NDA Coaching Chandigarh

Sets , relation & Functions Theory

Set, Relation and FunctionSETS

In Mathematical Language all loving and non living things in universe are known as objects.

A collection of well defined objects is known as a set, Generally sets are denoted by A, B, C.and its elements are denoted by a, b, c..

Let A is a non empty set. If x is an element of A , then we write x A and read as x is an element of A or x belongs to A. If x is not an element of A, then we write , x A and read as x is not an element of A or x does not belong to A.

REPRESENTATION OF SETS

There are two methods to represent a set

1. Listing method

2. Set builder Method

Listing Method : In this method, we make a list of the elements of the set and put it within braces().

This method is also known as Tabular method or Roster method

e.g. A = Set of vowels of English alphabet

= (a, e, t, o, u)

Set Builder Method: In this method , we list the property or properties, satisfied by the elements of set.

A= [x : p(x)]Where P(x) is a property of set which is satisfied by each elements of set.

e.g., A = [1,2,3,4] = [x : x N and x < 5]

This method is known as Rule method or Property method

TYPES OF SETS

Empty Set

A set which has no element is called an empty set. It is denoted by or []

e.g. A= [x : x N and 3 < x < 4] =

Such sets which have at least one element are called non void set.

Singleton Set

A set which has only one element is called a singleton set.

e.g, A = [x : x N and 3 < x < 5]Finite set

A set is which the process of counting of elements surely comes to an end, is called a finite set

e.g, [x : x N, x < 5 ] is a finite set.Cardinal Number of a Finite Set

The number of elements in a finite set is called cardinal number of a finite set. It is denoted by 0 (A) or n (A).

e.g. If A = (1,2 ,3, 4,5100] is finite set, then n (A) = 100Infinite Set

A set in which the process of counting of elements does not come to an end, is called infinite set.

e.g, A = Set of all points on a plane

and B= Set of students present in a class

Set A is infinite set while B is a finite set.Equivalent Sets

If cardinal number of two finite sets A and B are equal i.e, n (A) = n (B) then both sets will be equivalent sets.

Equal Sets

If A and B are two sets and each elements of A is an element of set B and each element of set B is an element of set A, then sets A and B will be equal sets.SubsetLet A and B are two non empty sets. If each elements of A is an element of B, then set A is known as subset of set B. It is denoted by A B and read as A is subset of B

If x A and A B, then

= x B

If A is not a subset of B, then we write as A B and read as A is not a subset of B.

Proper Subset

Set A is known as proper subset of set B, if each element of A is set B but set B has at least one element which is not in A. It is denoted by A B and read as set is a proper subset of BSuperset

If set A is a subset of set B, then set B is known as superset of set A and is denoted by bA.

Universal Set

A superset of each set is known as universal set. It is denoted by U.

Power set

Let A be a set, then collection of all subsets of set A is known as power set, It is denoted by P(A).

e.g, If A = [1,2,3], then

P(A) =

VENN DIAGRAM

Geometrical representation of various types of set and their operations are called Venn diagram types of set and their operations are called Venn diagram. These are represented by figure or graph. Universal set is denoted by a rectangular figure and its subsets are denoted by circles.OPERATIONS ON SETS

Union of Sets

Let A and B are two sets then union of sets A and B is denoted by A B and A B is a set contains all elements which are any of A or of B or both A and B .

A B=

x A B

x A or x B

and if

x A B

x A and x B

Intersection of Sets

If A and B are two sets, then intersections of A and B is denoted by A B and A B contains those elements which are in both A and B.

A B =

x A B

x A and x B

Disjoint Sets

Two sets A and B are disjoint sets, if A B = .

Difference of Sets

If A and B are two non empty sets, then difference of two sets A and B is denoted by A- B and it contain those elements which are in A but not in B. Similarly, B A is a set which contains those elements which are in B but not in A.

Hence, A B = [x : x A but x B] And b A = [x : x B but x A]

Symmetric Difference of Sets

If A and B are two sets, then set (A B) (B A ) is known as symmetric difference of sets A and B and is denoted by A B .Complement Set

I U is universal set and A U, then complement of A is denoted by A or U A.

A = U A = [x : x U but x A]

It is clear that if

x A x B

Example If A = [1,2,3,4] and B {4,5,6,7] find the value of A B.

Solution A = and B =

A B =

And

B A =

A B = (A B) (B A) =

LAWS OF ALGEBRA OF SETS

If A, B and C are three sets, the

1. Idempotent laws

(a) A A = A

(b) A A = A

2. Identify laws

(a) A

EMBED Equation.3 = A

(b) A U = A

3. Commutative laws

(a) A B = B A (b) A B = B A

4. Associative laws(a) (A B) C = A (B C)

(b) (A B) C = A (B C)5. Distributive laws

(a) A( BC) = (A B) (A C)

(b) A ( BC) = (A B) (A C)6. De- morgans laws

(a) (A B) = A B

(b) (A B) = A B

7. (a) A B = A B(b) B A = B A

(c) A B = A A B =

(d) (A B) B = A B

(e) (A B) B =

(f) (A B) (B A ) = (A B ) (AB)

8. (a) A (B C) = (A B) (A C)(b) A - (B C) = (A B) (A C)(c) A (B -C) = (A B) (A C)

(d) A (B C) = (A B) (A C)

Example 2 A class has 175 students . The following is the description showing the number of students studying one or more of the following subjects in this class.

Mathematics 100; Physics 70; Chemistry 46; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics , Physics and chemistry 18.Find:

(i) How many students are enrolled in Mathematics alone, Physics alone and chemistry alone?

(ii) The number of students who have not offered any of these three subjects

Solution : Let A, B, C denote the sets of students enrolled in Mathematics, Physics and chemistry respectively.Let us denote the number of elements contained in each bounded region by small letters a,b,c,d,e,f,g as shown in the figure.

Using the given data, we have

a + b + c + d =100;

b + c + e + f = 70; c + d + f + g = 46;

b + c = 30;

c + d = 28;

c + f + 23;

c = 18.

Solving these questions, we get;

c = 18; f = 5; d =10;

b = 12; g = 13; e = 35

and a = 60

(i) Number of students enrolled in:

Mathematics alone = a = 60;

Physics alone = e = 35

Chemistry alone = g = 13

(ii) Number of students who have not offered any of these three subjects

= [175 (a + b + c+ d + e + f +g]

= [175 (60 +12+18+10+35+5+13)]

= [175 153] = 22

ORDERED PAIR

Two elements a and b, listed in a specified order , form an ordered pair, denoted b (a, b). In an ordered pair (a, b); a is regarded as the first element and b is the second element.

It is evident from the definition that

(i) (a, b) (b, a)

(ii) (a, b) = (c, d) iff a = c, b = d

CARTESIAN PRODUCTLet A and B be two non empty sets. The Cartesian product of A and B is denoted by A x B is defined as the set of all ordered pairs (a, b) where a A and b B.

Symbolically A x B =

e.g, Suppose A = (1,2,3) and B = (x , y)

Then

A x B =

B x A =

e.g. If there are three sets A, B , C and a A, b B, c C, then we form an ordered triplet (a,b,c). The set of all ordered triplets (a,b,c) is called the Cartesian product of these sets A, B and C,

i.e., A x B x C =

Facts Related to Cartesian product

If A, B and C are three sets , then

1. (a) A x (B C) = (A x B) (A x C)

(b) A x (B C) = (A x B) (A x C)2. A x (B C) = ( A x B) (A x C)

3. A x B = B x A A = B

4. If A B A x B ( A x B) ( B x A)

5. If A B A x C B x C6. A B and C D A x C B x D7. (A x B) (C x D) = ( A C) x (BD)

8. A x (B C) = (A x B) (A x C)

9. A x (B C) = (A x B) ( A x C)

10. If A and B have n common elements, then A x B and B x A will have n2 elements common.

RELATIONS

Let A and B be two non empty sets, Then a relation R from A to B is a subset of A x B.Thus, R is a relation from A to B R A x B. if R is a relation from a non empty set A to a non empty set B and if (a, b) R, Then we write aRb which is read as a is related to b by the relation R. If (a, b) R, then we write aRb and we say that a is not related to b by the relation R.

SOME PARTICULAR TYPS OF RELATIONS

1. Vold relation: Since it follows that is a relation on A called the empty or vold relation.

2. Universal relation: Since it follows that A x A is a relation o A called the universal relation.3. Identity relation: The relationIA =

Is called the identity relation on A .

e.g., if A = (1,2,3) then the identity relation on A is given by IA = [(1,1) (2,2), (3,3)]

INVERSE RELATIONS

If R is a relation on set A, then the relation R-1 on A defined by R-1= [b, a) : (a, b) R] is called an inverse relation to A.Clearly , domain (R-1) = range (R)

And range (R-1)= domain (R)

e.g., Let A = (1,2,3) and let

R = [ (1,2) (2,2), (3,1), (3,2)]

Then, R being a subset of A x A, it is a relation on A. Clearly 1R2 : 2R2 : 3R1 and 3R2

Domain (R) = ( 1,2,3) and Range (R) = (2,1)

Also, R-1 = [(2,1), (2,2), (1,3), (2,3)]

Domain (R-1) = (2,1) and Range (R-1) = (1,2,3)COMPOSTIONS OF RELATIONS

Let R A x B, S B x C be two relations. Then compositions of the relations R and S by SOR A x C and is defined by (a,c) SOR, iff b B s.t.

(a, b) R, (b, c) S

e.g., Let A = (1,2,3), B = (a,b,c,d), C = ()

R (A x B) = ( 1 , a), ( 1,c), (2 , d)

S ( B x C) = [(a, ) (c,), (d, )]

Then SOR (A x C) = [(1, ) (1,), (2, )]

One should be careful in computing the relation RoS, Actually SoR starts with R and RoS starts with S.

In general SoR RoS

Also (SoR)-1 = R-1oS-1, known as reversal rule.

Reflexive relations

R is a reflexive relation if (a,a) R, aR,. It should be noted if any a A, such that (a, a) R is not is reflexive e.g, Let A = [1,2,3] and R = [(1,1), (2,2)]

Then R is not reflexive since 3 A but (3,3)

Symmetric Relations

R is called a symmetric relation on A if

(x, y) R (y, x) R. That is, if x related to y, then y is also related to x.. It should be noted that R is symmetric , iff R-1 = R

Anti symmetric Relations

R is called an anti-symmetric relation, if (a, b) R and (b, a) R a = b. Thus if a b, then a may be related to b or b may be related to a, but never both.

e.g., Let N be the set of natural . A relation R N x N is defined by xRy. Iff x divides y .

Then xRy , yRx x divides y, y divides x.Transitive Relations

R is called a transitive relation, if

(a, b) R, (b, c) R (a, c ) R.In other words, if a is related to b, b is related to c, then a is related to c.

Transitivity falls only when there exists a, b, c such that aRb, bRc but aRb.

e.g., Consider set A = [1,2,3] and the relation

R = [(1,2), (2,1), (1,1)].

Then R is not transitive since (2,1) R, (1,2) R but (2,2) R.

EQUIVALENCE RELATION

A relation R in a set A is called an equivalent relation . if

(i) R is reflexive i.e., (a,a) R, a A

(ii) R is symmetric i.e., (a,b) R, (b, a) R

(iii) R is transitive i.e, (a, b), (b, c), R (a, c) RExample:3 Let N be the set of all natural numbers and let R = [(a, b) : a N, b N and 2 a + b = 10]

The find R-1,

Solution: Clearly, r = [(1, 8), (2,6), (3,4), (4,2)]Here, Dom (R) = [1,2,3,4]

And Range (R) = [8,6,4,2]

R-1= [(8,1), (6,2), (4,3), (2,4)]

De-morgans Laws: If A and B are any two sets, then (i)(A ( B)( = A( ( B(

(ii)(A ( B)( = A( ( B(CONGRUENCES

Let m be a positive integers, then the two integer a and b said to be congruent modulo m(( if a b is divisible by m, i..e, a b = m (, where ( is an positive integer.

The congruent modulo m( is defined on all a, b ( I by a = b (mod m) iff a b = (, ( ( I+Example: The congruent solution of 8x = 6 (mod 14) is

(a) x = 6, 7

(b)x = 6, 13

(c) x = 2, 13 (d) none of these

COMPREHENSIVE APPROACH

Number of subsets of a set of n elements is 2n. Number of proper subsets of a set of n elements is 2n -1.

CONSTANT

Those quantities which are unaltered under any mathematical operation, are called constant.

VARIABLES

Those quantities which are altered under any mathematical operation and can opt any value, are known as variables.

Variables are of two types:

1.Independent variable

2.Depending variable 1.Independent variable

Those variables which can opt any value are called independent variable.

2.Dependent Variable

Those variables whose value depend upon the order variables, are called dependent variable.

e.g., Let y = x2 + 7x + 2

Here, x is an independent variable whereas y is a dependent variable. Since, the value of y depends upon the value of x.

DOMAIN

The domain of y = f(x) is the set of all real x for which f(x) is defined (real).

RULES FOR FINDING DOMAIN

(i)Expression under even root (i.e., square root, fourth root etc.) ( 0

(ii)Denominator ( 0.

(iii)If domain of y = f(x) and y = g(x) are D1 and D2 respectively, then the domain of f(x) ( g(x) or f(x). g(x) is D1 ( D2. While domain of is D1 ( D2 ( {g(x) = 0}.

RANGE

Range of y = f(x) is collection of all outputs f(x) corresponding to each real number in the domain. RULE FOR FINDING RANGE:

First of all find the domain of y = f(x)

(i)If domain ( finite number of points

( range ( set of corresponding f(x) values

(ii)If domain ( R or R ( (some finite points)

Then express x in terms of y. From this find y for x to be defined, (i.e., find the values of y for which x exists).

(iii)If domain ( a finite interval, find the least and greatest value for range using monotonically.

FUNCTION

Let A and B are two non-empty sets, then a subset f of A ( B is known as a function from A to B, if for each element x of A there exists a unique element y in B such that (x, y) ( f.

OR

A function f from a set A to a set B is a rule that assigns a unique element f(x) in B to each element in A.

The terms map, mapping, corresponding are used as synonyms for function and it is denoted by f : A ( B.

Set A is known as domain of function f and set B is known as co-domain of function of f. Set {y ( B : y = f(x)} is known as range of f.

EXPLICIT AND IMPLICIT FUNCTIONS

A function is said to be an explicit function, if the dependent variable y is expressible completely in term of independent variable.

A function is said to be an implicit function, if the dependent variable is not directly expressible in terms of the independent variable x.

e.g., y = 3ex ( x2 + 2, x ( R is an explicit function while + = is an implicit function.

Vertical parallel Line Test (VPL Test)

This is a geometrical test to check whether a relation is a function or not.

If we draw a vertical parallel line i.e., any line parallel to y-axis, then, if this line interests the graph of the expression in more than one point, then the expression is a relation else, if it intersects at only one point, the expression is a function.

In figure, the vertical parallel line intersects the curve at two points thus the expression is a relation whereas in figure, the vertical parallel line intersects the curve at one point. Thus, the expression is a function.

CLASSIFICATION OF FUNCTIONS

Constant Function

If in the range of the function f there is only single element,

then f is known as constant function.

OR

Let k be a constant, then function f(x) = k, ( x is known

as constant function.

Figure shows a graph of y = k i.e., f(x) = k.

Domain of f(x) = R

and

Range of f(x) ={k}

Polynomial Function

The function y = f(x) = a0xn +..+ an, where a0, a1, a2,.., an are real coefficient and n is a non-negative integer, is known as a polynomial function. If a0 ( 0, then degree of polynomial function is n.

Domain of f(x) = R

Rational Function

If P(x) and Q(x) are polynomial function, then function f(x) = is known as rational function.

Domain of f(x) = R ( {x : Q(x) = 0)

Irrational Function

The function containing one or more term having non-integral rational powers of x are called irrational function.

e.g.

y = f(x) =

Identity Function

Function f(x) = x, ( x is known as identity function.

Domain of (x) = R and Range of f(x) = R figure shows a graph of

identity function in which curve passes through origin and inclined

at an angle 45o to x-axis.

Exponential Function

Function f(x) = ax, a > 0, a ( 1, a ( constant, is known as exponential function.

Domain of f(x) = R

and

Range of f(x) = 0, (Logarithmic Function

Function f(x) = loga x, (x, a > 0) and a ( 1, is known as logarithmic function.

Domain of f(x) = (0, ()

and

Range of f(x) = R

Trigonometric Function

Functions of trigonometric ratios are known as trigonometric functions.

FunctionDomainRange

sin xR[(1, 1]

cos xR[(1, 1]

tan xR (

R

cot xR - n( : n ( IR

sec xR (

(((, (1] ( [1, ()

cosec xR ( [n( : n ( I](((, (1] ( [1, ()

Inverse Trigonometric Function

The functions involving inverse trigonometric ratio are known as inverse trigonometric functions.

Modulus Function

Function y = f(x) = |x| is known as modulus function.

y = f(x) =

Domain of f(x) = x ( R

and

Range of f(x) = [0, ()

Figure shows a graph of modulus function which is also an even function.

Properties of Modulus Function

1.|x| ( a ( (a ( x ( a; (a ( 0)

2.|x| ( a ( x ( (a or x ( a; (a ( 0)

3.|x ( y| ( |x| + |y|

4.|x + y| ( ||x| - |y||

Signum Function

Function f(x) = Sgn (x) is known as Signum function n

f(x) = Sgn(x)

=

=

Domain of f(x) = R

and

Range of f(x) = {(1, 0, 1}

Figure shows the graph of Signum function.

Greatest Integer Function

The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function.

f(x) = [x]

Domain of f(x) = R

and

Range of f(x) = 1

Properties of Greatest Integer Function

1.[x + I] = [x] + I, if I is an integer.

2.[x + y] ( [x] + [y]

3.If [( (x)] ( I, then ( (x) ( I

4.If [( (x)] ( I, then ( (x) < I + 1

5.[(x] =

6.If [x] > n ( x ( n + 1, n ( I

7.If [x] < n ( x < n, n ( I

8.[x + y] = [x] + [y x + [x]], ( x, y ( R

9.[x] + = [nx], n ( N

Least Integer Function

The function whose value of any number x is the smallest integer greater than or equal to x is called least integer function or the integer floor function.

f(x) = (x)

Domain of f(x) = R

and

Range of f(x) = I

Figure shows the graph of least integer function.

Fractional Part Function

y = {x} is known as fractional part function.

If x = I + f, where I = [x] and f = {x}

y = x ( [x]

Domain of f(x) = R

and

Range of f(x) = [0, 1)

Properties of Fractional Part Function

1.If 0 ( 1, then {x} = x

2.If x ( I, then {x} = 0

3.If x ( I, then {(x} = 1 ( {x}

Example: find the domain of definition f(x) =

Solution: For f to be defined x ( (6, 6 and log0.4 ( 0,

0 < ( 1 ( 1 < x < ( but x ( 6.

(Domain of f(x) = (1, () ( {6}

Example: If f is a function such that f(0) = 2, f(1) = 3 and f(x + 2) = 2f(x) ( f(x + 1) for every real x, then find the value of f(5).

Solution:At x = 0 ( f(2) = 2f(0) ( f(1) = 2 ( 2 ( 3 = 1

At x = 1 ( f(3) = 2f(1) ( f(2) = 6 ( 1 = 5

At x = 2 ( f(4) = 2f(2) f(3) = 2 ( 1 ( 5 = (3

At x = 3 ( f(5) = 2f(3) f(4) = 2(5) ((3) = 13

Even functions and ODD function

Even Function

A real function f(x) is an even function, if f((x) = f(x) e.g., f(x) = cos x is an even function.

Odd Function

A real function f(x) is an odd function, if f((x) = (f(x) e.g., f(x) = sin x or x3 are odd functions.

Properties of Even and Odd Functions

1.Product of two odd functions or two even functions is an even function.

2.Product of an even function and an odd function is in odd function.

3.Each function can be expressed as the sum of an even function and an odd function.

4.Derivative of an even function is an odd functions and derivative of an odd function is an even function.

5.Square of an even or an odd function is always an even function.

6.The graph of an odd function is symmetrical about origin or symmetrical in opposite quadrants.

7.The graph of an even function is symmetrical about y-axis.

Example: Show that f(x) = , where x is not on integral multiple of ( and [(] denotes greatest integer functions an odd function.

Solution: Clearly =

(f(x) is an odd function. even extension and odd extension of a function

Let y = f(x) is defined in [a, b], then y = f((x) is its even extension in [(b, (a] and y = (f((x) is its odd extension in [(b, (a].

Periodic Functions

A function f : X ( Y is said to be a periodic function, if there exist a positive real number T such that f(x + T) = f(x) ( x ( X. The least value of T is called the fundamental period of a function. In general, the fundamental period (principal period) is called the period of a function. Graphically graph gets repeated after each period of the function.

Standard Results on Some Periodic Functions

S.No.FunctionsPeriod

1.sinn x, cosn x, secn x, cosecn x ( (If n is an even number) 2( (If n is an odd number)

2.tann x, cotn x (

3.|sin x |, |cos x|, |tan x|, |sec x|, |cosec x|, |cot x|(

4.x ( [x]1

Properties of Periodic Functions

1.If f(x) is a periodic function with fundamental period T, then will also be a periodic function with same fundamental period T.

2.If f(x) is a periodic function with period T, then f(ax + b) is also a periodic function with fundamental period .

3.If f(x) is a periodic function with period T, then af(x) + b is also a periodic function with same fundamental period T.

4.If f(x) and g(x) are two functions with fundamental periods T1 and T2 respectively, then f(x) + g(x) is a periodic function with fundamental period LCM to T1 and T2, provided f(x) and g(x) cannot be inter-changed by adding a positive number in x which is less than LCM of T1 and T2, in that case period becomes that number, and also LCM to T1 and T2 should exist otherwise this is not a periodic function. 5.If f(x) is a periodic function with period T and g(x) is a monotonic function, then g[f(x)] is also a periodic function with same period T as that of f(x)

Example: If f(x) = sin () x, [(] denotes greatest integer has ( as its fundamental period, then find the value of a.

Solution:

= ( (

= 2 ( [a] = 4

(

a ( [4, 5)

Composite functions

Let f(x) and g(x) be two functions whose domains are D1 and D2 respectively.

If range (f) ( domain (g),

then (gof) (x) = g{f(x)}, ( x ( D1,

and, if range (g) ( domain (f), then

(fog) (x) = f{g(x)}, ( x ( D2

Generally, (fog) (x) ( (gof) (x)

Let f : x ( Y and g : Y ( Z be two functions we define a function h : X ( Z such that

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

h = gof

h(x)

Properties of Composition of Function

1.If f and g are even functions ( fog is an even function.

2.If f is odd and g is an even function ( fog is an odd function.

3.If f is an even and g is an odd function ( gof is an even function.

4.The composition of functions is not commutative

fog ( gof

5.If f, g, h are any three function such that (fog) oh and fo(goh) both exist, then

(fog) (oh) = fo(goh)

6.The composition of any function with the identity function is the function itself.

Example: If f(x) = - tan , (1 < x < 1 and g(x) = , then find the domain of gof.

Solution: Since, domain of f and gof are same,

and f(x) = ( tan , (1 < x < 1

(Domain of gof = ((1, 1)

One-one Function

A function f : A ( B is said to one-one. If different elements in A have different images in B.

This type of functions are known as injective functions.

Many-One Function

If there exist at least two distinct elements in domain having same image, then it is many-one function.

Methods to find One-One and Many-One Functions

1.If x1 ( x2 ( f(x1) ( f(x2), then f(x) is one-one. 2.If f(x) = f(x2) ( x1 = x2, then f(x) is one-one.

3.Any function, which increases monotonically or decreases monotonically is one-one function i.e., f((x) > 0 for all x in domain or f((x) < 0 for all x in domain.

4.Any continuous function f(x) which has at leas tone local maximum or local minima, is many-one.

5.If there is any line parallel to x-axis intersecting the graph of y = f(x) in more than one point, then function in many-one.

Horizontal Parallel Lines Test (HPL Test)

This is also a geometrical test to chock whether a function is one-one or not.

If we draw a horizontal parallel line i.e., any line parallel to x-axis, then, if this line intersects the graph of the function in at least two points, then the function is not one-one function else, if it intersects at only one point, the function is one-one.

In figure, the horizontal parallel line intersects the graph at one point, thus the function is one-one whereas in figure, the line intersects the graph at two points thus the function is not one-one function.

Differential Test for Injectivity

If the function is strictly increasing or strictly decreasing through out its domain, then the function is one-one.

Onto Functions

If the function f : A ( B is such that each and every elements in B is the f image of at least one element in A, then f is a function of A onto B.

OR

If each element in co-domain have at least one pre-image in the set of domain, then function is onto.

OR

If range is same as co-domain, then function is onto.

In figure, each element B is image of at least one element in A.

Onto function is also called surjective function .

Whether a Function is Onto or Not

For any function y = f(x), find the range of y. If the range of y equals to the co-domain of the function, then function is onto.

Into Functions

If the function f : A ( B is such that there is at least one element in B which is not f image of any element in A, then f is called a into function.

OR

If there is even single element in the set of co-domain which does not have its pre-image, then it is known as into function.

OR

If range of a function f is proper subset of co-domain, then function f is into.

In figure, there is an element (e) which is not image of any element in A. Thus, f is into function.

Bijective Functions

Function which are one-one functions and onto functions are called bijective functions.

EQUAL FUNCTION

Two functions f and g are called equal functions, if

(a) Domain of f = domain of g

(b)Range of f = range of g

(c) f(x) = g(x), ( x ( Domain of f or g.

Inverse Function

Let f : A ( B is a bijective function, then there exists a unique function g : B ( A is a function such that f(x) = y ( g(y), x, ( x ( A and y ( B, then g is called inverse function of f.

Hence,

g = f(1 : B ( A

Properties of inverse Function

1.Inverse of bijective function is unique.

2.Inverse of bijective function is also bijective function.

3.If f : A ( B is bijective function and g : B ( A, is inverse of f, then fog = IB and gof = IA where IA and IB are identity function of sets A and B respectively. 4.If f : A ( B and g : B ( A are two bijective functions, then gof : A ( C is also bijective function and (gof)(1 = f(1. 5.fog ( gof, but if fog = gof, then either f(1 = g or g(1 = f and (fog) (x) = (gof) (x) = x.

Example: Show that f(x) = sin from f : A (A are invertible, where A = [(1, 1].

Solution: Since, f(x) = sin is both one-one and onto.

Its inverse exist. Exercise

Q1. Which of the following is a null set? (a) {0}

(b) {x : x > 0 or x < 0}

(c) {x : x2 = 4 or x = 3} (d) {x : x2 + 1 = 0, x ( R}

Q2. If A = P((1, 2)) where P denotes the power set, then which one of the following is correct? (a) {1, 2} ( A

(b) 1 ( A

(c) ( ( A

(d) {1, 2} ( A

Q3. If A = {x : s is a multiple of 3} and B = {x : x is a multiple of 5}, then A B is (where, means complement of A) (a) ( B

(a) A (

(c) (

(c)

Q4. The shaded region in the given figure is

(a) A ( (B ( C)

(b) A ( (B ( C)

(c) A ( (B ( C)

(d) A ( (B ( C)Q5. Consider the following Venn diagram

If |E| = 42, |A| = 15, |B| = 12 and |A ( B| = 22, then the area represented by shaded portion in the above Venn diagram is

(a) 25

(b) 27

(c) 32

(d) 37 Q6. Consider the set of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with values 1. Let C be the subset of the set of all determinants with value (1. Then, (a) C is empty

(b) B has as many elements as C

(c) A = B ( C

(d) B has twice as many elements as C Q7. Let n(U) = 700, n(A) = 200, n(B) = 300, n(A ( B) = 100, then n(A( ( B() is equal to (a) 400

(b) 600

(c) 300

(d) None of these Q8. For real numbers x and y, we write xRy ( x y + is an irrational number. Then, the relation R is (a) reflexive

(b) symmetric

(c) transitive

(d) none of these Q9. If R ( A ( B and S ( B ( C be two relations, then (SoR)(1 is equal to (a) S(1oR-1

(b) R-1oS-1(c) SoR

(D) RoS Q10. If R is a relation on a finite set having n elements, then the number of relations on A is (a) 2n

(b)

(c) n2

(d) nnQ11. Which one of the following functions, f : R ( R is injective? (a) f(x) = |x|, ( x ( R (b) f(x) = x2, ( x ( R

(c) f(x) = 11, ( x ( R (d) f(x) = (x, ( x ( R Q12. The domain of the function f(x) = + is (a) [1, ()

(b) (((, 6)

(c) [1, 6]

(d) None of these Q13. If ((x) = ax, then [((p)]3 is equal to (a) ((3p)

(b) 3((p)

(c) 6((p)

(d) 2((p) Q14. If f(x) = x2 - x(2, then f is equal to (a) f(x)

(b) (f(x)

(c)

(d) [f(x)]2Q15. Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey, 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is (a) 128

(b) 216

(c) 240

(d) 160 Q16. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is (a) at least 30

(b) at most 20

(c) exactly 25

(d) None of these

Directions: The following questions are Assertion Reason type Questions. Each of these question contains two statements. Statement I (Assertion) and Statement II (Reason) Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice. Q17. Statement I: A ( B = A ( C and A ( B = A ( C, then B = C.

Statement II: A ( (B ( C) = (A ( B) ( (A ( C).

(a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

(b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I

(c) Statement I is true, Statement II is false

(d) Statement I is false, Statement II is true Q18. Statement I: If a set A has a n elements, then the number of binary relation on A = . Statement II: Number of possible relations from A to A = .




(d) Statement I is false, Statement II is trueQ19. Statement I: Sets A and B have three and six elements respectively, then the minimum number of elements in A ( B is 6. Statement II: A ( B = 3




(d) Statement I is false, Statement II is trueQ20. Consider the following statements. I. All poets (P) and learned (L)

II. All learned (L) and happy (H).

Which one of the following Venn diagrams correctly represents both the above statements taken together?

Directions: In a city, 25% of the families have phone and 15% of the families have car, 65% of the families have neither phone nor car. 2000 families have both phone and car. Q21. Percentage of families having phone and car both, is (a) 5%

(b) 10%

(c) 20%

(d) 25%Q22. Percentage of families having either phone or car is

(a) 10%

(b) 30%

(c) 35%

(d) 40%Q23. What is the number of families in the city? (a) 30000

(b) 40000

(c) 20000

(d) 10000Q24. If A = {1, 2}, B = {2, 3} and C = {3, 4}, then what is the cardinality of (A ( B) ( (A ( C)? (a) 8

(b) 6

(c) 2

(d) 1 Q25. If A is a finite set having n elements, then the number of relations which can be defined in A is (a) 2n

(b) n2(c)

(d) nnQ26. Which one of the following is an example of non-empty set? (a) Set of all even prime numbers

(b) (x : x2 2 = 0 and x is rational)

(c) {x : x is a natural number, x < 8 and simultaneously x > 12}

(d) {x : x is a point common to any two parallel lines}Q27. The relation R in the set Z of integers given by R = {(a, b) : a b is divisible by 5} is (a) reflexive

(b) reflexive but not symmetric

(c) symmetric and transitive

(d) an equivalence relation Q28. Let A = {a, b, c, d} and B = {x, y, z}. What is the number of elements in A ( B? (a) 6

(b) 7

(c) 12

(d) 64Q29. If f be a function from the set of natural numbers to the set of even natural numbers given by f(x) = 2x. then, f is (a) one-to-one but not onto

(b) onto but not one-one

(c) Both one-one and onto

(d) Neither one-one nor onto Q30. If A = {1, 3, 5, 7}, then what is the cardinality of the power set P(A)? (a) 8

(b) 15

(c) 16

(d) 17 Q31. What is the range of the function f(x) = , x ( 0? (a) Set of all real numbers (b) Set of all integers

(c) {(1, 1}

(d) {(1, 0, 1}Q32. Which one of the following is a null set? (a) {0}

(b) {{{}}}

(c) {{}}

(d) {x | x2 + 1| = 0, x ( R}Q33. Let N be the set of natural numbers and f : N ( N be a function given by f(x) = x + 1 for x ( N. Which one of the following is correct? (a) f is one-one and onto

(b) f is one-one but not onto

(d) f is only onto

(d) f is neither one-one nor onto Q34. The relation has the same father as over the set of children is (a) only reflexive

(b) only symmetric

(c) only transitive (d) an equivalence relation Q35. Let P = {1, 2, 3} and a relation on set P is given by the set R = {(1, 2), (1, 3), (2, 1), (1, 1), (2, 2), (3, 3), (2, 3)}. Then R is (a) reflexive, transitive but not symmetric

(b) symmetric, transitive but not reflexive

(c) symmetric, reflexive but not transitive

(d) None of the above Q36. Let A = {x : x is a square of a natural number and x is less than 100} and B is a set of even natural numbers. What is the cardinality of A ( B? (a) 4

(b) 5

(c) 9

(d) None of these Q37. If ( is a null set, then which one of the following is correct? (a) ( = 0

(b) ( = {0}

(c) ( = {(}

(d) ( = {}Q38. If A = {1, 2, 5, 6} and B = {1, 2, 3}, then what is (A ( B) ( (B ( A) equal to? (a) {(1, 1), (2, 1), (6, 1), (3, 2)}

(b) {(1, 1), (1, 2), (2, 1), (2, 2)}

(c) {(1, 1), (2, 2)}

(d) {(1, 1), (1, 2), (2, 5), (2, 6)}Q39. Let M be the set of men and R is a relation is son of defined on M. Then, R is (a) an equivalence relation

(b) a symmetric relation

(c) a transitive relation

(d) None of these Q40. If the cardinality of a set A is 4 and that of a set B is 3, then what is the cardinality of the set A ( B? (a) 1

(b) 5

(c) 7

(d) Cannot be determined Q41. Consider the function f : R ( {0, 1} such that

f(x) =

Which one of the following is correct?

(a) The function of one-one into

(b) The function is many-one into

(c) The function is one-one onto

(d) The function is many-one onto Q42. If f(x) = , x ( R, then what is f(1 (x) equal to? (a)

(b)

(c)

(d)

Q43. If f(x) = 2x + 7 and g(x) = x2 + 7, x ( R, then which value of x will satisfy fog (x) = 25? (a) (1, 1

(b) (2, 2

(c) (,

(d) None of these Q44. Out of 32 persons, 30 invest in National Savings Certificates and 17 invest in shares. What is the number of persons who invest in both? (a) 13

(b) 15

(c) 17

(d) 19Q45. If Na = {ax | x ( N}, then what is N12 ( N8 equal to? (a) N12

(b) N20(c) N24

(d) N48Q46. A mapping f : R R which is defined as f(x) = cos x; x ( R is (a) only one-one

(b) only onto

(c) one-one onto

(d) Neither one-one nor onto Q47. In an examination out of 100 students, 75 passed in English, 60 passed in Mathematics and 45 passed in both English and Mathematics. What is the number of students passed in exactly one of the two subjects? (a) 45

(b) 60

(c) 75

(d) 90 Answers

Q1.(d)Q2.(d)Q3.(b)Q4.(c)Q5.(a)Q6.(b)Q7.(c)

Q8.(a)Q9.(b)Q10.(b)Q11.(d)Q12.(c)Q13.(a)Q14.(b)

Q15.(d)Q16.(c)Q17.(a)Q18.(b)Q19.(a)Q20.(d)Q21.(a)

Q22.(c)Q23.(b)Q24.(c)Q25.(c)Q26.(a)Q27.(d)Q28.(c)

Q29.(c)Q30.(c)Q31.(c)Q32.(d)Q33.(b)Q34.(d)Q35.(a)

Q36.(a)Q37.(d)Q38.(b)Q39.(d)Q40.(d)Q41.(d)Q42.(b)

Q43.(c)Q44.(b)Q45.(c)Q46.(d)Q47.(a)

Complex numberImaginary numbers

Square roots of negative numbers are known as imaginary number. e.g., etc., are imaginary numbers.

Let a is a positive real number, then

= ( = ( i ,

where i =

Each imaginary number can be expressed as a product of real number and i.

Example: Prove that (1 + i)4 ( = 16.

Solution:(1 + i)4 ( = (1 + i)4 ( (1 + i)4

= (1 ( i2)4 = (1 + 1)4 = 24 = 16

complex numbers

The numbers of the form x + iy, where x and y are real numbers and i = , are known as complex numbers.

It is denoted by z.

(

z = x + iy

Here, x is known as real part of z and is denoted by Re (z) and y is known as imaginary part of z and is denoted by lm(z).

If y = 0 ( z = x i.e., z is a purely real number.

If x = 0 ( z = iy i.e., z is a purely imaginary number.

Equality of complex numbers

Let z1 = x1 + iy1 and z2 = x2 + iy2 are two complex numbers, then these two numbers are equal, if

x1 = x2 and y1 = y2

i.e.,

Re(z1) = Re(z2)

and

Im(z1) = Im(z2)

operations on complex numbers

Addition of Complex Numbers

Let z1= x1 + iy1 and z2 = x2 + iy2 are two complex numbers, then

z1 + z2 = x1 + iy1 + x2 + iy2

= (x1 + x2) + i(y1 + y2)

Re(z1 + z2) = Re(z1) + Re(z2)

and Im(z1 + z2) = Im(z1) + Im(z2)

Properties of addition of complex

1.z1 + z2 = z2 + z1

(Commutative law)

2.z1 + (z2 + z3) = (z1 + z2) + z3(Associative law)

3.z + 0 = 0 + z

(where 0 = 0 + i0)

Example: Find the value of x and y, if 2 + (x + iy) = 3 ( i.

Solution: (2 + (x + iy) = 3 ( i

((2 + x) + iy = 3 i

On equating real and imaginary parts separately.

(

2 + x = 3 and y = (1

(

x = 1 and y = -1

Subtraction of Complex Numbers

Let z1 = x1 + iy1 and z2 = x2 + iy2 are two complex numbers, then

z1 ( z2 = (x1 + iy1) ( (x2 + iy2)

= (x1 ( x2) + i(y1 ( y2)

( Re(z1 ( z2) = Re(z1) ( Re(z2)

and Im(z1 z2) = Im(z1) ( Im(z2)

Example: Simplify 3(1 ( 2i) ( ((4 ( 5i) + (-8 + 3i) .

Solution: 3(1 2i) ( ((4 ( 5i) + ((8 + 3i)

= 3 ( 6i + 4 + 5i 8 + 3i

= (3 + 4 ( 8) + ((6 + 5 + 3)i

= (1 + 2i

Multiplication of Complex Numbers


z1z2 = (x1 + iy1) (x2 + iy2)

= (x1x2 ( y1y2) + i(x1y2 + x2y1)

(z1z2 = [Re(z1) Re(z2) ( lm(z1)Im(z2)] + i[Re(z1) Im(z2) + Re(z2) Im(z1)]

Properties of multiplication of complex numbers

1.z1z2 = z2z1

(Commutative law)

2.(z1z2) z3 = z1(z2z3)

(Associative law)

3.If z1z2 = 1 = z2z1, then z1 and z2 are multiplication inverse of each other.

4.(a) z1(z2 + z3) = z1z2 + z1z3 (Left distribution law)

(b) (z2 + z3) z1 = z2z1 + z3z1 (Right distribution law)

Example: Find the real values of x and y, if = i .

Solution:

+ = i

({(1 + i) x 2i} (3 i) + {(2 3i) y + i} (3 + i) = i (3 + i) (3 i)

((1 + i) (3 i) x ( 2i (3 i) + (2 3i) (3 + i) y + i (3 + i) = 10i

((4 + 2i) x 6i 2 + (9 ( 7i) y + 3i ( 1 = 10i

((4x 2 + 9y 1) + i (2x ( 6 ( 7y + 3) = 10i

((4x + 9y 3) + i(2x 7y 3) = 10i

(4x + 9y 3 = 0 and 2x 7y 3 = 10

(Equating real and imaginary parts separately)

Solving these equations, we get x = 3, y = (1.

Division of Complex Numbers


=

= [(x1x2 + y1y2) + i(x2y1 ( x1y2)]

conjugate complex numbers

Let z = x + iy is a complex number, then conjugate of z is denoted by and is equal to x iy.

Properties of Conjugate of Complex Numbers

If z, z1, z2 are complex numbers, then

1.

= z

2.z + = 2 Re (z)

3.z ( = 2 lm (z)

4.z = ( z is purely real

5.z = ( ( z is purely imaginary

6.z = {Re(z)}2 + {lm (z)}27.

= +

Example: If (x + iy) = , the prove that (x iy) = and (x2 + y2) = (

Solution: (x + iy) =

=

=

= + i

Equating real and imaginary part on both sides of (i), we get

x = and y =

((x iy) =

=

=

Now, x2 + y2 = (x + iy) (x ( iy)

=

Modulus of a complex number

Let z = x + iy is a complex number, then modulus of complex number z is denote by |z|.

(|z| = =

Properties of Modulus of a Complex Number

1.|z| ( 0 ( |z| = 0, iff z = 0 and |z| > 0, iff z ( 0

2.(|z| ( Re (z) ( |z| and (|z| ( lm (z) ( |z|

3.|z| = || = |(z| = |(|

4.

= |z|25.|z1z2| = |z1| |z2| In general |z1z2z3zn| = |z1| |z2| |z3|.|zn|

Example: If z = x + iy and ( = , then show that |(| = 1 implies that in the complex plane, z lies on x-axis.

Solution: ( =

As |(| = 1 (|z i| = |1 iz| = |z + i|

(z lies on the right bisector of the line segment connecting the points 1 and (1. Thus z lies on the real axis.

Argument of a complex number

Let z = x + iy is a complex number, then argument of complex number is denoted by arg(z) or amp(z)

arg(z) = tan(1

Properties of Argument of a Complex Number

If z1, z2 and z3 are three complex numbers, then

1.arg (z1z2) = arg (z1) + arg(z2) + 2k( (k = 0 or 1 or (1) in general arg (z1z2z3zn) = arg (z1) + arg (z2) + arg(z3) ++ arg(zn) + 2k(2.arg = arg (z1) ( arg (z2) + 2k( + 2k( (k = 0 or 1 or -1)

3.arg = 2 arg (z) + 2k(

(k = 0 or 1 or (1)

4.arg (zn) = n arg (z) + 2k(

(k = 0 or 1 or (1)

5.If arg = (, then arg = 2k( - ( where k ( I 6.arg () = (arg (z)

7.If arg(z) = 0 ( z is real

8.arg (z1) = arg(z1) ( arg(z2) Example: If z = x + iy satisfy amp(z 1) = amp(z + 3i), then fin the value of (x 1) : y.

Solution: amp [(x 1) + iy] = amp [x + i(y + 3)]

(

tan(1 = tan(1

(xy = (x 1) (y + 3) ( 3(x 1) = y

(

Representation of Complex Number

Complex numbers can be represented as follows:

Geometrical Representation of Complex Numbers

A complex number z = x + iy can be denoted on a complex plane as a point (x, y)

Two lines XOX( and YOY( perpendicular to each other cut at the point O. If horizontal line X(OX represents a axis of real numbers and vertical line YOY( represents a axis of imaginary numbers, where OX and OY are in positive direction, then any complex number x + iy can be represented by a point P(x, y) whose distance from imaginary axis is x and its distance from real axis is y.

Length of line segment OP is known as modulus of z or absolute value of z and it is denoted as |z|. It is always a positive number.

Hence,

OP2= OM2 + MP2

(

OP2 = x2 + y2

OP =

Hence,

|z| =

=

A line OP subtends an angle with OX axis in anticlockwise direction, is known as argument or amplitude of complex number z.

Hence,

tan ( =

(

( = tan(1

The least value of ( which lies in (( ( ( ( (, is known as principal value of argument.

Vector Representation of Complex Numbers

Any complex number can be represented as a point P(x, y) of vector OP in two dimensional plane because a complex number depends upon two values.

1.its modulus

2.its argument

z = z + iy is denoted by a vector OP and length of OP is |z| and arg(z) is an angle which is made by OP with positive x-axis.

Trigonometrical or Polar Representation of Complex Numbers

Let z = x + iy is a complex number which is denoted by a point P(x, y) in a complex plane, then

OP = |z| and (POX = ( = arg(z)

In ( POM,

cos ( =

(

x = |z| cos (

and

sin ( =

(

y = |z| sin (

(z = x + iy ( z = |z| cos ( + i|z| sin (

(

z = |z| (cos ( + i sin ()

(

z = r (cos ( + i sin ()

where r = |z| and ( = arg (z)

This form of z is known as polar form.

In general, polar form is

z = r [cos (2n( + () + i sin (2n( + ()]

where r = |z|, ( = arg (z) and n ( N

D.E. MOIVRES THEOREM

1.If n is a rational number, then (cos ( + i sin ()n = cos n( + i sin n(.

2.If z = (cos (1 + i sin (1) (cos (2 + i sin (2) .. (cos (n + i sin (n) then z = cos ((1 + (2 + .+ (n) + i sin ((1 + (2 + + (n)

3.If z = r (cos ( + i sin () and n is a positive integer, then (z)1/n = r1/n where k = 0, 1, 2, 3,, (n - 1)

4.(cos ( - i sin ()n = cos n( ( i sin n(5.

= (cos ( + i sin ()(1 = cos ( ( i sin (6.(sin ( ( i cos ()n ( sin n( ( i cos n(7.(sin ( + i cos ()n = =

8.(cos ( + i sin ()n ( cos n( ( i sin n(Example: If xr = cos + i sin , then find the value of x1, x2, x3(.

Solution: (

xr = cos + i sin

(

x1 = cos + i sin

x2 = cos = i sin

(x1 x2 x3 ( .

= cos + i sin

= cos

= cos ( + i sin ( = (1

Example: Express (1 cos ( + i sin () in modulus amplitude form.

Solution: Let (1 cos ( + i sin () = r (cos ( + i sin ()

Then, r cos ( = 1 ( 1 cos ( and r sin ( = sin (

On squaring and adding, we get

r2 = 2 (1 cos () = 4 sin2

(

r = 2 sin

(cos ( =

and sin ( =

= cos

(

tan ( = cot = tan

(

( =

So, (1 ( cos ( + i sin ()

= 2 sin

Square Root of a Complex Number

If a + ib is a complex number such that = x + iy, where x and y are real numbers.

Now,

= x + iy ( (x + iy)2 = a + ib

(

(x2 ( y2) + 2ixy = a + ib

(

x2 ( y2 = a

(i)

and

2xy = b

(ii)

Now

(x2 + y2)2 = (x2 y2)2 + 4x2y2

(

(x2 + y2)2 = a2 + b2

(

(x2 + y2) =

(iii)

[( x2 + y2 > 0, ( (x2 + y2)2 = a2 + b2 ( x2 + y2 = ]

On solving Eqs. (i) and (ii)

x2 = []

and

y2 =

(

x = (

and

y = (

If b is positive, then the sign of x and y from Eq. (ii) will be same i.e.,

= (

If b is negative, then the sign of x and y will be opposite.

i.e.,

= ( ( i

Example: Evaluate : (4 + 3 +

Solution: We may write

(4 + 3 ) = (4 + 6i )

Let (4 + 3)1/2 = (x + iy). Then,

(4 + 6i)1/2 = (x + iy)

(4 + 6i = (x2 ( y2) + (2xy)i

(x2 ( y2 = 4 and 2xy = 6

((x2 + y2) =

= = 14

On solving the equation x2 + y2 = 14 and x2 y2 = 4,

We get

x2 + 9and y2 = 5

(

x = ( 3 and y = (

Since xy > 0, it follows that x and y are of the same sign.

((x = 3, y =) or (x = (3, y = ()

So,

(4 + 3)1/2 = (4 + 6i)1/2

= ( (3 + i)

(i)

(

(4 ( 3)1/2 = ( (3 ( i)

(ii)

Hence, (4 + 21/2 + (4 ( 3)1/2 = ( 6

[adding Eqs. (i) and (ii)]

CUBE ROOTS OF UNITY

Let

x = ( x3 ( 1 = 0

(

(x 1) (x2 + x + 1) = 0

Therefore, x = 1,

If second, root be represented by (, then third root will be (2.

(Cube roots of unit are 1, (, (2 and (, (2 are called the imaginary cube roots of unity.

Properties of Cube Roots of Unity

1.1 + (r + (2r = 0, if r is not a multiple of 3 = 3, if r is multiple of 3.

2.(2 = 1 or (3r = 1

3.(3r+1 = (, (3r+2 = (2

4.It always forms an equilateral (.

nth ROOTS OR UNITY

Let z = 11/n, then

z = (cos 0o = i sin 0o)1/n

(z = (cos 2r( + i sin 2rn)1/n, r ( I

(z = cos + i sin , r = 0, 1, 2, 3,, (n ( 1) [Using De-Moiveres theorem]

(z = , r = 0, 1, 2, , (n ( 1)

(z = , r = 0, 1, 2, ., (n 1)

(z = (r, ( = , r = 0, 1, 2, ., (n - 1)

Thus, nth roots of unit are

1, (, (2,., (n(1,

where ( = = cos + i sin

Important Identities

(i)x2 + x + 1 = (x - () (x - (2)

(ii)x2 ( x + 1 = (x + () (x + (2)

(iii) x2 + xy + y2 = (x - y() (x - y(2)

(iv)x2 xy + y2 = (x + (y) (x + y(2)

(v)x2 + y2 = (x + iy) (x iy)

(vi)x3 + y3 = (x + y) (x + y() (x + y(2)

(vii)x2 + y2 xy yz + zx = (x + y( + z(2) (x + y(2 + z()

or (x( + y(2 + z) (x(2 + y( + z)

or (x( + y + z(2) (x(2 + y + z()

(ix)x3 + y3 + z3 ( 3xyz = (x + y + z) (x + (y + (2z) (x + (2y + (z)

(x)Two points P(z1) and Q(z2) lie on the same side or opposite side of the line z1 + + b and + b have same sign or opposite sign.

Properties of nth Roots of Unity

1.nth roots of unity form a GP with common ratio .

2.Sum of nth roots of unity is always zero.

3.Sum of pth powers of nth roots of unity is zero, if p is not a multiple

4.Sum of pth powers of nth roots of unity is n, if p is a multiple of n.

5.Product of nth roots of unity is ((1)n(1. 6.nth roots of unity lies on the unity circle |z| = 1 and divided its circumference into n equal parts.

ExerciseQ1. The argument of is (a)

(b)

(c)

(d) None of these

Q2. The modulus of the complex number z = (a)

(b)

(c)

(d) None of these

Q3. is equal to (a) 32

(b) 64

(c) (64

(d) None of these

Q4. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) is equal to (a) A2 ( B2

(b) A2 + B2(c) A4 + B4

(d) A4 ( B4Q5. If = 2 represents a circle, then its radius is (a) 1

(b)

(c)

(d)

Q6. The points z1, z2, z3 and z4 in the complex plane are the vertices of a parallelogram taken in order if and only if (a) z1 + z4 = z2 + z3(b) z1 + z3 = z2 + z4(c) z1 + z2 = z3 + z4(d) None of these Q7. is equal to (a) 1

(b) 0

(c) 2

(d) (1

Q8. If x satisfies the equation x2 2x cos ( + 1 = 0, then the value of xn + is equal to (a) 2n cos n(

(b) 2n cosn ((c) 2 cos n(

(d) 2 cosn (Q9. If z = , then (a) Re (z) = 0

(b) Im (z) = 0

(c) Re (z) > 0, Im (z) = 0 (d) Re (z) > 0, Im (z) < 0

Q10. If zr = cos , r = 1, 2, (, then z1. z2. z3 . ( is equal to (a) 1

(b) (i

(c) i

(d) (1

Q11. If ( = 299 (a + ib), then a2 + b2 is equal to (a)

(b) 4

(c)

(d) None of these

Q12. If z + z-1 = 1, then z100 + z(100 is equal to (a) i

(b) (i

(c) 1

(d) (1

Q13. If (( ( 1) is a cube root of unity and (1 + ()7 = A + B(, then A and B are respectively then numbers (a) 0, 1

(b) 1, 1

(c) 1, 0

(d) (1, 1

Q14. If ( is an nth root of unity other than unity itself, then the value of 1 + ( + (2 + . + (n(1 is equal to (a) 0

(b) 1

(c) (1

(d) None of these

Q15. The value of is equal to (( is an imaginary cube root of unity)

(a) 0

(b) 2(

(c) 2(2

(d) (3(2Q16. In which quadrant of the complex plane, the point lies? (a) First

(b) Second

(c) Third

(d) Fourth

Q17. 1 + i2 + i4 + .. + i2n is (a) positive

(b) negative

(c) zero

(d) cannot be determined

Q18. The smallest positive integer for which (1 + i)2n = (1 i)2n is (a) 4

(b) 8

(c) 2

(d) 12

Q19. is equal to (a)

(b) (, (/2)

(c) (, 3(/4)

(D) None of these

Q20. The locus of the point z = x + iy satisfying = 1 is (a) X-axis

(b) Y-axis

(c) y = 2

(d) z = 2

Direction: If z1 = and z2 = . Then, Q21. amp (z1) + amp (z2) is equal to (a)

(b)

(c)

(d) (

Q22. Which of the following is correct? (a) z1 < z2

(b) z1 > z2(c) z1 ( z2

(d) None of these Q23. What is the value of , where i = ? (a) (

(b) (

(c) (

(d) (

Q24. If A + iB = , where i = , then what is the value of A? (a) (8

(b) 0

(c) 4

(d) 8

Q25. If z = , then what is the value of z2 = ? (i = ) (a) 0

(b) (1

(c) 1

(d) 8

Q26. If z = 1 + cos , then what is |z| equal to? (a) 2 cos

(b) 2 sin

(c) 2 cos

(d) 2 sin

Q27. What is the value of +1? (a) (1

(b) 0

(c) 1

(d) 2

Q28. If 2x = 3 + 5i, then what is the 2x3 + 2x2 7x + 72? (a) 4

(b) (4

(c) 8

(d) (8 Answers

Q1.(c)Q2.(a)Q3.(c)Q4.(b)Q5.(d)Q6.(c)Q7.(d)

Q8.(c)Q9.(b)Q10.(d)Q11.(b)Q12.(d)Q13.(b)Q14.(a)

Q15.(d)Q16.(b)Q17.(d)Q18.(c)Q19.(c)Q20.(a)Q21.(b)

Q22.(d)Q23.(a)Q24.(b)Q25.(d)Q26.(c)Q27.(b)Q28.(a)Quadratic equationPolynomial and polynomial equation

Let a0, a1, a2,.,an are real numbers, then f(x) = a0 + a1x + a2x2 + . + anxn is known as polynomial.

Polynomial a0 + a1x + a2x2 + .+ anxn is known as a polynomial of degree n if an ( 0.

If f(x) is a real or complex polynomial, then f(x) = 0 is known as a polynomial equation.

If f(x) is a polynomial of degree 2, then f(x) = 0 is known as a quadratic equation. General quadratic equation is ax2 + bx + c, where a, b and c ( C or R.

A quadratic equation of the form ax2 + c = 0 is known as pure quadratic equation.

Identity and equation

A statement of equality of two expressions which is satisfied for each value of variable is called identity.

e.g., (x 4)2 + 8x = x2 = 16 is an identity.

A statement of equality between two expression which is satisfied for definite value of variable, is known as equation.

e.g., x2 ( 5x + 6 = 0 is an equation, which is not satisfied for any value of x respect 2 and 3.

Definition or a root

Values of variables of an equation which satisfied the given equation are known as roots of the equation.

i.e., if f(x) = 0 is a polynomial equation and f(a) = 0, then a is a root of polynomial equation f(x) = 0.

Example: If a, b, c, d ( R such that a < b < c < d, then show that the roots of the equations

(x a) (x c) + 2(x b) (x d) = 0 are real and distinct.

Solution: Let f(x) = (x a) (x c) + 2 (x b) (x d).

Thus, f(a) = 2 (a b) (a d) > 0

[( a b < 0 and a d < 0]

f(b) = (b a) (b c) < 0

[( b a > 0 and b c < 0]

and f(d) = (d a) (d b) > 0

[( d a > 0 and d b > 0]

So, root of f(x) = 0 lies between a and b and another root lies between b and d.

Hence, the roots of the given equation are real and distinct.

Descartes Rules of Signs

The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of signs from positive to negative and negative to positive in f(x).

The maximum number of negative real roots of a polynomial equation f(x) = 0 is the number of changed of signs from positive to negative and negative to positive in f((x). Example: Consider the equation x3 + 6x2 + 11x 6 = 0. The signs of the various terms are:

Solution: Clearly, there is only one change of sign in the expression x3 + 6x2 + 11x ( 6. So, the given equation has at most one positive real root.

Roots of quadratic Equation

Roots of quadratic equation ax2 + bx + c = 0 are and , where b2 4ac is known as discriminant and it is denoted by D.

Relation between coefficient and roots of an equation

Quadratic Equation

If ( and ( are the roots of the quadratic equation ax2 + bx + c = 0, then

( + ( = and (( =

Cubic Equation

If (, ( and ( are the roots of a cubic equation ax3 + bx2 + cx + d = 0, then

( + ( + ( =

(( + (( + (( =

and

((( =

Example: If the roots of the equation x2 + px + q = 0 are in the same ratio as those of the equation x2 + lx + m = 0, prove that p2m = l2q.

Solution: Let the roots of each equation be in the ratio k : 1.

Let k( and ( be the roots of x2 + px + q = 0

Then,

k ( + ( = (p

and

k( . ( = q

( =

and (2 =

(

= (

(i)

Again, let k( and ( be the roots of x2 + lx + m = 0.

Then,

k( + ( = (l and k( . ( = m

(

( = and (2 =

(

i.e.,

(ii)

Thus, from Eqs. (i) and (ii), we get

= , i.e. p2m = l2 q.

Equations of Given roots

Quadratic Equation

If ( and ( are the roots of a quadratic equation, then the equation will be x2 ((( + () x + (( = 0.

Cubic Equation

If (, ( and ( are the roots of a cubic equation, then the equation will be

x3 ( (( + ( + () x2 + ((( + (( + (() x - ((( = 0

Example: If (, ( are the roots of equation 2x2 ( 5x + 7 = 0, then find the equation whose roots are 2( + 3(, 3( + 2(.

Solution: Clearly,

( + ( =

and (( =

Sum of roots = (2( + 3() = (3( + 2()

= 5(( + () =

Product of roots = (2( + 3() (3( + 2()

= 6((2 + (2) + 13((

= 6 [(( + ()2 ( 2((] + 13((

= = 41

The required equation is

x2 ( x + 41 = 0

(

2x2 ( 25x + 82 = 0

Nature of Roots

1.The roots are real and distinct, iff D > 0.

2.The roots are real and equal, iff D = 0.

3.The roots are complex with non-zero imaginary part, iff D < 0.

4.The roots are rational, iff a, b, c are rational and D is perfect square.

5.The roots are of the form p + (p, q ( Q), iff a, b, c are rational and D is not a perfect square.

6.If a = 1, b, c ( I and the roots are rational numbers then these roots must be integers.

7.If a quadratic equation in x has more than two roots then it is an identity in x that is a = b = c = 0.

Example: Show that the roots of the equation 2(a2 + b2) x2 + 2(a + b) x + 1 = 0 are imaginary, when a ( b.

Solution: We have,

D = 4 (a + b)2 ( 8(a2 + b2)

= 4 ((a2 b2 + 2ab)

= (4 (a2 + b2 ( 2ab)

= (4 (a b)2 < 0

[( a b ( 0]

Hence, the roots of the given equation are imaginary.

Example: If the roots of the equation p(q r) x2 + q(r p)x + r (p q) = 0 be equal, show that

Solution: Since, the roots are equal, we have

D = 0

(

q2 (r p)2 ( 4pr (q r) (p q) = 0

(

q2 (r2 + p2 ( 2rp) ( 4pr (pq ( q2 rp + rq) = 0

(

q2r2 + p2q2 + 4r2p2 + 2pq2r 4pqr2 ( 4p2rq = 0

(

(pq + qr 2rp)2 = 0

(

pq + qr 2rp = 0

(

pq + qr = 2rp

(

[on dividing both sides by pqr]Symmetric function

Let ( and ( be the roots of the equation ax2 + bx + c = 0 then

( + ( = ( and (( =

1.(2 + (2 = (( + ()2 ( 2((2.(2 + (2 = (( + ()3 ( 3(( (( + () 3.(4 + (4 = [(( + ()2 ( 2((]2 2((()24.(( ( () =

5.(2 ( (2 = (( + ()

Example: If (, ( are the roots of the equation px2 + qx + r = 0, find the value of (3( + (3(.

Solution: Clearly, ( + ( = and (( =

((3( + (3( = (( ((2 + (2)

= ((() [(( ( ()2 ( 2((]

=

=

maximum and minimum value of ax2 + bx + c

(ax2 + bx + c = a

Case I. When a > 0

(ax2 + bx + c (

(Minimum value of ax2 + bc + c is .

Case II. When a < 0

(

ax2 + bc + c (

(Maximum value of ax2 + bx + c is

Example: Find all the values of a for which the roots of the equation (a 3)x2 ( 2ax + 5a = 0 are positive.

Solution: Let f(x) = (a ( 3)x2 ( 2ax + 5a. For the roots of f(x) = 0 to be positive, we must have

1.Discriminant ( 0

2.Sum of the roots > 0, and

3. (a 3) f(0) > 0

Now,

Discriminant ( 0

4a2 ( 20a (a 3) > 0

(16a2 + 60 a ( 0

4a(4a 15) ( 0 ( 0 ( a (

(i)

Sum of the roots > 0 (

> 0 (

> 0

(

a < 0 or a > 3

(ii)

and

(a 3) f(0) > 0

(

(a 3) 5a > 0

(

a(a 3) > 0

(

a < 0 or a > 3

(iii)

From eqs. (i), (ii) and (iii), we get

3 < a ( i.e., a (

COMMON ROOTS

Let a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 be two quadratic equations such that a1, a2 ( 0 and a1b2 ( a2b1. Let ( be the common root of these two equations. Then,

(b1c1 b2c1) (a1b2 a1h1) = (c1a2 c2a1)2

TRANSFORMATION OF EQUATIONS

(i)To obtain an equation whose roots are reciprocals of the roots of a given equation is obtained by replacing x by 1/x in the given equation.

(ii)Transformation of an equation to another equation whose roots are negative of the roots of a given equation replace x by (x.

(iii)Transformation of an equation to another equation whose roots are square of the roots of a given equation replace x by .

(iv)Transformation of an equation to another equation whose roots are cubes of the roots of a given equation replace x by x1/3.

e.g. Form an equation whose roots are cubes of the roots of equation ax3 + bx2 + cx + d = 0.

Exercise

Q1. The roots of the equation (a)

(b)

(c)

(d)

Q2. If x2 2x + sin2 ( = 0, then x belongs to (a) [(1, 1]

(b) [0, 2]

(c) [(2, 2]

(d) [1, 2]

Q3. The condition that the roots of the equation ax2 + bx + c =0 be such that one root is n times the other (a) na2 = bc (n + 1)2(b) nb2 = ca (n + 1)2(c) nc2 = ab (n + 1)2(d) None of these Q4. If the roots of the equation ax2 + bx + c = 0 are of the form , then (a + b + c)2 is equal to (a) b2 4ac

(b) b2 2ac

(c) 2b2 ac

(d) (a2Q5. If the roots of ax2 + bx + c = 0 are in the ratio m : n, then (a) mna2 = (m + n) c2(b) mnb2 = (m + n) ac

(c) mnb2 = (m + n)2 ac (d) None of these Q6. If p and q are non-zero constants, the equation x2 + px + q = 0 has roots ( and (, then the equation qx2 + px + 1 = 0 bas roots (a) ( and

(b)

(c)

(d) None of these

Q7. If a, b and are positive and are in AP, the roots of the quadratic equation ax2 + bx + c = 0 are real for (a) ( 4

(b)

(c) all a and c

(d) no a and c

Q8. If the roots of the equation, x2 + 2ax + b = 0 are real and distinct and they differ by at most 2m, then b lies in the interval (a) (a2 m2, a2)

(b) [a2 m2, a2)

(c) (a2, a2 + m2)

(d) None of these Q9. The value of k for which one of the roots of x2 x + 3k = 0 is double of one of the roots of x2 x + k = 0 is(a) 1

(b) (2

(c) 2

(d) None of these

Q10. If the equations x2 + 2x + 3( = 0 and 2x2 + 3x + 5( = 0 have a non-zero common root, then ( is equal to (a) 1

(b) (1

(c) 3

(d) None of these

Q11. If the equations ax2 + bx + c = 0 and x2 + x + 1 = 0 have common root, then (a) a + b + c = 0

(b) a = b = c

(c) a = b or b = c or c = a (d) None of these

Q12. If the quadratic equations ax2 + 2cx + b = 0 and ax2 + 2bx + c = 0 (b ( c) have a common root, then a + 4b + 4c is equal to (a) (2

(b) (1

(c) 0

(d) 1

Q13. If 2x 2x(1 = 4, then xx is equal to (a) 1

(b) 2

(c) 256

(d) None of these

Q14. Let (, ( be the roots of the equation x2 3x + p = 0 and let (, ( be the roots of the equation x2 12x + q = 0. If the numbers (, (, (, ( (in order) form an increasing GP, then (a) p = 2, q = 16

(b) p = 2, q = 32

(c) p = 4, q = 16

(d) p = 4, q = 32

Q15. Find the equation whose roots are and (a) cax2 bx + 1 = 0 (b) cax2 + bx + 1 = 0

(c) cax2 + bx 1 = 0 (d) None of these Q16. What is positive square root of 7 + 4? (a)

(b)

(c)

(d)

Q17. If ( and ( are the roots of the equation x2 + x + 2 = 0, then what is equal to? (a) 4096

(b) 2048

(c) 1024

(d) 512

Q18. If ( and ( are the roots of the equation ax2 + bx + b = 0, then what is the value of

= ?

(a) (1

(b) 0

(c) 1

(d) 2

Q19. How many real roots does the quadratic equation f(x) = x2 + 3 |x| + 2 = 0 have? (a) One

(b) Two

(c) Four

(d) No real root

Q20. If ( and ( are the roots of the equation x2 + bx + c = 0, then what is the value of ((1 + ((1 ?

(a) (

(b)

(c)

(d)

Q21. If the roots of a quadratic equation ax2 + bx + c = 0 are ( and (, then the quadratic equation having roots (2 and (2 is (a) x2 (b2 2ac) x + c = 0

(b) a2x2 ( (b2 2ac) x + c = 0

(c) ax2 ( (b2 2ac) x + c2 = 0

(d) a2x2 ( (b2 2ac) x + c2 = 0 Q22. If the roots of the equation 3ax2 + 2bx + c = 0 are in the ratio 2 : 3, then which one of the following is correct? (a) 8ac = 25b

(b) 8ac = 9b2(c) 8b2 = 9ac

(d) 8b2 = 25ac Q23. If the roots of a quadratic equation are m + n and m ( n, then the quadratic equation will be (a) x2 + 2mx + m2 mn + n2 = 0

(b) x2 + 2mx + (m n)2 = 0

(c) x2 2mx + m2 n2 = 0

(d) x2 + 2mx + m2 n2 = 0 Q24. If (, ( are the roots of x2 + px q = 0 and (, ( are the roots of x2 px + r = 0, then what is the value of (( + () (( + ()? (a) p + r

(b) p +q

(c) q + r

(d) p q

Q25. If the roots of the equation x2 4ax ( log3 N = 0 are real, then what is the minimum value of N ? (a)

(b)

(c)

(d)

Q26. If the equations x2 px + q = 0 and x2 ax + b = 0 have a common root and the roots of the second equation are equal, then which one of the following is correct? (a) aq = 2 (b + p)

(b) aq = b + p

(c) ap = 2 (b + q)

(d) ap = b + q

Q27. If one of the roots of the equation a(b c) x2 + b(c a) x + c(a b) = 0 is 1, then which is the second root? (a)

(b)

(c)

(d)

Q28. What is the value of Y = ?

(a) 10

(b) 8

(c) 6

(d) 4

Q29. One of the roots of the quadratic equation ax2 + bx + c = 0, a ( 0 is positive and the other root is negative. The condition for this to happen is (a) a > 0, b > 0, c > 0 (b) a > 0, b > 0, c > 0

(c) a < 0, b > 0, c < 0 (d) a < 0, c > 0

Q30. If 3 is the root of the equation x2 8x + k = 0, then what is the value of k? (a) (15

(b) 9

(c) 15

(d) 24

Q31. If ( and ( are the roots of the equation x2 2x + 4 = 0, then what is the value of (3 + (3 ? (a) 16

(b) 16

(c) 8

(d) (8

Q32. If (, ( are the roots, of the quadratic equation x2 x + 1 = 0, then which one of the following is correct? (a) ((4 ( (4) is real (b) 2((5 + (5) = ((()5(c) ((6 ( (6) = 0

(d) ((8 + (8) = ((()8Q33. If is one of the roots of ax2 + bx + c = 0, where a, b, and c are real, then what are the values of a, b, c respectively (a) 6, (4, 1

(b) 4, 6, (1

(c) 3, (2, 1

(d) 6, 4, 1

Q34. If ( is a complex cube roots of unity and x = (2 ( ( - 2, then what is the value of x2 + 4x + 7? (a) (2

(b) (1

(c) 0

(d) 1

Q35. If a, b and c are real numbers, then roots of the equation (x a) (x b) + (x b) (x c) + (x c) (x a) = 0 are always (a) real

(b) imaginary

(c) positive

(d) negative

Q36. If x = 2 + 21/3 + 22/3, then what is the value of x3 6x2 + 6x? (a) 1

(b) 2

(c) 3

(d) (2

Q37. If (, ( are the roots of ax2 + bx + b = 0, then what is equal to? (a) 0

(b) 1

(c) 2

(d) 3

Q38. Which one of the following is one of the roots of the equation (b c) x2 + (c a) x + (a b) = 0?

(a)

(b)

(c)

(d)

Q39. If ( and ( are the roots of the equation 2x2 2(1 + n2) x + (1 + n2 + n4) = 0, then what is the value of (2 + (2? (a) 2n2

(b) 2n4(c) 2

(d) n2Answers

Q1.(c)Q2.(b)Q3.(b)Q4.(a)Q5.(c)Q6.(c)Q7.(a)

Q8.(b)Q9.(b)Q10.(b)Q11.(b)Q12.(c)Q13.(d)Q14.(b)

Q15.(a)Q16.(d)Q17.(c)Q18.(b)Q19.(d)Q20.(a)Q21.(d)

Q22.(d)Q23.(c)Q24.(c)Q25.(d)Q26.(c)Q27.(c)Q28.(d)

Q29.(d)Q30.(c)Q31.(b)Q32.(d)Q33.(a)Q34.(c)Q35.(a)

Q36.(b)Q37.(a)Q38.(b)Q39.(d)

Sequence and seriesArithmetic progression

A sequence is said to be an arithmetic progression, if the difference of a term and the previous term is always same, i.e.,

an+1 ( an = Constant (= d) ( n ( N.

The constant difference, generally denoted by d, is called the common difference.

In the sequence a, a + d, a + 2d, , the difference between two consecutive terms is constant, then this sequence is an arithmetic progression. Here first term is a and common difference is d.

nth term of series,

Tn = a + (a ( 1) d.

Last term of series,

l = a + (n 1) d

Sum of n terms of series,

Sn = [(2a + (n 1)d] = (a + l)

Example: In an AP, the pth term is (q and the (p + q)th term is zero, then find the qth term.

Solution: (Tp = (q and Tp+q = 0

(a + (p ( 1) d = (q

(i)

and a + (p + q ( 1) d = 0

(ii)

On solving Eqs. (i) and (ii), we get

d = 1 and a = (q + 1 ( p

(

Tq = a + (q ( 1) d = (p

Important Facts 1.If a fixed constant C is added to (or subtracted from) each term of a given AP, then the resulting sequence is also an AP, with the same common difference as that of the given AP.

2.If each term of an AP is multiplied by (or divided by a non-zero) fixed constant C, then the resulting sequence is also an AP with common difference C times the previous.

3.If a1, a2, a3,, and b1, b2, b3, are two AP(s then a1 ( b1, a2 ( b2, a3 ( b3, is also an AP with common difference d1 ( d2. 4.If the pth term of an AP is q and the qth term is p, then its (p + q)th term is (p + q n)

5.If the pth term of AP is and the qth term is , then its pqth term is 1 and sum of pq terms (Spq) is (pq + 1) 6.If in an AP, Sp = q and Sq = p, then Sp+a = ((p + q). 7.If the sum of three consecutive terms of an AP is given, it is convenient to assume them is a d, a, a + d, where the common difference is d.

8.If the sum of four consecutive terms of an AP is given it is convenient to assume that as a 3d, a d, a + d, a + 3d, where the common difference is 2d.

9.In an AP, the sum of terms equidistant from the beginning and end is constant and equal to the sum of first and last term, i.e.,

a1 + an = a2 + an(1 = a3 + an(2 = . 10. Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it, i.e.,

an = (an(k + an+k), k < n

and for k = 1, an = (an(1 + an+1)

11.Tn = Sn ( Sn(1 (n ( 2). 12.If Tn = pn + q, then it will form an AP of common difference p and first term p + q. 13.a sequence is an AP, iff its nth term is of the temr An + B, i.e., a linear expression in n.

14.A sequence obtained by multiply or division of corresponding terms of two AP is not an AP.

SELECTION OF TERMS IN AN A.P.

It should be noted that in case of an odd number of terms, the middle term is a and the common difference is d while in case of an even number of terms the middle terms are a d, a + d and the common differences is 2d. i.e.,

(1)Selecting odd number of terms of A.P.

(i)Selecting 3 terms of A.P.:

a d, a, a + d

(ii)Selecting 5 terms of A.P.:

a 2d, a d, a, a + d, a + 2d.

(2)Selecting even number of terms of A.P.

(i)Selecting 2 terms of A.P.:

a d, a + d

(ii)Selecting 4 terms of A.P.:

a 3d, a d, a + d, a + 3d and so on.

SOME USEFUL RESUTLS

(i)

r = 1 + 2 + + n =

(ii)

= 12 + 22 +..+ n2 =

(iii)

r3 = 13 + 23 ++ n3 =

(iv)

= 14 + 24 + + n4 =

Example: The ratio between the sum of two arithmetic progressions is (7n + 1) : (4n + 27). Find the ratio of their 11th terms.

Solution: Let a1, a2 be the first terms and d1 and d2 be the common differences of the given APs. Then their sums of n terms are given by

Sn= [2a1 + (n 1)d1]

and

Sn( = [2a2 + (n ( 1)d2]

On dividing Eqs. (i) by (ii), we get

=

(The ratio of their 11th term

=

=

=

SUM OF n TERMS OF AN A.P.

The sum Sn of n terms of an A.P. with first term a and common difference d is given by

Sn = [2a + (n 1)d]

Also, Sn = [a + l], where l = last term = a + (n 1) d.

Arithmetic Mean of two quantities

Let a and b are two quantities, then their arithmetic mean, ( = .

NOTE:

*If there are n arithmetic means between a and b, then they will be A1 =

*The sum of n arithmetic means between a and b is n times of arithmetic means of a and b.

i.e.

A1 + A2 +..+ An = n = nA

Example: m arithmetic means have been inserted between 1 and 31 in such a way that the ratio of the 7th and the (m ( 1)th manes is 5 : 9, Find the value of m.

Solution: Let x1, x2,.., xm be the m arithmetic means between 1 and 31. .

Then 1, x1, x2,., xm, 31 are in AP.

Let d be the common difference of this AP.

Then tm+2 = 31 ( 1 + (m + 2 ( 1) d = 31

(

d =

(i)

Also,

= (

(

(

d =

(ii)

From Eqs. (i) and (ii), we get

= ( m = 14.

Geometric progression

A sequence of non-zero number is called a geometric progression, if the ratio of a term and the term preceding to it is always a constant quantity.

The constant ratio is called the common ratio of the GP

If a1, a2, a3,., an are in GP, then

= r

r is known as geometric ratio of GP.

i.e., a, ar, ar2, are in GP, here a is first term of r is common ratio.

nth term of GP, Tn = arn(1

Last term of a GP, l = an(1

Sum of n terms of GP,

Sn =

where r > 1

=

when r < 1

Sum of infinite terms of GP,

S( =

where |r| < 1.

Example: If ax = by = cz and x, y, z are in GP, prove that logb a = logc b.

Solution: Let ax = by = cz = k.

Then loga k = x, logb k = y and logc k = z.

Now, x, y, z are in GP.

(

(

(logb a = logc b

Example: If p, q, r are in AP, show that the pth, qth and rth terms of any GP are in GP.

Solution: Let us consider a GP with first term a and the common ratio R. Then

Tp = aRp(1, Tq = aRq(1 and Tr = aRr(1

(

= R(q ( p)

and

= R(r(q)

But p, q, r being in AP, we have (q p) = (r q)

So,

This shows that Tp, Tq, Tr are in GP.

IMPORTANT FACTS

1.If each term of a GP is multiplied (or divided) by a non-zero constant C((0), then the resulting sequence is also a GP with same common zero.

2.If a, ar, ar2, and a(, a( r(, a( r(2,. Are two GPs, then the sequence aa(, aa( (rr(), aa( (rr()2,. and , obtained by the products and quotients of the corresponding terms of the given GPs are also GPs with common ratios rr( and respectively. 3.The reciprocals of the terms of a GP also form a GP.

4.If a1, a2, a3, is a GP of positive terms (that is, ai > 0, for all values of i), then log a1, log a2, log a3, is an AP and the converse is also true in this case. 5.The geometric mean G of two non-zero numbers a and b is given by . It is to be noted that a, G and b are in GP. If a1, a2,., an are n non-zero numbers, then their geometric means is given by G = (a1a2 .an)1/n. 6.The product of n geometric means between a and is 1.

7.The odd number of terms in a GP should be taken as . ar3, ar2, ar, a, , ,. while the even number of terms in a GP should be taken as .ar5, ar3, ar, ,8.Let the first term of a GP be positive, then if r > 1, then it is an increasing GP, but, if r is positive and less than 1 i.e., 0 < r < 1, then it is a decreasing GP.

9.Let the first term of a GP be negative, then if r > 1 then it is a decreasing GP, but if 0 < r < 1, then it is a increasing GP.

10.If a1, a2, a3,, an are in GP, then will also be in GP whose common ratio is rk. 11.If a1, a2, a3,., an are in AP, the aa1, aa2, aa3,., aan will be GP whose common ratio is ad.

geometric mean of two quantities

Let a and b are two quantities, then Geometric means of a and b is G = .

Example: The arithmetic means between two positive number a and b, where a > b is twice their geometric mean. Prove that a : b = 2 (2 + ) : (2 ( )

Solution: AM = (a + b) and GM =

(

AM = 2 (GM)

(

(a + b) = 2

(

=

(

[by componendo and dividendo]

(

(

=

(

[by compnendo and dividendo]

(

(

HARMONIC PROGRESSION

A sequence a1, a2,., an of non-zero numbers is called a harmonic progression, if the sequence is an AP.

OR

A sequence is said be a harmonic progression, if reciprocal of terms of a sequence form an AP.

nth term of HP from beginning.

Tn=

=

nth term of HP from end

Tn( =

=

Example:If a, b, c are in GP, prove that loga n, logb n and logc n are in HP.

Solution: Let loga n = x, logb n = y and logc n = z. Then

n = ax , n = by and n = cx

(

ax = by = cx = n

a = n1/x, b = n = n1/y and c = n1/z

Now, a, b, c are in GP,

(

ac = b2(

n1/x . n1/z = (n1/y)2

(

(

(

are in AP.

x, y, z are in HP.

Hence, loga n, logb n and logc n are in HP.

HARMONIC MEAN OF TWO QUANTITIES

Let a and b are two quantities, then harmonic mean of a and b, H =

Relation among arithmetic mean, geometric mean and harmonic mean

If a and b are two real numbers and A, G, H are AM, GM and HM respectively.

(A = , G = and H =

(

A ( G ( H and G2 = AH

1.The equation having a and b as its roots is

x2 2Ax + G2 = 0

2.If A, G, H are arithmetic, geometric and harmonic means between three given numbers a, b and c, then the equation having a, b, c as its roots is

x3 3Ax2 + x G3 = 0

Example: If and H are respectively the arithmetic mean and the harmonic mean between a and b, prove that .

Solution: We known that

AH = G2 = ab

(i)

(

=

=

[using (i)]

=

Hence,

Arithmetico-Geometric Series

A sequence whose each term is obtained by multiplying corresponding terms of AP and GP is called an arithmetico-geometric series.

If a + (a + d) + (a + 2d) + + (a + (n 1) d is an AP and 1 + r + r2 + ..+ rn(1 is a GP, then an multiplying corresponding terms of AP and GP a series.

a + (a + d) r + (a + 2d) r2 +.+ (a + (n ( 1) d) rn(1 is formed which is called a arithmetico-geometric series.

Sum of a n terms of arithmetico-geometric series.

Sn =

Sum of infinite terms of arithmetico-geometric series.

S( = ,

|r| < 1

To Find nth Term by the Difference Method

If T1, T2,, Tn are terms of any series and their difference (T2 ( T1), (T3 ( T2), (T4 ( T3),., (Tn ( Tn(1) are either in AP or in GP, then Tn can be easily find out. Thus on finding Tn, sum of series can be calculated by ( notation.

This method of finding Tn is known as difference method. Some times this method is used several times.

Example: Find the sum of the infinite series 1 + + + (.

Solution: This clearly not a arithmetico-geometic series, since, 1, 4, 9, 16, are not in AP. However, their successive difference (4 ( 1), (9 4), (16 9), are in AP.

Let

S( = 1 + + + .(

+ . (

On subtraction, we get

S( = 1 + + + + (

+ (

[multiplying throughout by ]

On subtracting the two series, we get

+ (

= 1 + = 2

(S( = =

Sigma Notion

Sum of terms which satisfy the same relation or condition is denoted by sigma ((). Important Facts 1.

a = a + a + upto m = am.

2.

3.Sum of n natural numbers= 1 + 2 + 3 + 4 +.+ n = (n =

4.Sum of squares of n natural numbers.

= 12 + 22 + 32 + 42 +.+ n2

= ( n2 =

5.Sum of cubes of n natural numbers

= 13 + 23 + 33 + . + n3

= (n3 = = ((n)26.Sum of cubes of natural numbers is divisible by sum of those numbers.

7.If nth term of any series is (an3 + bn2 + cn + d), then the sum of series,

S = ( (an3 + bn2 + cn + d)

= a(n3 + b(n2 + c(n + dn

= a + c + dn

Example: Find the sum of n terms of the series 12 + 32 + 52 +. to n terms.

Solution: tn = [1 + (n 1) ( 2]2

= (2n ( 1)2 = (4n2 ( 4n + 1)

Sn = (4k2 ( 4k + 1)

=

= 4 . n (n + 1) (2n + 1) ( 4. n (n + 1) + n

= [2 (n + 1) (2n + 1) (6 (n + 1) + 3] = (4n2 1)

Exercise

Q1. If = 7 the value of n is (a) 35

(b) 36

(c) 37

(d) 40 Q2. If the sum of the series 2, 5, 8, 11,. Is 60100, then n is equal to (a) 100

(b) 200

(c) 150

(d) 250 Q3. Let Sn denotes the sum of first n terms of an AP, if S2n = 3 Sn, then the ratio S3n/Sn is equal to (a) 4

(b) 6

(c) 8

(d) 10 Q4. If sum of n terms of an AP is 3n2 + 5n and Tm = 164, then m is equal to (a) 26

(b) 27

(c) 28

(d) None of these Q5. The sum of the first n terms of the series + . Is (a) 2n n ( 1

(b) 1 2(n

(c) n + 2(n 1

(d) 2n ( 1 Q6. If x = 1 + a + a2 + a3 +. to ( (|a| < 1) and y = 1 + b + b2 + b3 + .. to ( (|a| < 1), then 1 + ab + a2b2 + a3b3 + . ( is equal to (a)

(b)

(c)

(d) None of these Q7. In a GP, if the (m + n)th term be p and (m n)th term be q, then its mth term is (a)

(b)

(c)

(d)

Q8. The three harmonic means between 5 and 6 are (a)

(b)

(c)

(d) None of these Q9. If S be the sum to infinity of a GP, whose first term is a, then the sum of the first n terms is (a) S

(b)

(c)

(d) None of these Q10. If a, b, c are in GP, then is equal to (a) 1/(c2 b2)

(b) 4b2 c2(c) 1/(c2 a2)

(d) 1/(b2 c2) Q11. If the non-zero numbers a, b, c are in AP and tan-1 a, tan(1 b, tan(1 c are also in AP, then (a) a = b = c

(b) b2 = 2ac

(c) a2 = bc

(d) c2 = ab Q12. The geometric and harmonic means of two numbers x1 and x2 are 18 and 16, respectively. The value of |x1 ( x2| is equal to(a) 5

(b) 10

(c) 15

(d) 20 Q13. The value of x + y + z is 15, if a, x, y, z, b are in AP while the value of is , if a, x, y, z, b are in HP. Then, a and b are (a) 1, 9

(b) 3, 7

(c) 7, 3

(D) None of these Q14. If the harmonic mean between two positive numbers is to their GM as 12 : 13, the numbers are in the ratio (a) 4, 9

(b) 7, 6

(c) 6, 5

(d) None of these Q15. Two AMs A1 and A2, two GMs. G1 and G2 and two HMs H1 and H2 are inserted between any two numbers, then H1(1 + H2(1 is equal to

(a) A1-1 + A2-1

(b) G1(1 + G2(1(c)

(d)

Q16. If is the arithmetic mean between a and b, then n is equal to (a) (1

(b) (2

(c) 0

(d) 1Q17. The sum of 1 + upto n terms is (a)

(b)

(c)

(d) None of these Q18. The harmonic mean of and is equal to (a)

(b)

(c) a

(d)

Q19. If (m + 1) th, (n +1 )th and (r + 1)th terms of an AP arein GP and m, n, r are in HP, then the ratio of the first term of the AP to its common difference in terms of n is equal to (a) n/2

(b) (n/2

(c) n/3

(d) (n/3

Direction: Given two series

S1 = 1 + 2 +4 + 8 + . to 100 terms

and S1 = 1 + 4 + 7 + 10 + to 100 terms Q20. Find last terms of series S1 (a) 298

(b) 299(c) 2100

(d)

Q21. Find the number of common terms in both of the series (a) 4

(b) 5

(c) 6

(d) 7 Q22. Find the sum of all the terms of series S2. (a) 14450

(b) 14590

(c) 14950

(d) 19450 Direction: Read the following information carefully and answer the question that follow: Consider a sequence whose sum to n terms is given by quadratic function, Sn= 3n2 + 5n. Q23. The nature of the given series is (a) AP

(b) GP

(c) HP

(d) AGP Q24. For the given sequence the number 5456 is the (a) (153)th term

(b) (932) th term

(c) (707)th term

(d) (909) th term

Q25. Sum of the squares of the first 3 terms of the given series is

(a) 1100

(b) 660

(c) 799

(d) 1000Directions: The following questions are Assertion Reason type Questions. Each of these question contains two statements. Statement I (Assertion) and Statement II (Reason) Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice. Q26. Statement I: If sum of n terms of a series is 5n2 + 3n + 1, then the series is an AP. Statement II: Sum of n terms of an AP is always of the form an2 + bn.




(d) Statement I is false, Statement II is trueQ27. Statement I: If a, b, c are 3 positive numbers in GP. Then

Statement II: (Arithmetic mean) (Harmonic mean) = (Geometric mean)2.



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