction cosines of the line.

(i) We always have, l m n2 2 2 1+ + = .

(ii) Angle between two lines If is the angle betweentwo lines with direction cosines l m n1 1 1, , andl m n2 2 2, , , then cos = l1 l m m n n2 1 2 1 2+ + .

Numbers proportional to the direction cosines of a line arecalled the direction ratios of the line.

(i) If a b, and c are the direction ratios of a line, then

l =+ +

a

a b c2 2 2, m

b

a b c=

+ +2 2 2

and nc

a b c=

+ +2 2 2

(ii) In any line, direction cosines are unique but directionratios are not unique.

Direction Cosines and Direction Ratios of aLineThe direction cosines and direction ratios of theline segment joining P x y z( , , )1 1 1 and Q x y z( , , )2 2 2 arerespectively given by

x x

PQ2 1 , y y

PQ

z z

PQ2 1 2 1 , and ( , , )x x y y z z2 1 2 1 2 1

where, PQ x x y y z z= + + ( ) ( ) ( )2 12

2 12

2 12

LineA straight line is the locus of the intersection of two planes.A line is uniquely determined, if

(i) it passes through a given point and has givendirection or

(ii) it passes through two given points.

Equation of Line in Vector Form(i) Equation of a line through a given point A with

position vector a

and parallel to a given vector b

is

given by r a b

= + t , where t is a scalar.

If b

= + +a b c$ $ $i j k, then a b, and c are direction ratiosof the line and conversely, if a b c, and are direction

ratios of a line, then b i j k

= + +a b c$ $ $ will be theparallel to the line.

(ii) The vector equation of a line passing through two

points with position vectors a

and b

is given by

r a b a

= + t ( ), where t is a scalar.

(iii) Angle between two lines If is the angle between

two lines r a b

= + t and r c d

= + s , then

cos

| || |

=

b d

b d

.

(iv) Skew-lines If two lines do not meet and notparallel, then they are known as skew-lines.

(v) Shortest distance between two skew-lines Theshortest distance between two skew-lines is thelength of perpendicular to both the lines.

(a) Shortest distance between the skew-lines

r a b

= +1 1 and r a b

= +2 2 is given by

SD =

|( ) ( )|

| |

a a b b

b b

2 1 1 2

1 2

(b) The shortest distance between the parallel lines

r a b

= +1 and r a b

= +2 is given by

SD =

b a a

b

( )2 1

Equation of Line in Cartesian Form(i) The equation of a line passing through a point

A x y z( , , )1 1 1 and having direction ratios a b, and c is

x x

a

y y

b

z z

c

= = 1 1 1

(ii) If l, m and n are the direction cosines of the line, thenequation of the line is

x x

l

1 = = y ym

z z

n1 1

(iii) The equation of a line passing through two pointsA x y z( , , )1 1 1 and B x y z( , , )2 2 2 is

x x

x x

y y

y y

z z

z z

=

=

1

2 1

1

2 1

1

2 1

(iv) If a b c1 1 1, , and a b c2 2 2, , are direction ratios of twolines respectively, then the angle between the linesis given by

cos = + +

+ + + +

a a b b c c

a b c a b c

1 2 1 2 1 2

12

12

12

22

22

22

or sin = + +

+ +

( ) ( ) ( )a b a b b c b c c a c a

a b c

1 2 2 12

1 2 2 12

1 2 2 12

12

12

12 a b c2

222

22+ +

(v) Two lines with direction ratios a b c1 1 1, , and a b c2 2 2, ,are (a) perpendicular, if a a b b c c1 2 1 2 1 2 0+ + = .

(b) parallel, ifa

a

b

b

c

c1

2

1

2

1

2

= = .


(vi) The shortest distance between the lines

x x

a

y y

b

z z

c

= = 11

1

1

1

1

andx x

a

y y

b

z z

c

= = 22

2

2

2

2

is

x x y y z z

a b c

a b c

bc b c c a c a

2 1 2 1 2 1

1 1 1

2 2 2

1 2 2 12

1 2 2 1

+ ( ) ( )

( )

2

1 2 2 12+ a b a b

PlaneA plane is a surface such that a line segment joining anytwo points on it lies wholly on it. A plane is determineduniquely, if any one of the following is known.

(i) The normal to the plane and its distance from theorigin is given, i.e., equation of a plane in normal form.

(ii) It passes through a point and is perpendicular to agiven direction.

(iii) It passes through three given non-collinear points.

Now, we shall find vector and cartesian equations ofthe planes.

Equation of Plane in Cartesian Form(i) The general equation of a plane is

ax by cz d+ + + = 0.The direction ratios of the normal to this plane area b c, and .

(ii) If a plane cuts (intercepts) a b, and c with thecoordinate axes, then the equation of the plane isx

a

y

b

z

c+ + =1.

(iii) Equation of a plane passing through a point andperpendicular to a given vector The equation of aplane passing through a point ( , , )x y z1 1 1 is

a x x b y y( ) ( ) + 1 1 + =c z z( )1 0where, a, b and c are direction ratios of perpendicularvector.

(iv) Equation of a plane passing through threenon-collinear points The equation of a planepassing through three non-collinear points A x y z( , , )1 1 1 ,B x y z( , , )2 2 2 and C x y z( , , )3 3 3 is given by

x x y y z z

x x y y z z

x x y y z z

=1 1 1

2 1 2 1 2 1

3 1 3 1 3 1

0

(v) Equation of a plane through the intersection of twoplanes The equation of a plane through theintersection of the planes

a x b y c z d1 1 1 1 0+ + + =and a x b y c z d2 2 2 2 0+ + + = is( )a x b y c z d1 1 1 1+ + + + ( )a x b y c z d2 2 2 2 0+ + + =

(vi) Distance of a point from a plane Distance of apoint P x y z( , , )1 1 1 from a plane ax by cz d+ + + = 0 isgiven by

pax by cz d

a b c= + + +

+ +

| |1 1 12 2 2

(vii) Angle between two planes The angle between twoplanes is the angle between their normals.

If is the angle between the planesa x b y c z d1 1 1 1 0+ + + =

and a x b y c z d2 2 2 2 0+ + + = , then

cos = + +

+ + + +

a a b b c c

a b c a b c

1 2 1 2 1 2

12

12

12

22

22

22

If two planes are perpendicular, thena a b b c c1 2 1 2 1 2 0+ + = and if they are parallel, then

a

a

b

b

c

c1

2

1

2

1

2

= = .

(viii) Equation of the bisector plane The equation of thebisector plane to the planes a x b y c z d1 1 1 1 0+ + + =and a x b y c z d2 2 2 2 0+ + + = are given by

a x b y c z d

a b c

a x b y c z d

a b c

1 1 1 1

12

12

12

2 2 2 2

22

22

+ + +

+ += + + +

+ + 22

Coplanarity of Two Lines

Two linesx x

a

y y

b

z z

c

= = 11

1

1

1

1

andx x

a

y y

b

z z

c

= = 22

2

2

2

2

are coplanar, if and only if

x x y y z z

a b c

a b c

2 1 2 1 2 1

1 1 1

2 2 2

0

=

The equation of plane containing the above lines isx x y y z z

a b c

a b c

=

1 1 1

1 1 1

2 2 2

0

Equation of Plane in Vector Form(i) Let O be the origin and $n be a unit vector in the

direction of the normal ON to the plane and let

ON p= . Then, the equation of the plane is r n

=$ p.

If r

. ( $ $ $a b c di j k+ + =) is the vector equation of a plane,then ax by cz d+ + = is the cartesian equation of the plane,where a b, and c are the direction ratios of the normal to theplane.

(ii) Equation of a plane perpendicular to a given vectorand passing through a given point The equation of a

plane passing through the point a

and perpendicular

to the given vector n

is

(r a

) n

=0

19


20

(iii) Equation of plane passing through three non-collinearpoints The equation of plane passing through three

non-collinear points a

, b

and c

is

( ) [( ( )]r a b a c a

=) 0(iv) Plane passing through the intersection of two given

planes The equation of plane passing through the

intersection of two planes r n

=1 1d and r n

=2 2d is

r n n

+ = +( )1 2 1 2 d d .

(v) Two lines r a b

= +1 1 and r a b

= +2 2 are coplanar, if

and only if (a a

2 1 ) = ( )b b1 2 0.

(vi) Angle between the planes r n

=1 1p and r n

=2 2p is given

by cos

| || |

=

n n

n n

1 2

1 2

.

Two planes are perpendicular to each other, if n n 0

=1 2

and parallel, if n n 01 2

= .

(vii) Distance of a point from a plane The

perpendicular distance of a point a

from the

plane r n

= d , where n

is normal to the plane, is

a n

n

d

(viii) The length of the perpendicular from origin O to

the plane r n

= d isd

| |n

.

(ix) The angle between line r a b

= + and plane

r n

= d is cos =

b n

b n

and sin =

b n

b n

, where = 90 .


Linear Programming

Linear Programming ProblemA linear programming problem is one that is concerned withfinding the optimal value (maximum or minimum value) of a linearfunction (called objective function) of several variables (say x andy called decision variable), subject to the conditions that thevariables are non-negative and satisfy a set of linear inequalities(called linear constraints).

Some Terms Related to LPP

1. ConstraintsThe linear inequations or inequalities or restrictions on thevariables of a linear programming problem are called constraints.The conditions x y 0 0, are called non-negative restrictions.

2. Optimisation Problem

A problem which seeks to maximise or minimise a linear function

subject to certain constraints determined by a set of linear

inequalities is called an optimisation problem. Linear

programming problems are special type of optimisation

problems.

3. Objective Function

A linear function of two or more variables which has to be

maximised or minimised under the given restrictions in the form

of linear inequations (or linear constraints) is called the

objective function.The variables used in the objective function

are called decision variables.

4. Optimal ValuesThe maximum or minimum value of an objective function isknown as its optimal value.

5. SolutionA set of values of the variables which satisfy theconstraints of given linear function (objective function)of several variable, subject to the conditions that thevariables are non-negative is called a solution of thelinear programming problem.

6. Feasible Solution

Any solution to the given linear programming

problem which also satisfies the non-negative

restrictions of the problem is called a feasible

solution. Any point outside the feasible region is

called an infeasible solution.

7. Feasible Region

The set of all feasible solutions constitutes a region

which is called the feasible region. Each point in this

region represents a feasible choice. The region other

than feasible region is called an infeasible region.

8. Bounded RegionA feasible region of a system of linear inequalities issaid to be bounded, if it can be enclosed within acircle. Otherwise, it is said to be unbounded region.

9. Optimal Solution

A feasible solution at which the objective function has

optimal value is called the optimal solution of the

linear programming problem.

10. Optimisation Technique

The process of obtaining the optimal solution is

called optimisation technique.

Corner Point MethodIt is a graphical method to solve the LPP.

The following steps are given below

Step I Find the feasible region of the linear programmingproblem and determine its corner points (vertices)either by inspection or by solving the two equationsof the lines intersecting at that point.

Step II Evaluate the objective function z ax by= + at eachcorner point. Let M and m respectively denote thelargest and smallest values of these points.

Step III When the feasible region is bounded, M and m arethe maximum and minimum values of z.

Step IV In case, the feasible region is unbounded, wehave

(a) M is the maximum value of z, if the open halfplane determined by ax by M+ > has no pointin common with the feasible region. Otherwise,z has no maximum value.

(b) Similarly, m is the minimum value of z, if theopen half plane determined by ax by m+ < hasno point in common with the feasible region.Otherwise, z has no minimum value.

21


Probability

Some Basic Definitions

1. Experiment

An operation which results in some well-defined outcomes, iscalled an experiment.

2. Random Experiment

An experiment in which total outcomes are known in advance butoccurrence of specific outcome can be told only after completionof the experiment, is known as a random experiment.

3. Trial

When a random experiment is repeated under identicalconditions and it does not give the same result each time butmay result in one of the several possible outcomes, then suchexperiment is called a trial.

4. Sample Space

The set of all possible outcomes of a random experiment iscalled its sample space. It is usually denoted by S.

5. Discrete Sample Space

A sample space is called a discrete sample space, if S is a finiteset.

EventA subset of the sample space associated with a random

experiment is called an event or case.

Type of Events

Equally Likely EventsThe given events are said to be equally likely, if none of them isexpected to occur in preference to the other.

Mutually Exclusive EventsA set of events is said to be mutually exclusive, if the happening ofone excludes the happening of the other i.e., if A and B aremutually exclusive, then ( )A B = .

Exhaustive EventsA set of events is said to be exhaustive, if the performance of theexperiment always results in the occurrence of atleast one ofthem.

If E E En1 2, , , are exhaustive events, then E E E Sn1 2 = .

Complement of an Event

Let some A be an event in a sample space S, then

complement of A is the set of all sample points of

the space other than the sample point in A and it

is denoted by A or A.i e A n n S n. ., { : , A} =

Some Basic Terms

CoinA coin has two sides, head and tail. If an eventconsists of more than one coin, then coins areconsidered as distinct, if not otherwise stated.

(i) Sample space of one coin= { }H, T

(ii) Sample space of two coins

= {( }H, T T, H H, H T, T), ( ), ( ), ( )(iii) Sample space of three coins

= {( ) ( ) ( ), ( )H, H, H H, H, T H, T, H T, H, H, , ,( ) ( ) ( )H, T, T T, H, T T, T, H, , , ( )}T, T, T

Die

A die has six faces marked 1, 2, 3, 4, 5 and 6. If we

have more than one die, then all dice are

considered as distinct, if not otherwise stated.

(i) Sample space of a die = { , , , , , }1 2 3 4 5 6(ii) Sample space of two dice

=

( , ), ( , ), ( , ), ( , ), ( , ), ( , )

( , ), ( , ), (

1 1 1 2 1 3 1 4 1 5 1 6

2 1 2 2 2, ), ( , ), ( , ), ( , )

( , ), ( , ), ( , ), ( , ), ( , )

3 2 4 2 5 2 6

3 1 3 2 3 3 3 4 3 5 , ( , )

( , ), ( , ), ( , ), ( , ), ( , ), ( , )

( , ), ( ,

3 6

4 1 4 2 4 3 4 4 4 5 4 6

5 1 5 2 5 3 5 4 5 5 5 6

6 1 6 2 6 3 6 4

), ( , ), ( , ), ( , ), ( , )

( , ), ( , ), ( , ), ( , ), ( , ), ( , )6 5 6 6

Playing CardsA pack of playing cards has 52 cards. There are 4 suits(spade, heart, diamond and club), each having 13 cards.There are two colours, red (heart and diamond) and black(spade and club), each having 26 cards.

In 13 cards of each suit, there are 3 face cards namelyking, queen and jack, so there are in all 12 face cards.Also, there are 16 honour cards, 4 of each suit namely ace,king, queen and jack.

ProbabilityIn a random experiment, let S be the sample space and Ebe the event. Then,

P EE

( ) = Number of distinct elements inNumber of distinct elements inS

n E

n S= ( )

( )

= Number of outcomes favourable toNumber of all p

E

ossible outcomes

(i) If E is an event and S is the sample space, then

(a) 0 1 P E( ) (b) P( ) =0

(c) P S( ) = 1

(ii) P E P E( ) ( )= 1

Addition Theorem of Probability(a) For two events A and B,

P A B P A P B P A B( ) ( ) ( ) ( ) = + If A and B are mutually exclusive events, then

P A B P A P B( ) ( ) ( ) = +[for mutually exclusive, P A B( ) ] = 0

(b) For three events A B, and C,

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

If A B, and C are mutually exclusive, then

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

for mutually exclusive events,

P A B P B C( ) ( ) = =

=

P C A

P A B C =

( )

( ) 0

(c) If a set of events A A An1 2, , , are mutually exclusive,then A A A An1 2 3 = . P A A A An( )1 2 3

= + ++P A P A P An( ) ( ( )1 2 )and P A A A An( )1 2 3 0 =

If a set of events A A An1 2, , , are exhaustive, thenP A A An( )1 2 1 = .

Booleys Inequality(i) If A A An1 2, , , are n events associated with a

random experiment, then

(a) P (A A A )1 2 n =i 1

n

iP(A ) (n 1)

(b) P ( A ) P(A )i 1

n

ii 1

n

i = =

(ii) If A and B are two events associated to a randomexperiment, then

P A B P A P A B( ) ( ) ( ) +P A P B( ) ( )(iii) If A and B are two events associated with a random

experiment, then

(a) P A B P B P A B( ) ( ) ( ) = (b) P A B P A P A B( ) ( ) ( ) = (c) P A B A B P A P B[( ) ( )] ( ) ( ) = +

2P A B( )(d) P A B P A B( ) ( ) = 1(e) P A B P A B( ) ( ) = 1

(f) P A P A B P A B( ) ( ) ( )= + (g) P B P A B P B A( ) ( ) ( )= +

(iv) (a) P (exactly one of A B, occurs)

= + P A P B P A B( ) ( ) ( )2= P A B P A B( ) ( )

(b) P (neither A nor B) = P (A B ) = 1 P(A B )(v) If A B, and C are three events, then

P (exactly one of A B C, , occurs)

= + + P A P B P C P A B( ) ( ) ( ) ( )2 2 2P B C P A C( ) ( ) + 3P A B C( )

(vi) P(atleast two of A B C, , occurs)

= + + P A B P B C P C A P A B C( ) ( ) ( ) ( )2(vii) P (exactly two of A B C, , occurs)

= + + P A B P B C P A C P A B C( ) ( ) ( ) ( )3

(viii) (a) P A P A( ) ( )= 1(b) P A A P S P( ) ( ), ( ) = = 0

Conditional ProbabilityLet E and F be two events associated with a randomexperiment. Then, probability of occurrence of event E,when the event F has already occurred, is calledconditional probability of event E over F and is denoted byP E F( / ).

P E FP E F

P F( / )

( )

( )= , where P F( ) 0.

Properties of Conditional ProbabilityLet A,B and C be the events of a sample space S. Then,

(i) P S A P A A( / ) ( / )= =1(ii) P A B C P A C P B C P A B C{( )/ } ( / ) ( / ) {( )/ }; = +

P C( ) 0(iii) P A B P A B( / ) ( / ), = 1 where A is complement of A.

22


Cards52

Red26

Black26

Colours

Heart13

Diamond13

Club13

Spade13

23

Multiplication Theorem on ProbabilityLet A and B are two events associated with a randomexperiment, then

P A BP A P B A P A

P B P A B P A( )

( ) ( / ), ( )

( ) ( / ), ( =

where

where

0

)

0The above result is known as the multiplication rule ofprobability.

Multiplication Probability for more thanTwo EventsLet E, F and G be three events of sample space S, then

P E F G P E PF

EP

G

E F( ) ( ) =

Independent EventsTwo events A and B are said to be independent, if theoccurrence or non-occurrence of one event does not affectthe occurrence or non-occurrence of another event.

Then, P A B P A P B( ) ( ) ( ) = ; P A B P A( / ) ( )=and P B A P B( / ) ( )=

Partition of Sample SpaceA set of events E E En1 2, , ..., is said to represent a partitionof the sample space S, if it satisfies the followingconditions

(i) E E i j i j ni j = =, , , , ,...,1 2(ii) E E E Sn1 2 =...(iii) P E i , ni( ) , ,...,> =0 1 2for all

Theorem of Total ProbabilityLet S be the sample space and E E E En1 2 3, , ,..., be n mutuallyexclusive and exhaustive events associated with a randomexperiment.

If E is any event which occurs with E E E En1 2 3, , ,..., .

Then, P E P E P E E P E P E E( ) ( ) ( / ) ( ) ( / )= + 1 1 2 2

+ + + P E P E E P E P E En n( ) ( / ) ... ( ) ( / )3 3

or P E P E PE

Ei

ii

n

( ) ( )=

=

1

Bayes TheoremLet S be the sample space and E E En1 2, ,..., be n mutuallyexclusive and exhaustive events associated with a randomexperiment. If A is any event which occurs with E1, E2, ..., En,then probability of occurrence of Ei when A occurred,

P E AP E P A E

P E P A E

i nii i

i i

i

n( / )

( ) ( / )

( ) ( / )

, , ,...,= =

=

1

1 2

If P E P E P E P En( ) ( ) ( ) ... ( ),1 2 3= = = = then

P E EP E E

P E E

ii

i

i

n( / )

( / )

( / )

=

=

1

Probability Distribution of aRandom VariableThe system in which the value of a random variable aregiven along with their corresponding probability is calledprobability distribution. If X is a random variable and takesthe value x x x xn1 2 3, , ,..., with respective probabilitiesp p p pn1 2 3, , ,..., . Then, the probability distribution of X isrepresented by

X x1 x2 x3 ... xn

P X( ) p1 p2 p3 ... pn

such that pi = 1

If xi is one of the possible values of a random variable X,the statement X x i= is true only at some point(s) of thesample space. Hence, the probability that X takes value x iis always non-zero, i.e., P X x i( ) .= 0

Mean and Variance of a ProbabilityDistribution

Mean of a probability distribution is x pi ii

n

=

1

. It is also

called expectation of X, i.e., E X x pi ii

n

( ) ==

1

.

Variance is given by V X x p x pi i i ii

n

i

n

( ) =

== 2

1

2

1

or V ( ) ( ) [ ( )] ,X E X E X= 2 2 where E X x p xi ii

n

( ) ( )2 2

1

= =

Bernoulli TrialsThe independent trials which have only two outcomes i.e.,success or failure, is called Bernoulli trial.

Conditions for Bernoulli Trials(i) There should be a finite number of trials.

(ii) The trials should be independent.

(iii) Each trial has exactly two outcomes success or failure.

(iv) The probability of success remains the same in eachtrial.

Binomial DistributionThe probability distribution of number of successes in anexperiment consisting n Bernoulli trials obtained by thebinomial expression ( )p q n+ , is called binomialdistribution.

This distribution can be represented by the following table

X 0 1 2 ...r ... n

P X( )n nC q0

n nC q p11 1 n nC q p2

2 2 ...n rn r rC q p ... n n

nC p

Here, P X r C q pn rn r r( )= = is called the probability function

of the binomial distribution.

where, p = Probability of successq = Probability of failure,n = Number of trials and p q+ =1


Revision Notes Maths XII

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