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Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1 -

MATHEMATICS IMPORTANT FORMULAE

AND CONCEPTS

for

Final Revision

CLASS – XII

2016 – 17

CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION

Prepared by

M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed.

Kendriya Vidyalaya GaCHIBOWLI


CHAPTER – 1: RELATIONS AND FUNCTIONS

QUICK REVISION (Important Concepts & Formulae) Relation Let A and B be two sets. Then a relation R from A to B is a subset of A × B. R is a relation from A to B R A × B. Total Number of Relations Let A and B be two nonempty finite sets consisting of m and n elements respectively. Then A × B consists of mn ordered pairs. So, total number of relations from A to B is 2nm. Domain and range of a relation Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R, while the set of all second components or coordinates of the ordered pairs in R is called the range of R. Thus, Dom (R) = a : (a, b) R and Range (R) = b : (a, b) R. Inverse relation Let A, B be two sets and let R be a relation from a set A to a set B. Then the inverse of R, denoted by R–1, is a relation from B to A and is defined by R–1 = (b, a) : (a, b) R. Types of Relations Void relation : Let A be a set. Then A × A and so it is a relation on A. This relation is called the void or empty relation on A. It is the smallest relation on set A. Universal relation : Let A be a set. Then A × A A × A and so it is a relation on A. This relation is called the universal relation on A. It is the largest relation on set A. Identity relation : Let A be a set. Then the relation I A = (a, a) : a A on A is called the identity relation on A. Reflexive Relation : A relation R on a set A is said to be reflexive if every element of A is related to itself. Thus, R reflexive (a, a) R a A. A relation R on a set A is not reflexive if there exists an element a A such that (a, a) R. Symmetric relation : A relation R on a set A is said to be a symmetric relation iff (a, b) R (b, a) R for all a, b A. i.e. aRb bRa for all a, b A. A relation R on a set A is not a symmetric relation if there are atleast two elements a, b A such that

(a, b) R but (b, a) R. Transitive relation : A relation R on A is said to be a transitive relation iff (a, b) R and (b, c) R (a, c) R for all a, b, c A. i.e. aRb and bRc aRc for all a, b, c A. Antisymmetric relation : A relation R on set A is said to be an antisymmetric relation iff (a, b) R and (b, a) R a = b for all a, b A.

Equivalence relation : A relation R on a set A is said to be an equivalence relation on A iff It is reflexive i.e. (a, a) R for all a A. It is symmetric i.e. (a, b) R (b, a) R for all a, b A.


It is transitive i.e. (a, b) R and (b, c) R (a, c) R for all a, b, c A. Congruence modulo m Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m if a – b is divisible by m and we write a b(mod m). Thus, a b (mod m) a – b is divisible by m. Some Results on Relations If R and S are two equivalence relations on a set A, then R S is also an equivalence relation on A. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set. If R is an equivalence relation on a set A, then R–1 is also an equivalence relation on A. Composition of relations Let R and S be two relations from sets A to B and B to C respectively. Then we can define a relation SoR from A to C such that (a, c) SoR b B such that (a, b) R and (b, c) S. This relation is called the composition of R and S. Functions Let A and B be two empty sets. Then a function 'f ' from set A to set B is a rule or method or correspondence which associates elements of set A to elements of set B such that (i) All elements of set A are associated to elements in set B. (ii) An element of set A is associated to a unique element in set B. A function ‘f ’ from a set A to a set B associates each element of set A to a unique element of set B. If an element a A is associated to an element b B, then b is called 'the f image of a or 'image of a

under f or 'the value of the function f at a'. Also, a is called the preimage of b under the function f. We write it as : b = f (a).

Domain, CoDomain and Range of a function Let f : AB. Then, the set A is known as the domain of f and the set B is known as the codomain of f. The set of all f images of elements of A is known as the range of f or image set of A under f and is denoted by f (A). Thus, f (A) = f (x) : x A = Range of f. Clearly, f (A) B. Equal functions Two functions f and g are said to be equal iff (i) The domain of f = domain of g (ii) The codomain of f = the codomain of g, and (iii) f (x) = g(x) for every x belonging to their common domain. If two functions f and g are equal, then we write f = g. Types of Functions (i) Oneone function (injection) A function f : A B is said to be a oneone function or an injection if different elements of A have different images in B. Thus, f : A B is oneone a b f (a) f (b) for all a, b A f (a) = f (b) a = b for all a, b A. Algorithm to check the injectivity of a function Step I : Take two arbitrary elements x, y (say) in the domain of f. Step II : Put f (x) = f (y) Step III : Solve f (x) = f (y). If f (x) = f (y) gives x = y only, then f : A B is a oneone function (or an injection) otherwise not.


Graphically, if any straight line parallel to x-axis intersects the curve y = f (x) exactly at one point, then the function f (x) is oneone or an injection. Otherwise it is not.

If f : R R is an injective map, then the graph of y = f (x) is either a strictly increasing curve or a

strictly decreasing curve. Consequently, 0 0dy dyordx dx

for all x.

Number of oneone functions from A to B , ,0,

nmP if n m

if n m

where m = n(Domain) and n = n(Codomain) (ii) Ontofunction (surjection) A function f : AB is said to be an onto function or a surjection if every element of B is the fimage of some element of A i.e., if f (A) = B or range of f is the codomain of f. Thus, f : A B is a surjection iff for each b B, a A that f (a) = b. Algorithm for Checking the Surjectivity of a Function Let f : A B be the given function. Step I : Choose an arbitrary element y in B. Step II : Put f (x) = y. Step III : Solve the equation f (x) = y for x and obtain x in terms of y. Let x = g(y). Step IV : If for all values of y B, for which x, given by x = g(y) are in A, then f is onto. If there are some y B for which x, given by x = g(y) is not in A. Then, f is not onto. Number of onto functions :If A and B are two sets having m and n elements respectively such that

1 n m, then number of onto functions from A to B is 1( 1) .

nn r n m

rr

C r

(iii) Bijection (oneone onto function) A function f : A B is a bijection if it is oneone as well as onto. In other words, a function f : A B is a bijection if

(i) It is oneone i.e. f (x) = f (y) x = y for all x, y A. (ii) It is onto i.e. for all y B, there exists x A such that f (x) = y. Number of bijections : If A and B are finite sets and f : A B is a bijection, then A and B have the same number of elements. If A has n elements, then the number of bijections from A to B is the total number of arrangements of n items taken all at a time i.e. n! (iv) Manyone function A function f : A B is said to be a manyone function if two or more elements of set A have the same image in B. f : AB is a manyone function if there exist x, y A such that x y but f (x) = f ( y).


(v) Into function A function f : AB is an into function if there exists an element in B having no preimage in A. In other words f : A B is an into function if it is not an onto function. (vi) Identity function Let A be a nonempty set. A function f : AA is said to be an identity function on set A if f associates every element of set A to the element itself. Thus f : A A is an identity function iff f (x) = x, for all x A. (vii) Constant function A function f : A B is said to be a constant function if every element of A has the same image under function of B i.e. f (x) = c for all x A, where c B. Composition of functions Let A, B and C be three nonvoid sets and let f : A B, g : B C be two functions. For each x A there exists a unique element g( f (x)) C.

The composition of functions is not commutative i.e. fog gof. The composition of functions is associative i.e. if f, g, h are three functions such that (fog)oh and

fo(goh) exist, then (fog)oh = fo(goh). The composition of two bijections is a bijection i.e. if f and g are two bijections, then gof is also a

bijection. Let f : AB. The foIA = IB of = f i.e. the composition of any function with the identity function is the

function itself.

Inverse of an element Let A and B be two sets and let f : A B be a mapping. If a A is associated to b B under the function f, then b is called the f image of a and we write it as b = f (a). Inverse of a function If f : A B is a bijection, we can define a new function from B to A which associates each element y B to its preimage f –1(y) A.


Algorithm to find the inverse of a bijection Let f : A B be a bijection. To find the inverse of f we proceed as follows : Step I : Put f (x) = y , where y B and x A. Step II : Solve f (x) = y to obtain x in terms of y. Step III : In the relation obtained in step II replace x by f –1(y) to obtain the inverse of f. Properties of Inverse of a Function (i) The inverse of a bijection is unique. (ii) The inverse of a bijection is also a bijection. (iii) If f : A B is a bijection and g : B A is the inverse of f, then fog = IB and gof = IA , where IA and IB are the identity functions on the sets A and B respectively. If in the above property, we have B = A. Then we find that for every bijection f : A A there exists a bijection g : A A such that fog = gof = IA . (iv) Let f : A B and g : B A be two functions such that gof = IA and fog = IB . Then f and g are bijections and g = f –1. Binary Operation Let S be a nonvoid set. A function f from S × S to S is called a binary operation on S i.e. f : S × S S

is a binary operation on set S. Generally binary operations are represented by the symbols *, ,. etc. instead of letters f, g etc. Addition on the set N of all natural numbers is a binary operation. Subtraction is a binary operation on each of the sets Z, Q, R and C. But, it is a binary operation on N. Division is not a binary operation on any of the sets N, Z, Q, R and C. However, it is not a binary

operation on the sets of all nonzero rational (real or complex) numbers. Types of Binary Operations (i) Commutative binary operation A binary operation * on a set S is said to be commutative if a * b = b * a for all a, b S

Addition and multiplication are commutative binary operations on Z but subtraction is not a commutative binary operation, since 2 –3 3 –2.

Union and intersection are commutative binary operations on the power set P(S) of all subsets of set S. But difference of sets is not a commutative binary operation on P(S).

(ii) Associative binary operation A binary operation * on a set S is said to be associative if (a * b) * c = a * (b * c) for all a, b, c S. (iii) Distributive binary operation Let * and o be two binary operations on a set S. Then * is said to be (i) Left distributive over o if a*(b o c) = (a * b) o (a * c) for all a, b, c S (ii) Right distributive over o if (b o c) * a = (b * a) o (c * a) for all a, b, c S. (iv) Identity element Let * be a binary operation on a set S. An element e S is said to be an identity element for the

binary operation * if a * e = a = e * a for all a S.

For addition on Z, 0 is the identity element, since 0 + a = a = a + 0 for all a R.

For multiplication on R, 1 is the identity element, since 1 × a = a = a × 1 for all a R.

For addition on N the identity element does not exist. But for multiplication on N the identity element is 1.


(v) Inverse of an element Let * be a binary operation on a set S and let e be the identity element in S for the binary operation *.

An element a S is said to be an inverse of a S, if a * a= e = a* a. Addition on N has no identity element and accordingly N has no invertible element. Multiplication on N has 1 as the identity element and no element other than 1 is invertible. Let S be a finite set containing n elements. Then the total number of binary operations on S is

2nn . Let S be a finite set containing n elements. Then the total number of commutative binary operation

on S is ( 1)2

n nn

.


CHAPTER – 2: INVERSE TRIGONOMETRIC FUNCTIONS

QUICK REVISION (Important Concepts & Formulae) Inverse Trigonometrical Functions A function f : A B is invertible if it is a bijection. The inverse of f is denoted by f –1 and is defined as f –1(y) = x f (x) = y. Clearly, domain of f –1 = range of f and range of f –1 = domain of f. The inverse of sine function is defined as sin–1x = sinq = x, where [– /2, /2] and

x [–1, 1]. Thus, sin –1 x has infinitely many values for given x [–1, 1]

There is one value among these values which lies in the interval [–/2, /2]. This value is called the

principal value. Domain and Range of Inverse Trigonometrical Functions

Properties of Inverse Trigonometrical Functions

sin–1(sin) = and sin(sin–1x) = x, provided that 1 1x and 2 2

cos–1(cos) = and cos (cos–1 x) = x, provided that 1 1x and 0

tan–1(tan) = and tan(tan–1 x) = x, provided that x and 2 2

cot –1(cot) = and cot(cot –1 x) = x, provided that – < x < and 0 < < . sec –1(sec) = and sec(sec –1 x) = x cosec –1(cosec) = and cosec(cosec–1 x) = x,

1 1 1 11 1sin cos cos sinx ec or ec xx x


1 1 1 11 1cos s s cosx ec or ec xx x

1 1 1 11 1tan cot cot tanx or xx x

2

1 1 2 1 1 1 1

2 2

1 1 1sin cos 1 tan cot sec cos1 1

x xx x ecx xx x

2

1 1 2 1 1 1 1

2 2

1 1 1cos sin 1 tan cot cos s1 1

x xx x ec ecx xx x

2

1 1 1 1 1 2 1

2 2

1 1 1tan sin cos cot sec 1 cos1 1

x xx x ecx xx x

1 1sin cos , 1 12

x x where x

1 1tan cot ,2

x x where x

1 1sec cos , 1 12

x ec x where x or x

1 1 1tan tan tan , 11x yx y if xy

xy

1 1 1tan tan tan , 11x yx y if xy

xy

1 1 1tan tan tan1x yx y

xy

1 1 1 2 2 2 2sin sin sin 1 1 , , 0, 1x y x y y x if x y x y




1 1 1 2 2 2 2cos cos cos 1 1 , , 0, 1x y xy x y if x y x y





1 1 1 1sin ( ) sin , cos ( ) cosx x x x 1 1 1 1tan ( ) tan , cot ( ) cotx x x x

1 1 2 1 1 22sin sin 2 1 , 2cos cos 2 1x x x x x

2

1 1 1 12 2 2

2 2 12 tan tan sin cos1 1 1

x x xxx x x

1 1 3 1 1 33sin sin 3 4 , 3cos cos 4 3x x x x x x

3

1 12

33tan tan1 3

x xxx


CHAPTER – 3: MATRICES

QUICK REVISION (Important Concepts & Formulae) Matrix

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters.

Order of a matrix

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). In general, an m × n matrix has the following rectangular array:

11 12 1

21 22 2

1 2

.....

...... . . .. . . .

.....

n

n

m m mn

a a aa a a

a a a

or A = [aij]m × n, 1≤ i ≤ m, 1 ≤ j ≤ n i, j N Thus the ith row consists of the elements ai1, ai2, ai3,..., ain, while the jth column consists of the elements a1j, a2j, a3j,..., amj , In general aij, is an element lying in the ith row and jth column. We can also call it as the (i, j)th element of A. The number of elements in an m × n matrix will be equal to mn.

We can also represent any point (x, y) in a plane by a matrix (column or row) as ,x

or x yy

Types of Matrices

(i) Column matrix A matrix is said to be a column matrix if it has only one column. In general, A = [aij]m × 1 is a column matrix of order m × 1. (ii) Row matrix A matrix is said to be a row matrix if it has only one row. In general, B = [bij]1 × n is a row matrix of order 1 × n. (iii) Square matrix A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. In general, A = [aij]m × m is a square matrix of order m. If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, ..., ann are said to

constitute the diagonal, of the matrix A. (iv) Diagonal matrix A square matrix B = [bij]m × m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [bij]m × m is said to be a diagonal matrix if bij = 0, when i ≠ j. (v) Scalar matrix


A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij]n × n is said to be a scalar matrix if bij = 0, when i ≠ j bij = k, when i = j, for some constant k. (vi) Identity matrix A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity

matrix. In other words, the square matrix A = [aij] n × n is an identity matrix, if 10ij

if i ja

if i j

We denote the identity matrix of order n by In. When order is clear from the context, we simply write it as I. Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix. (vii) Zero matrix A matrix is said to be zero matrix or null matrix if all its elements are zero. We denote zero matrix by O.

Equality of matrices

Two matrices A = [aij] and B = [bij] are said to be equal if (i) they are of the same order (ii) each element of A is equal to the corresponding element of B, that is aij = bij for all i and j.

Operations on Matrices Addition of matrices

The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrices. Furthermore, the two matrices have to be of the same order.

Thus, if A = 11 12 13

21 22 23

a a aa a a

is a 2 × 3 matrix and B = 11 12 13

21 22 23

b b bb b b

is another 2×3 matrix. Then,

we define

A + B = 11 11 12 12 13 13

21 21 22 22 23 23

a b a b a ba b a b a b

.

In general, if A = [aij] and B = [bij] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m × n, where cij = aij + bij, for all possible values of i and j. If A and B are not of the same order, then A + B is not defined.

Multiplication of a matrix by a scalar

If A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k. In other words, kA = k [aij]m × n = [k (aij)]m × n, that is, (i, j)th element of kA is kaij for all possible values of i and j.

Negative of a matrix The negative of a matrix is denoted by –A. We define –A = (– 1) A. Difference of matrices If A = [aij], B = [bij] are two matrices of the same order, say m × n, then

difference A – B is defined as a matrix D = [dij], where dij = aij – bij, for all value of i and j. In other words, D = A – B = A + (–1) B, that is sum of the matrix A and the matrix – B.


Properties of matrix addition (i) Commutative Law If A = [aij], B = [bij] are matrices of the same order, say m × n, then A + B = B + A. (ii) Associative Law For any three matrices A = [aij], B = [bij], C = [cij] of the same order, say m × n, (A + B) + C = A + (B + C). (iii) Existence of additive identity Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition. (iv) The existence of additive inverse Let A = [aij]m × n be any matrix, then we have another matrix as – A = [– aij]m × n such that A + (– A) = (– A) + A= O. So – A is the additive inverse of A or negative of A.

Properties of scalar multiplication of a matrix

If A = [aij] and B = [bij] be two matrices of the same order, say m × n, and k and l are scalars, then (i) k(A +B) = k A + kB, (ii) (k + l)A = k A + l A

Multiplication of matrices

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = [aij]m × n, B = [bjk]n ×

p, then the ith row of A is [ai1 ai2 ... ain] and the kth column of B is

1

2

.

.

k

k

nk

bb

b

then

cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ain bnk = The matrix C = [cik]m × p is the product of A and B. If AB is defined, then BA need not be defined. In the above example, AB is defined but BA is not defined because B has 3 column while A has only 2 (and not 3) rows. If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m. In particular, if both A and B are square matrices of the same order, then both AB and BA are defined.

Non-commutativity of multiplication of matrices

Now, we shall see by an example that even if AB and BA are both defined, it is not necessary that AB = BA.

Zero matrix as the product of two non zero matrices

We know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

Properties of multiplication of matrices

The multiplication of matrices possesses the following properties: The associative law For any three matrices A, B and C. We have (AB) C = A (BC), whenever both

sides of the equality are defined. The distributive law For three matrices A, B and C.

A (B+C) = AB + AC (A+B) C = AC + BC, whenever both sides of equality are defined.

The existence of multiplicative identity For every square matrix A, there exist an identity matrix of same order such that IA = AI = A.


Transpose of a Matrix If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (AT). In other words, if A = [aij]m × n, then A′ = [aji]n × m.

Properties of transpose of the matrices

For any matrices A and B of suitable orders, we have (i) (A′)′ = A, (ii) (kA)′ = kA′ (where k is any constant) (iii) (A + B)′ = A′ + B′ (iv) (A B)′ = B′ A′

Symmetric and Skew Symmetric Matrices

A square matrix A = [aij] is said to be symmetric if A′ = A, that is, [aij] = [aji] for all possible values of i and j.

A square matrix A = [aij] is said to be skew symmetric matrix if A′ = – A, that is aji = – aij for all

possible values of i and j. Now, if we put i = j, we have aii = – aii. Therefore 2aii = 0 or aii = 0 for all i’s. This means that all the diagonal elements of a skew symmetric matrix are zero.

Theorem 1 For any square matrix A with real number entries, A + A′ is a symmetric matrix and A –

A′ is a skew symmetric matrix. Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a skew symmetric

matrix. Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

(i) The interchange of any two rows or two columns. Symbolically the interchange of ith and jth

rows is denoted by Ri ↔ Rj and interchange of ith and jth column is denoted by Ci ↔ Cj.

(ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the

multiplication of each element of the ith row by k, where k ≠ 0 is denoted by Ri → k Ri.

The corresponding column operation is denoted by Ci → kCi

(iii) The addition to the elements of any row or column, the corresponding elements of any other row

or column multiplied by any non zero number.

Symbolically, the addition to the elements of ith row, the corresponding elements of jth row

multiplied by k is denoted by Ri → Ri + kRj.

The corresponding column operation is denoted by Ci → Ci + kCj.

Invertible Matrices

If A is a square matrix of order m, and if there exists another square 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 by A–1. In that case A is said to be invertible.


A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

If B is the inverse of A, then A is also the inverse of B. Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. Theorem 4 If A and B are invertible matrices of the same order, then (AB)–1 = B–1A–1. Inverse of a matrix by elementary operations

If A is a matrix such that A–1 exists, then to find A–1 using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find A–1 using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB. In case, after applying one or more elementary row (column) operations on A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S., then A–1 does not exist.


CHAPTER – 4: DETERMINANTS

QUICK REVISION (Important Concepts & Formulae) Determinant

If A = a bc d

, then determinant of A is written as |A| = a bc d

= det (A) or Δ

(i) For matrix A, |A| is read as determinant of A and not modulus of A. (ii) Only square matrices have determinants.

Determinant of a matrix of order one

Let A = [a ] be the matrix of order 1, then determinant of A is defined to be equal to a Determinant of a matrix of order two

Let A = 11 12

21 22

a aa a

be a matrix of order 2 × 2, then the determinant of A is defined as:

det (A) = |A| = Δ = 11 12

21 22

a aa a

= 11 22 21 12a a a a

Determinant of a matrix of order 3 × 3

Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and C3) giving the same value as shown below. Consider the determinant of square matrix A = [aij]3 × 3

i.e., | A | = 11 12 13

21 22 23

31 32 33

a a aa a aa a a

Expansion along first Row (R1)

Step 1 Multiply first element a11 of R1 by (–1)(1 + 1) [(–1)sum of suffixes in a11] and with the second order determinant obtained by deleting the elements of first row (R1) and first column (C1) of | A | as a11 lies in R1 and C1,

i.e., 22 231 111

32 33

( 1)a a

aa a

Step 2 Multiply 2nd element a12 of R1 by (–1)1 + 2 [(–1)sum of suffixes in a12] and the second order determinant obtained by deleting elements of first row (R1) and 2nd column (C2) of | A | as a12 lies in R1 and C2,

i.e., 21 231 212

31 33

( 1)a a

aa a

Step 3 Multiply third element a13 of R1 by (–1)1 + 3 [(–1)sum of suffixes in a13] and the second order determinant obtained by deleting elements of first row (R1) and third column (C3) of | A | as a13 lies in R1 and C3,


i.e., 21 221 313

31 32

( 1)a a

aa a

Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by

22 23 21 23 21 221 1 1 2 1 311 12 13

32 33 31 33 31 32

| | ( 1) ( 1) ( 1)a a a a a a

A a a aa a a a a a

or |A| = a11 (a22a33 – a32 a23) – a12 (a21a33 – a31a23) + a13 (a21a32 – a31a22) Expansion along second row (R2)

| A | = 11 12 13

21 22 23

31 32 33

a a aa a aa a a

Expanding along R2, we get 12 13 11 13 11 122 1 2 2 2 3

21 22 2332 33 31 33 31 32

| | ( 1) ( 1) ( 1)a a a a a a

A a a aa a a a a a

Expansion along first Column (C1)

| A | = 11 12 13

21 22 23

31 32 33

a a aa a aa a a

By expanding along C1, we get 22 23 12 13 12 131 1 2 1 3 1

11 21 3132 33 32 33 22 23

| | ( 1) ( 1) ( 1)a a a a a a

A a a aa a a a a a

For easier calculations, we shall expand the determinant along that row or column which contains

maximum number of zeros. While expanding, instead of multiplying by (–1)i + j, we can multiply by +1 or –1 according as (i + j)

is even or odd. If A = kB where A and B are square matrices of order n, then | A| = kn| B |, where n = 1, 2, 3

Properties of Determinants Property 1

The value of the determinant remains unchanged if its rows and columns are interchanged. if A is a square matrix, then det (A) = det (A′), where A′ = transpose of A. If Ri = ith row and Ci = ith column, then for interchange of row and columns, we will

symbolically write Ci↔ Ri Property 2

If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes. We can denote the interchange of rows by Ri ↔ Rj and interchange of columns by Ci ↔ Cj.


Property 3 If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

Property 4

If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

o By this property, we can take out any common factor from any one row or any one column of a given determinant.

o If corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then its value is zero.

Property 5

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

Property 6

If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + k Cj.

If Δ1 is the determinant obtained by applying Ri → kRi or Ci → kCi to the determinant Δ, then Δ1

= kΔ. If more than one operation like Ri→ Ri + kRj is done in one step, care should be taken to see that

a row that is affected in one operation should not be used in another operation. A similar remark applies to column operations.

Area of triangle

Area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3), is given by the expression 1 1

2 2

3 3

11 12

1

x yx yx y

…………………… (1)

Since area is a positive quantity, we always take the absolute value of the determinant in (1). If area is given, use both positive and negative values of the determinant for calculation. The area of the triangle formed by three collinear points is zero.

Minors and Cofactors Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element a= is denoted by Mij.

Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n – 1. Cofactor of an element aij, denoted by Aij is defined by Aij = (–1)i + j Mij , where Mij is minor of aij. If elements of a row (or column) are multiplied with cofactors of any other row (or column), then

their sum is zero. Adjoint and Inverse of a Matrix

The adjoint of a square matrix A = [aij]n × n is defined as the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A.


For a square matrix of order 2, given by 11 12

21 22

a aa a

The adj A can also be obtained by interchanging a11 and a22 and by changing signs of a12 and a21, i.e.,

Theorem 1 If A be any given square matrix of order n, then A(adj A) = (adj A) A = |A| I, where I is

the identity matrix of order n A square matrix A is said to be singular if |A| = 0. A square matrix A is said to be non-singular if |A| ≠ 0 Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA are also

nonsingular matrices of the same order. Theorem 3 The determinant of the product of matrices is equal to product of their respective

determinants, that is, |AB| = |A| |B|, where A and B are square matrices of the same order If A is a square matrix of order n, then |adj(A)| = |A|n – 1. Theorem 4 A square matrix A is invertible if and only if A is nonsingular matrix.

Let A is a non-singular matrix then we have |A| ≠ 0 1 1A is invertible and

| |A adjA

A

Applications of Determinants and Matrices Application of determinants and matrices for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations:

Consistent system A system of equations is said to be consistent if its solution (one or more) exists. Inconsistent system A system of equations is said to be inconsistent if its solution does not exist. Solution of system of linear equations using inverse of a matrix

Consider the system of equations a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3

1 1 1 1

2 2 2 2

3 3 3 3

,a b c x d

Let A a b c X y and B da b c z d

Then, the system of equations can be written as, AX = B, i.e.,


1 1 1 1

2 2 2 2

3 3 3 3

a b c x da b c y da b c z d

Case I If A is a nonsingular matrix, then its inverse exists. Now AX = B

or A–1 (AX) = A–1 B (premultiplying by A–1) or (A–1A) X = A–1 B (by associative property) or I X = A–1 B or X = A–1 B

This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.

Case II

If A is a singular matrix, then |A| = 0. In this case, we calculate (adj A) B. If (adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent. If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.


CHAPTER – 5: CONTINUITY & DIFFERENTIABILITY

QUICK REVISION (Important Concepts & Formulae) Continuity at a Point : A function f (x) is said to be continuous at a point x = a of its domain, if and

only if lim

( ) ( )f x f ax a

.

Continuity on an open interval : A function f (x) is said to be continuous on an open interval (a, b) if and only if it is continuous at every point on the interval (a, b). Continuity on a closed interval : A function f (x) is said to be continuous on a closed interval [a, b] if and only if (i) f is continuous on the open interval (a, b).

(ii) lim

( ) ( )f x f ax a

(iii) lim

( ) ( )f x f ax a

In other words, f (x) is continuous on [a, b] if and only if it is continuous on (a, b) and it is continuous at a from the right and at b from the left. Continuous Function : A function f (x) is said to be continuous, if it is continuous at each point of its domain. Everywhere Continuous Function : A function f (x) is said to be everywhere continuous if it is continuous on the entire real line (–). Theorem Suppose f and g be two real functions continuous at a real number c. Then (1) f + g is continuous at x = c. (2) f – g is continuous at x = c. (3) f . g is continuous at x = c.

(4) fg

is continuous at x = c, (provided g (c) ≠ 0).

Discontinuous Functions A function f is said to be discontinuous at a point a of its domain D if it is not continuous at a. The point a is then called a point of discontinuity of the function. The discontinuity may arise due to any of the following situations:

lim

( )f xx a

or lim

( )f xx a

both may not exist.

lim

( )f xx a

as well as lim

( )f xx a

may exist, but are unequal.

lim

( )f xx a

as well as lim

( )f xx a

both may exist, but either of the two or both may not be equal

to f (a).


Removable Discontinuity : A function f is said to have removable discontinuity at x = a if lim lim

( ) ( )f x f xx a x a

but their common value is not equal to f (a).

Differentiability at a Point Let f (x) be a real valued function defined on an open interval (a, b) and let ( , )c a b . Then f (x) is said

to be differentiable or derivable at x = c, if and only if lim ( ) ( )f x f c

x c x c

exists finitely.

f (x) is differentiable at x = c '( ) '( ).Lf c Rf c If '( ) '( )Lf c Rf c , we say that f (x) is not

differentiable at x = c. f (x) is differentiable at point P, if and only if there exists a unique tangent at point P. In other words,

f (x) is differentiable at a point P if and only if the curve does not have P as a corner point. f (x) is differentiable at x = c f(x) is continuous at x = c. Differentiability in a Set A function f (x) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b) if it is differentiable at each point of (a, b). Some Standard Results on Differentiability

I. Every polynomial function is differentiable at each x R .

II. The exponential function ax , a > 0 is differentiable at each x R .

III. Every constant function is differentiable at each x R .

IV. The logarithmic function is differentiable at each point in its domain.

V. Trigonometric and inverse trigonometric functions are differentiable in their domains.

VI. The sum, difference, product and quotient of two differentiable functions is differentiable.

VII. The composition of differentiable functions is a differentiable function.

Derivative : The rate of change of a function with respect to the independent variable. For the function y

= f (x) it is denoted by dydx

Differentiation : The process of obtaining the derivative of a function by considering small changes in the function and independent variable, and finding the limiting value of the ratio of such changes.

lim lim( ) ( ) ( ) ( )( ( )) , ( ( ))d f x h f x d f x h f xf x f xx c x cdx h dx h

Geometrically Meaning of Derivative at a Point Geometrically derivative of a function at a point x = c is the slope of the tangent to the curve y = f (x) at the point (c, f (c)).

Slope of tangent at P = lim ( ) ( ) ( )

x c

f x f c df xx c x c dx


Differentiation of a constant function is zero i.e. ( ) 0d c

dx

Let f (x) be a differentiable function and let c be a constant. Then c.f (x) is also differentiable such

that . ( ) . ( )d dc f x c f xdx dx

That is the derivative of a constant times a function is the constant times the derivatives of the function.

If f (x) and g(x) are differentiable functions, then f (x) ± g(x) are also differentiable such that

[ ( ) ( )] ( ) ( )d d df x g x f x g xdx dx dx

That is the derivative of the sum or difference of two functions is the sum or difference of their derivatives.

Product Rule : If f (x) and g(x) are two differentiable functions, then f (x).g(x) is also differentiable such

that [ ( ). ( )] ( ) ( ) ( ) ( )d d df x g x f x g x g x f xdx dx dx

That is, derivative of the product of two functions = [(First function) × (derivative of second function) + (second function) × (derivative of first function)].

Quotient Rule : If f (x) and g(x) are two differentiable functions and g(x) 0, then ( )( )

f xg x

is also

differentiable such that 2

( ) ( ) ( ) ( )( )( ) ( )

d dg x f x f x g xd f x dx dxdx g x g x


Differentiation of Implicit Functions When it is not possible to express y as a function of x in the form of y = f(x), then y is said to be an implicit function of x. To find the derivative in such case we differentiate both sides of the given relation with respect of x. Differentiation of Logarithmic Functions y = f (x)g(x) = eg(x).logf (x) and then differentiating with respect to x, we may get

( )log ( ) 1( ). ( ) log ( ) ( )( )

g x f xdy d de g x f x f x g xdx f x dx dx

( ) ( )( ) ( ) log ( ) ( )( )

g x g x d df x f x f x g xf x dx dx

Differentiation of Parametric Functions When x and y are given as functions of a single variable, i.e., x = f(t), y = g(t) are two functions and t is a

variable. Then dy dy dtdx dx dt

.

Differentiation of a Function with respect to Another Function Let u = f (x) and v = g(x) be two functions of x. Then to find the derivative of f (x) w.r.t. g(x) i.e., to find dudv

we use the formula du du dxdv dv dx

Thus, to find the derivative of f (x) w.r.t. g(x), we first differentiate both w.r.t. x and then divide the derivative of f (x) w.r.t. x by the derivative of g(x) w.r.t. x. Rolle's Theorem Statement : Let f be a real valued function defined on the closed interval [a, b] such that (i) It is continuous on the closed interval [a, b] (ii) It is differentiable on the open interval (a, b) (iii) f (a) = f (b). Then there exists a real number ( , )c a b such that f ‘(c) = 0. Algebraic Interpretation of Rolle's Theorem : Between any two roots of a polynomial f (x), there is always a root of its derivative f ‘(x). Lagrange's Mean Value Theorem Statement : Let f (x) be a function defined on [a, b] such that (i) It is continuous on [a, b]. (ii) It is differentiable on (a, b). (iii) f (a) f (b)

Then there exists a real number ( , )c a b such that f ‘(c) = ( ) ( )f b f ab a

Geometrical Interpretation of Lagrange's Mean Value Theorem : Let f (x) be a function defined on [a, b] such that the curve y = f (x) is a continuous curve between points A(a, f (a)) and B(b, f (b)) and at every point on the curve, except at the end points, it is possible to draw a unique tangent. Then there exists a point on the curve such that the tangent at this is parallel to the chord joining the end points of the curve.


CHAPTER – 6: APPLICATION OF DERIVATIVES

QUICK REVISION (Important Concepts & Formulae) Rate of Change of Quantities If a quantity y varies with another quantity x, satisfying some rule y = f (x) , then f ′(x) represents the rate of change of y with respect to x and f ′(h) represents the rate of change of y with respect to x at x = h .

dydx

is positive if y increases as x increases and is negative if y decreases as x increases.

Strictly Increasing Function : A function f (x) is said to be a strictly increasing function on (a, b) if

1 2 1 2( ) ( )x x f x f x for all 1 2, ( , )x x a b . Strictly Decreasing Function : A function f (x) is said to be a strictly decreasing function on (a, b) if

1 2 1 2( ) ( )x x f x f x for all 1 2, ( , )x x a b . Monotonic Function : A function f (x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing on (a, b). A function f (x) is said to be increasing (decreasing) at point x0 if there is an interval (x0 – h, x0 + h ) containing x0 such that f (x) is increasing (decreasing) on (x0 – h, x0 + h). A function f (x) is said to be increasing on [a, b] if it is increasing (decreasing) on (a, b) and it is also increasing at x = a and x = b. If f (x) is increasing function on (a, b), then tangent at every point on the curve y = f (x) makes an acute angle q with the positive direction of x-axis. Let f be a differentiable real function defined on an open interval (a, b). (a) If f ‘(x) > 0 for all ( , )x a b , then f (x) is increasing on (a, b). (b) If f'(x) < 0 for all ( , )x a b , then f (x) is decreasing on (a, b). Let f be a function defined on (a, b). (a) If f'(x) > 0 for all ( , )x a b except for a finite number of points, where f’(x) = 0, then f (x) is increasing on (a, b). (b) If f'(x) < 0 for all ( , )x a b except for a finite number of points, where f’(x) = 0, then f (x) is decreasing on (a, b). Slope of Tangent

If a tangent line to the curve y = f (x) makes an angle θ with x-axis in the positive direction, then dydx

=

slope of the tangent tan θ . Algorithm for Finding The Equation of Tangent and Normal to The Curve y = f (x) at the given Point (x0 , y0)

Step I : Find dydx

from the given equation y = f (x).

Step II : Find the value of dydx

at the given point P(x0 , y0).


Step III : The equation of the tangent at (x0, y0) to the curve y = f (x) is given by

0 0

0 0( , )

( )x y

dyy y x xdx

The equation of the normal to the curve y = f (x) at a point (x0 , y0) is given by

0 0

0 0

( , )

1 ( )

x y

y y x xdydx

If dydx

does not exist at the point (x0 , y0) , then the tangent at this point is parallel to the y-axis and its

equation is x = x0.

If tangent to a curve y = f (x) at x = x0 is parallel to x-axis, then 0

0x x

dydx

If dydx

at the point (x0 , y0) is zero, then equation of the normal is x = x0.

If dydx

at the point (x0 , y0) does not exist, then the normal is parallel to x-axis and its equation is y =

y0. Particular cases (i) If slope of the tangent line is zero, then tan θ = 0 and so θ = 0 which means the tangent line is parallel to the x-axis. In this case, the equation of the tangent at the point (x0, y0) is given by y = y0.

(ii) If 2 , then tan θ→∞, which means the tangent line is perpendicular to the x-axis, i.e., parallel to

the y-axis. In this case, the equation of the tangent at (x0, y0) is given by x = x0 Angle of Intersection of Two Curves The angle of intersection of two curves is defined to be the angle between the tangents to the two curves at their point of intersection. The other angle between the tangents is 180° – . Generally the smaller of these two angles is taken to be the angle of intersection. Orthogonal Curves If the angle of intersection of two curves is a right angle, the two curves are said to intersect orthogonally and the curves are called orthogonal curves. Approximation Let y = f (x), Δx be a small increment in x and Δy be the increment in y corresponding to the increment in x, i.e., Δy = f (x + Δx) – f (x). Then dy given by

'( )dy f x dx or dydy dxdx

is a good approximation of Δy when dx = Δx is relatively small and we

denote it by dy ≈ Δy. Maximum Let f (x) be a function with domain D R . Then f (x) is said to attain the maximum value at a point a D , if f (x) f (a) for all x D . In such a case, a is called point of maxima and f (a) is known as the maximum value or the greatest value or the absolute maximum value of f (x).


Minimum Let f (x) be a function with domain D R . Then f (x) is said to attain the minimum value at a point a D , if f (x) f (a) for all x D In such a case, a is called point of minima and f (a) is known as the minimum value or the least value or the absolute minimum value of f (x). Local Maximum : A function f (x) is said to attain a local maximum at x = a if there exists a neighbourhood ( , )a a of a such that, f (x) < f (a) for all ( , )x a a , x a or, f (x) – f (a) < 0 for all ( , )x a a ,x a. In such a case f (a) is called to attain a local maximum value of f (x) at x = a. Local Minimum : f (x) > f (a) for all ( , )x a a , x a or f (x) – f (a) > 0 for all ( , )x a a , x a. In such a case f(a) is called the local minimum value of f (x) at x = a.

If c is a point of local maxima of f , then f (c) is a local maximum value of f. Similarly, if c is a point of local minima of f , then f(c) is a local minimum value of f.

A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f. Note that if f is continuous at c and f ′(c) = 0, then there exists an h > 0 such that f is differentiable in the interval (c – h, c + h).

First Derivative Test for Local Maxima and Minima Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then (i) If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point

sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

(ii) If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

(iii)If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion.

Algorithm for Determining Extreme Values of a Function by using First Derivative Test Step I : Put y = f (x)

Step II : Find dydx

.

Step III : Put dydx

= 0 and solve this equation for x. Let c1 , c2 , c3 ,….be the roots of the equation. c1 , c2

, c3 ,…. are stationary values of x and these are the possible points where the function can attain a local maximum or a local minimum. So we test the function at each of these points.

Step IV : Consider x = c1 , if dydx

changes its sign from positive to negative as x increases through c 1 ,

then the function attains a local maximum at x = c1.

If dydx

changes its sign from negative to positive as x increases through c 1 , then the function attains

a local minimum at x = c1.

If dydx

does not change sign as x increase through c1 , then x = c1 is neither a points of local

maximum nor a point of local minimum. In this case x = c1 is a point inflexion.


Second Order Derivative Test : Let f be a function defined on an interval I and c I. Let f be twice differentiable at c. Then (i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0 The values f (c) is local maximum value of

f. (ii) x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0 In this case, f (c) is local minimum value of

f . (iii)The test fails if f ′(c) = 0 and f ″(c) = 0. In this case, we go back to the first derivative test and find

whether c is a point of maxima, minima or a point of inflexion. Algorithm for Determining Values of a Function by Using Second Derivative Test : From the I st derivative test criteria we obtain the following rule for determining maximum and minimum of f (x) : Step I : Find f ‘(x) Step III : Put f ‘(x) = 0 and solve this equation for x. Let c1 , c2 , c3 ,….be the roots of the equation. c1 , c2 , c3 ,…. are stationary values of x and these are the possible points where the function can attain a local maximum or a local minimum. So we test the function at each of these points. Step III : Find f ‘(x). Consider x = c1 if f (c1) < 0, then x = c1 is point of local maximum. if f ( c1) > 0, then x = c1 is point of local minimum. if f (c1) = 0, then use first order derivative test.


CHAPTER – 7: INTEGRALS

QUICK REVISION (Important Concepts & Formulae)

Integration is the inverse process of differentiation. Let ( ) ( )d F x f xdx

. Then we write

( ) ( )f x dx F x C . These integrals are called indefinite integrals or general integrals, C is called constant of integration. All these integrals differ by a constant.


Some Standard Results on Integration

(i) ( ) ( )d f x dx f xdx

= i.e. the differentiation of an integral is the integrand itself or differentiation

and integration are inverse operations. (ii) ( ) ( )kf x dx k f x dx , where k is a constant i.e., the integral of the product of a constant and a function = the constant integral of the function. (iii) ( ) ( ) ( ) ( )f x g x dx f x dx g x dx i.e., the integral of the sum or difference of a finite number of functions is equal to the sum or difference of the integrals of the various functions.


Definite Integrals

A definite integral is denoted by ( )b

a

f x dx , where a is called the lower limit of the integral and b

is called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the interval [a, b], then its value is the difference between the values of F at the end points, i.e., F(b) – F(a). Definite Integrals as the limit of a sum


Fundamental Theorem of Calculus First fundamental theorem of integral calculus Theorem 1 Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function. Then A′(x) = f (x), for all x [a, b]. Second fundamental theorem of integral calculus Theorem 2 Let f be continuous function defined on the closed interval [a, b] and F be an anti

derivative of f. Then ( ) [ ( )] ( ) ( )b

ba

a

f x dx F x F b F a

Properties of Definite Integrals


Evaluation of Definite Integrals by Substitution

To evaluate ( )b

a

f x dx , by substitution, the steps could be as follows:

1. Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce the given integral to a known form.

2. Integrate the new integrand with respect to the new variable without mentioning the constant of integration.

3. Resubstitute for the new variable and write the answer in terms of the original variable. 4. Find the values of answers obtained in (3) at the given limits of integral and find the

difference of the values at the upper and lower limits.


CHAPTER – 8: APPLICATION OF INTEGRALS

QUICK REVISION (Important Concepts & Formulae)


CHAPTER – 9: DIFFERENTIAL EQUATIONS

QUICK REVISION (Important Concepts & Formulae) Differential Equations An equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable is called a differential equation. Order of a Differential Equation The order of a differential equation is the order of the highest order derivative appearing in the equation. The order of a differential equation is a positive integer. Degree of a Differential Equation The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions.

If a differential equation, when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product of these, and also the coefficient of the various terms are either constants or functions of the independent variable, then it is said to be linear differential equation. Otherwise, is a non linear differential equation.

Solution of a Differential Equation The solution of a differential equation is a relation between the variables involved which satisfies the differential equation. Such a relation and the derivatives obtained from there, when substituted in the differential equation, makes left hand, and right hand sides identically equal. General Solution The solution which contains as many as arbitrary constants as the order of the differential equation is called the general solution of the differential equation. Particular Solution Solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equation is called a particular solution. Formulating a differential equation from a given equation representing a family of curves means finding a differential equation whose solution is the given equation. If an equation representing a family of curves, contains n arbitrary constants, then we differentiate the given equation n times to obtain n more equations. Using all these equations, we eliminate the constants. The equation so obtained is the differential equation of order n for the family of given curves.


CHAPTER – 10: VECTOR ALGEBRA

MARKS WEIGHTAGE – 06 marks

QUICK REVISION (Important Concepts & Formulae) Vector

The line l to the line segment AB, then a magnitude is prescribed on the line l with one of the two directions, so that we obtain a directed line segment. Thus, a directed line segment has magnitude as well as direction.

A quantity that has magnitude as well as direction is called a vector. A directed line segment is a vector, denoted as AB

or simply as | |a

, and read as ‘vector AB

’ or

‘vector | |a

’. The point A from where the vector AB

starts is called its initial point, and the point B where it ends

is called its terminal point. The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as | |AB

or | |a

. The arrow indicates the direction of the

vector. The vector OP

having O and P as its initial and terminal points, respectively, is called the position

vector of the point P with respect to O. Using distance formula, the magnitude of vector OP

is given by 2 2 2| |OP x y z

The position vectors of points A, B, C, etc., with respect to the origin O are denoted by

,a b and c

, etc., respectively Direction Cosines

Consider the position vector ( )OP or r

of a point P(x, y, z) in below figure. The angles α, β, γ made by the vector r

with the positive directions of x, y and z-axes respectively, are called its direction

angles. The cosine values of these angles, i.e., cosα, cosβ and cos γ are called direction cosines of the vector r

, and usually denoted by l, m and n, respectively.


The triangle OAP is right angled, and in it, we have cos xr

(r stands for | r

|). Similarly, from the

right angled triangles OBP and OCP, we may write cos yr

and cos zr

. Thus, the coordinates

of the point P may also be expressed as (lr, mr, nr). The numbers lr, mr and nr, proportional to the direction cosines are called as direction ratios of vector r

, and denoted as a, b and c, respectively.

l2 + m2 + n2 = 1 but a2 + b2 + c2 ≠ 1, in general. Types of Vectors Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null

vector), and denoted as 0

. Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors

,AA BB

represent the zero vector, Unit Vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector

in the direction of a given vector a

is denoted by a Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors. Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line,

irrespective of their magnitudes and directions.


Equal Vectors Two vectors a and b

are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as a b

.

Negative of a Vector A vector whose magnitude is the same as that of a given vector (say, AB

), but direction is opposite to that of it, is called negative of the given vector. For example, vector BA

is negative of the vector AB

, and written as BA

= − AB

.

The vectors defined above are such that any of them may be subject to its parallel displacement

without changing its magnitude and direction. Such vectors are called free vectors. Addition of Vectors Triangle law of vector addition

If two vectors a and b

are represented (in magnitude and direction) by two sides of a triangle taken in order, then their sum (resultant) is represented by the third side c

( a b

) taken in the

opposite order.

Subtraction of Vectors : To subtract b

from a

, reverse the direction of b

and add to a

. Geometrical Representation of Addition and Subtraction :

Parallelogram law of vector addition

If we have two vectors a and b

represented by the two adjacent sides of a parallelogram in magnitude and direction, then their sum a b

is represented in magnitude and direction by the

diagonal of the parallelogram through their common point. This is known as the parallelogram law of vector addition.

Properties of vector addition Property 1

For any two vectors a and b

, a b b a

(Commutative property) Property 2

For any three vectors ,a b and c

, a b c a b c

(Associative property)

Property 3


For any vector a

, we have 0 0a a a

, Here, the zero vector 0

is called the additive identity for the vector addition.

Property 4

For any vector a

, we have 0a a a a

Here, the vector a

is called the additive inverse for the vector addition. Multiplication of a Vector by a Scalar

Let a

be a given vector and λ a scalar. Then the product of the vector a

by the scalar λ, denoted as λ a

, is called the multiplication of vector a

by the scalar λ. Note that, λ a

is also a vector, collinear to the vector a

. The vector λ a

has the direction same (or opposite) to that of vector a

according as

the value of λ is positive (or negative). Also, the magnitude of vector λ a

is |λ| times the magnitude of the vector a

, i.e., | λ a

| = | λ | | a

|

Unit vector in the direction of vector a

is given by 1 .| |

a aa

Properties of Multiplication of Vectors by a Scalar For vectors a

, b

and scalars m, n, we have (i) m ( a

) = (–m) a

= – (m a

)

(ii) (–m) ( a

) = m a

(iii) m(n a

) = (mn) a

= n(m a

)

(iv) (m + n) a

= m a

+ n a

(v) m( a

+ b

) = m a

+ mb

. Vector joining two points

If P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2 is the vector

1 2PP

.

1 2PP

= Position vector of head – Position vector of tail = 2 1OP OP

= 2 1 2 1 2 1( ) ( ) ( )x x i y y j z z k

The magnitude of vector 1 2PP

is given by 2 2 22 1 2 1 2 1( ) ( ) ( )x x y y z z

SECTION FORMULA Internal Division : Position vector of a point C dividing a vector AB

internally in the ratio of m : n

is OC

= mb nam n


If C is the midpoint of AB

, then OC

divides AB

in the ratio 1 : 1. Therefore, position vector of C

is 1. 1.1 1 2b a a b

The position vector of the midpoint of AB

is 2

a b

Position vector of any point C on AB

can always be taken as c b a

where + = 1. . . ( ).n OA m OB n m OC

, where C is a point on AB

dividing it in the ratio m : n.

External Division : Let A and B be two points with position vectors a

and b

respectively and let C

be a point dividing AB

externally in the ratio m : n. Then the position vector of C is given by

OC

= mb nam n

Two vectors a

and b

are collinear if and only if there exists a nonzero scalar λ such that b

= λa. If

the vectors a

and b

are given in the component form, i.e. 1 2 3a a i a j a k

and

1 2 3b b i b j b k , then the two vectors are collinear if and only if

1 2 3 1 2 3b i b j b k a i a j a k

31 2

1 2 3

bb ba a a

If

1 2 3a a i a j a k , then a1, a2, a3 are also called direction ratios of a

.

In case if it is given that l, m, n are direction cosines of a vector, then li m j nk (cos ) (cos ) (cos )i j k is the unit vector in the direction of that vector, where α, β and γ are

the angles which the vector makes with x, y and z axes respectively. Product of Two Vectors Scalar (or dot) product of two vectors

The scalar product of two nonzero vectors a

and b

, denoted by .a b

, is defined as . | || | cosa b a b

where, θ is the angle between a

and b

, 0 ≤ θ ≤ π If either 0a

or 0b

, then θ is not defined, and in this case, we define . 0a b

Observations .a b

is a real number.

Let a

and b

be two nonzero vectors, then . 0a b

if and only if a

and b

are perpendicular to each other. i.e.

. 0a b a b

If θ = 0, then . | || |a b a b

. In particular, 2. | |a a a

, as θ in this case is 0.

If θ = π, then . | || |a b a b

In particular, 2.( ) | |a a a

, as θ in this case is π.

In view of the Observations 2 and 3, for mutually perpendicular unit vectors ,i j and k , we have . . . 1i i j j k k

. . . 0i j j k k i


The angle between two nonzero vectors a

and b

, is given by

1. .cos , cos| || | | || |

a b a bora b a b

The scalar product is commutative. i.e. . .a b b a

Property 1 (Distributivity of scalar product over addition) Let ,a b and c

be any three vectors,

then .( ) . .a b c a b a c

Property 2 Let a

and b

be any two vectors, and be any scalar. Then ( ). ( . ) .( )a b a b a b

Projection of a vector on a line If p is the unit vector along a line l, then the projection of a vector a

on the line l is given by .a p

.

Projection of a vector a

on other vector b

, is given by 1. . ( . )| | | |ba b or a or a bb b

If θ = 0, then the projection vector of AB

will be AB

itself and if θ = π, then the projection vector of AB

will be BA

If 2 or 3

2 , then the projection vector of AB

will be zero vector.

If α, β and γ are the direction angles of vector 1 2 3a a i a j a k

, then its direction cosines may be

given as 31 2.cos ,cos cos| | | | | || || |

aa aa ia a aa i

Vector (or cross) product of two vectors The vector product of two nonzero vectors a

and b

, is denoted by a

× b

and defined as | || | sina b a b n

, where, θ is the angle between a

and b

, 0 ≤ θ ≤ π and n is a unit vector

perpendicular to both a

and b

, such that a

, b

and n form a right handed system. If either 0a

or 0b

, then θ is not defined and in this case, we define 0a b

. Observations a

× b

is a vector. Let a

and b

be two nonzero vectors. Then 0a b

if and only if and a

and b

are parallel (or

collinear) to each other, i.e., 0 ||a b a b

In particular, 0a a

and ( ) 0a a

, since in the first situation, θ = 0 and in the second one, θ = π, making the value of sin θ to be 0.

If 2 then | || |a b a b

In view of the Observations 2 and 3, for mutually perpendicular unit vectors ,i j and k , we have 0i i j j k k

, ,i j k j k i k i j In terms of vector product, the angle between two vectors a

and b

may be given as

| |sin| || |a ba b


The vector product is not commutative, as a b b a

In view of the above observations, we have , ,j i k k j i i k j

If a

and b

represent the adjacent sides of a triangle then its area is given as 1 | |2

a b

.

If a

and b

represent the adjacent sides of a parallelogram, then its area is given by | |a b

.

The area of a parallelogram with diagonals a

and b

is 1 | |2

a b

| |a ba b

is a unit vector perpendicular to the plane of a

and b

.

| |a ba b

is also a unit vector perpendicular to the plane of a

and b

.

Property 3 (Distributivity of vector product over addition): If ,a b and c

are any three vectors

and λ be a scalar, then (i) ( )a b c a b a c

(ii) ( ) ( ) ( )a b a b a b

Let a

and b

be two vectors given in component form as

1 2 3a a i a j a k and

1 2 3b b i b j b k ,

respectively. Then their cross product may be given by

1 2 3

1 2 3

i j ka b a a a

b b b


CHAPTER – 11: THREE DIMENSIONAL GEOMETRY

QUICK REVISION (Important Concepts & Formulas) Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.


CHAPTER – 12: LINEAR PROGRAMMING

QUICK REVISION (Important Concepts & Formulae) A half-plane in the xy-plane is called a closed half-plane if the points on the line separating the half-

plane are also included in the half-plane. The graph of a linear inequation involving sign ‘’or ‘’is a closed half-plane. A half-plane in the xy-plane is called an open half-plane if the points on the line separating the half-

plane are not included in the half-plane. The graph of linear inequation involving sign ‘<‘or ‘>’ is an open half-plane. Two or more linear inequations are said to constitute a system of linear inequations. The solution set of a system of linear inequations is defined as the intersection of solution sets of

linear inequations in the system. A linear inequation is also called a linear constraint as it restricts the freedom of choice of the

values x and y. LINEAR PROGRAMMING In linear programming we deal with the optimization (maximization or minimization) of a linear function of a number of variables subject to a number of restrictions (or constraints) on variables, in the form of linear inequations in the variable of the optimization function. A Linear Programming Problem is one that is concerned with finding the optimal value (maximum

or minimum value) of a linear function (called objective function) of several variables (say x and y), subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints).

The term linear implies that all the mathematical relations used in the problem are linear relations

while the term programming refers to the method of determining a particular programme or plan of action.

Objective function Linear function Z = ax + by, where a, b are constants, which has to be

maximised or minimized is called a linear objective function. Variables x and y are called decision variables.

Constraints The linear inequalities or equations or restrictions on the variables of a linear

programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative restrictions.

Optimisation problem A problem which seeks to maximise or minimise a linear function (say of

two variables x and y) subject to certain constraints as determined by a set of linear inequalities is called an optimisation problem. Linear programming problems are special type of optimisation problems.


GRAPHICAL METHOD OF SOLVING LINEAR PROGRAMMING PROBLEMS Feasible region The common region determined by all the constraints including non-negative

constraints x, y ≥ 0 of a linear programming problem is called the feasible region (or solution region) for the problem. The region other than feasible region is called an infeasible region.

Feasible solutions Points within and on the boundary of the feasible region represent feasible

solutions of the constraints. Any point outside the feasible region is called an infeasible solution. Optimal (feasible) solution: Any point in the feasible region that gives the optimal value (maximum

or minimum) of the objective function is called an optimal solution. Theorem 1 Let R be the feasible region (convex polygon) for a linear programming problem and let

Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.

A corner point of a feasible region is a point in the region which is the intersection of two boundary

lines. Theorem 2 Let R be the feasible region for a linear programming problem, and let Z = ax + by be

the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.

A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within

a circle. Otherwise, it is called unbounded. Unbounded means that the feasible region does extend indefinitely in any direction.

If R is unbounded, then a maximum or a minimum value of the objective function may not exist.

However, if it exists, it must occur at a corner point of R. (By Theorem 1). The method of solving linear programming problem is referred as Corner Point Method. The method

comprises of the following steps:

1. Find the feasible region of the linear programming problem and determine its corner points (vertices)

either by inspection or by solving the two equations of the lines intersecting at that point.

2. Evaluate the objective function Z = ax + by at each corner point. Let M and m, respectively denote

the largest and smallest values of these points.

3. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.

(ii) In case, the feasible region is unbounded, we have:

4. (a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in

common with the feasible region. Otherwise, Z has no maximum value.

(b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no

point in common with the feasible region. Otherwise, Z has no minimum value.


DIFFERENT TYPES OF LINEAR PROGRAMMING PROBLEMS A few important linear programming problems are listed below:

1. Manufacturing problems In these problems, we determine the number of units of different

products which should be produced and sold by a firm when each product requires a fixed

manpower, machine hours, labour hour per unit of product, warehouse space per unit of the output

etc., in order to make maximum profit.

2. Diet problems In these problems, we determine the amount of different kinds of

constituents/nutrients which should be included in a diet so as to minimise the cost of the desired

diet such that it contains a certain minimum amount of each constituent/nutrients.

3. Transportation problems In these problems, we determine a transportation schedule in order to

find the cheapest way of transporting a product from plants/factories situated at different locations to

different markets.

SOLUTION OF LINEAR PROGRAMMING PROBLEMS There are two types of linear programming problems-

1. When linear constraints and objective functions are given.

2. When linear constraints and objective functions are not given.

1. When linear constraints and objective functions are given.

Working Rule

(i) Consider the linear equations of their corresponding linear inequations.

(ii) Draw the graph of each linear equation.

(iii) Check the solution region of each linear inequations by testing the points and then shade the

common region of all the linear inequations.

(iv) Determine the corner points of the feasible region.

(v) Find the value of objective function at each of the corner points obtained in above step.

(vi) The maximum or minimum value out of all the values obtained in above step is the maximum

or minimum value of the objective function.

2. When linear constraints and objective functions are not given.

Working Rule

(i) Identify the unknown variables in the given Linear programming problems. Denote them by

x and y.

(ii) Formulate the objective function in terms of x and y. Also, observe it is maximized or

minimized.


(iii) Write the linear constraints in the form of linear inequations formed by the given conditions.

(iv) Consider the linear equations of their corresponding linear inequations.

(v) Draw the graph of each linear equation.

(vi) Check the solution region of each linear inequations by testing the points and then shade the

common region of all the linear inequations.

(vii) Determine the corner points of the feasible region.

(viii) Evaluate the value of objective function at each corner points obtained in the above step.

(ix) As the feasible region is unbounded, therefore the value may or may not be minimum or

maximum value of the objective function. For this draw a graph of the inequality by equating

the objective function with the above value to form linear inequation i.e. < for minimum or >

for maximum. And check whether the resulting half plane has points in common with the

feasible region or not.


CHAPTER – 13: PROBABILITY

QUICK REVISION (Important Concepts & Formulae) Trial and Elementary Events : Let a random experiment be repeated under identical conditions.

Then the experiment is called a trial and the possible outcomes of the experiment are known as elementary events or cases.

Elementary events are also known as indecomposable events. Decomposable Events/Compound Events : Events obtained by combining together two or more

elementary events are known as the compound events or decomposable events. Exhaustive Number of Cases : The total number of possible outcomes of a random experiment in a

trial is known as the exhaustive number of cases. The total number of elementary events of a random experiment is called the exhaustive number of

cases. Mutually Exclusive Events : Events are said to be mutually exclusive or incompatible if the

occurrence of anyone of them prevents the occurrence of all the others, i.e., if no two or more of them can occur simultaneously in the same trial.

Equally Likely Events : Events are equally likely if there is no reason for an event to occur in

preference to any other events. The number of cases favourable to an events in a trial is the total number of elementary events such

that the occurrence of any one of them ensures the happening of the event. Independent Events : Events are said to be independent if the happening (or non-happening) of one

event is not affected by the happening (or non-happening) of others. Classical Definition of Probability of An Event : If there are n elementary events associated with a

random experiment and m of them are favourable to an event A, then the probability of happening of

A is denoted by P(A) and is defined as the ratio mn

.

( ) mP An

0 P(A) 1. A denotes not happening of A P( A ) = 1– P(A) P(A) + P(A) = 1 If P(A) = 1, then A is called certain event and A is called an impossible event if P(A) = 0.

The odds in favour of occurrence of the event A are defined by mn m

i.e., P(A) : P( A ) and the odds

against the occurrence of A are defined by n mm , i.e., P( A ) : P(A).


Sample Space : The set of all possible outcomes of a random experiment is called the sample space

associated with it and it is generally denoted by S. If E1 , E2 , ..., En are the possible outcomes of a random experiment, then S = E1 , E2 , ..., En . Each

element of S is called a sample point. Event : A subset of the sample space associated with a random experiment is called an event. Elementary Events : Single element subsets of the sample space associated with a random

experiment are known as the elementary events or indecomposable events. Compound Events : Those subsets of the sample space S associated to an experiment which are

disjoint union of single element subsets of the sample space S are known as the compound or decomposable events.

Impossible and Certain Event : Let S be the sample space associated with a random experiment.

Then f and S, being subsets of S are events.The event f is called an impossible event and the event S is known as a certain event.

Occurence or Happening of An Event : Let S be the sample space associated with a random

experiment and let A be an event. If w is an outcome of a trial such that w A, then we say that the event A has occurred. If w A , we say that the event A has not occured.

Algebra of Events

Mutually Exclusive Events : Let S be the sample space associated with a random experiment and

let A1 and A2 be two events.Then A1 and A2 are mutually exclusive events if A1 ∩ A2 = . Mutually Exclusive and Exhaustive System of Events : Let S be the sample space associated with

a random experiment. Let A1 , A2 , ..., An be subsets of S such that (i) Ai ∩ Aj = for i j, and (ii) A1 A2 ... An = S.

If E1 , E2 , ..., En are elementary events associated with a random experiment. Then (i) E i ∩ E j = f for i j and (ii) E1 E2 ….. En = S.

Favourable Events : Let S be the sample space associated with a random experiment and let AS. Then the elementary events belonging to A are known as the favourable events to A.


Experimentally Probability : Let S be the sample space associated with a random experiment, and let A be a subset of S representing an event. Then the probability of the event A is defined as

Number of elements in A ( )( )Number of elements in S ( )

n AP An s

P() = 0, P (S) = 1. Addition theorem for two events : If A and B are two events associated with a random experiment,

then P(A B) = P (A) + P(B) – P(A ∩ B). If A and B are mutually exclusive events, then P(A ∩ B) = 0, therefore P(A B) = P(A) + P(B).

This is the addition theorem for mutually exclusive events. Addition theorem for three events : If A, B, C are three events associated with a random

experiment then, 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, C are mutually exclusive events, then P (A ∩ B) = P (B ∩C) = P (A∩C) = P(A∩B∩C) = 0. P(A B C) = P(A) + P(B) + P(C)

Let A and B be two events associated with a random experiment.Then

(i) P( A ∩ B) = P(B) – P(A ∩ B) (ii) P(A ∩ B ) = P(A) – P(A ∩ B ) P ( A ∩ B) is known as the probability of occurence of B only. P(A ∩ B ) is known as the probability of occurence of A only. If B A, then (i) P(A B ) = P(A) – P(B) (ii) P(B) P(A). Conditional Probability

P (A/B) = Probability of occurrence of A given that B has already happened. P (B/A) = Probability of occurrence of B given that A has already happened.

Properties of conditional probability Property 1 P (S/F) = P(F/F) = 1 Property 2 If A and B are any two events of a sample space S and F is an event of S such that

P(F) 0, then P((A B)/F) = P(A/F) + P(B/F) – P ((A B)/F) Property 3 P(E’/F) = 1 P(E/F) Multiplication Theorem : If A and B are two events, then

P(A ∩ B) = P(A)P(B / A), if P(A) 0 P(A ∩ B) = P(B)P(A/ B), if P(B ) 0.

( ) ( )( / ) ( / )( ) ( )

P A B P A BP B A and P A BP A P B

Extension of multiplication theorem : If A1, A2 ,..., An are n events related to a random experiment, then

321 2 3 1

1 1 2 1 2 1

( .... ) ( ) ..........n

nn

A AAP A A A A P A P P PA A A A A A

where P(Ai / A1 A2 ... A i–1 ) represents the conditional probability of the event Ai , given that the events A1 , A2 , ..., Ai–1 have already happened.


Independent Events : Event are said to be independent, if the occurrence or nonoccurrence of one does not affect the probability of the occurrence or nonoccurrence of the other.

If A and B are two independent events associated with a random experiment then, P(A/B) = P(A) and P(B/A) = P(B) and viceversa. If A and B are independent events associated with a random experiment, then P(A B) = P(A) P(B)

i.e., the probability of simultaneous occurrence of two independent events is equal to the product of their probabilities.

If A1 , A2 , ..., An are independent events associated with a random experiment,

then P(A1 A2 A3 ... An) = P (A1) P (A2) ... P(An) If A1 , A2 , ..., An are n independent events associated with a random experiment, then

1 2 3 1 2( .... ) ( ) ( ).... ( )n nP A A A A P A P A P A Events A1, A2 ,..., An are independent or mutually independent if the probability of the simultaneous

occurence of (any) finite number of them is equal to the product of their separate probabilities while these events are pair wise independent if P(Ai Aj) = P(Aj ) P(Ai ) for all i j.

The Law of Total Probability : Let S be the sample space and let E1, E2 , ..., En be n mutually

exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or ... or En , then

1 21 2

( ) ( ) ( ) ......... ( )nn

A A AP A P E P P E P P E PE E E

Baye's Rule : Let S be the sample space and let E1 , E2 , ..., En be n mutually exclusive and

exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or ... or En , then

1

( ) ( / ) , 1, 2,...( ) ( / )

i i in

i ii

E P E P A EP i nA P E P A E

The events E1 , E2 , ..., En are usually referred to as 'hypothesis' and the probabilities P(E1), P(E2), ..., P(En) are known as the 'priori' probabilities as they exist before we obtain any information from the experiment.

The probabilities P(A/Ei); i = 1, 2, ..., n are called the likelihood probabilities as they tell us how

likely the event A under consideration occur, given each and every priori probabilities. The probabilities P (Ei /A); i = 1, 2, ..., n are called the posterior probabilities as they are determined

after the result of the experiment are known. Random Variable : A random variable is a real valued function having domain as the sample space

associated with a given random experiment. A random variable associated with a given random experiment associates every event to a unique

real number.

Probability Distribution : If a random variable X takes values x1 , x2 , ..., xn with respective, probabilities p1 , p2 , ..., pn , then

X : x1 x2 ... xn


P(X) : p1 p2 ... pn is called the probability distribution of X.

Binomial Distribution : A random variable X which takes values 0, 1, 2, ..., n is said to follow

binomial distribution if its probability distribution function is given by ( ) n r n rrP X r C p q , r = 0,

1, 2, ..., n where p, q > 0 such that p + q = 1. If n trials constitute an experiment and the experiment is repeated N times, then the frequencies of 0,

1, 2, ..., n successes are given by N×P(X = 0), N×P (X=1), N×P (X=2), ..., N×P (X=n).

If X is a random variable with probability distribution

X : x1 x2 ... xn P(X) : p1 p2 ... pn

then the mean or expectation of X is defined as 1

( )n

i ii

X E X p x

and the variance of X is defined

as and the variance of X is defined as

2 22

1 1( ) ( ) ( )

n n

i i i ii i

Var X p x E X p x E X

or Var (X) = E (X2) – [E(X)]2

The mean of the binomial variate X ~ B (n, p) is n p. The variance of the binomial variate X ~ B (n, p) is npq, where p + q =1 The standard deviation of a binomial variate X ~ B (n, p) is ( )Var X npq

mean > variance. Maximum Value of P(X = r) for given Values of n and p for a Binomial Variate X If (n + 1) p is an integer, say m, then ( ) (1 )n r n r

rP X r C p p is maximum when r = m or r = m – 1.

If (n + 1)p is not an integer, then P(X = r) is maximum when r = [(n + 1)p].

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