#### Chapters

Chapter 2 - Relations and Functions

Chapter 3 - Trigonometric Functions

Chapter 4 - Principle of Mathematical Induction

Chapter 5 - Complex Numbers and Quadratic Equations

Chapter 6 - Linear Inequalities

Chapter 7 - Permutations and Combinations

Chapter 8 - Binomial Theorem

Chapter 9 - Sequences and Series

Chapter 10 - Straight Lines

Chapter 11 - Conic Sections

Chapter 12 - Introduction to Three Dimensional Geometry

Chapter 13 - Limits and Derivatives

Chapter 14 - Mathematical Reasoning

Chapter 15 - Statistics

Chapter 16 - Probability

## Chapter 1 - Sets

#### Pages 4 - 24

Which of the following are sets? Justify our answer.

The collection of all months of a year beginning with the letter J.

Which of the following are sets? Justify our answer.

The collection of ten most talented writers of India.

Which of the following are sets? Justify our answer.

A team of eleven best-cricket batsmen of the world.

Which of the following are sets? Justify our answer.

The collection of all boys in your class.

Which of the following are sets? Justify our answer.

The collection of all natural numbers less than 100.

Which of the following are sets? Justify our answer.

A collection of novels written by the writer Munshi Prem Chand.

Which of the following are sets? Justify our answer.

The collection of all even integers.

Which of the following are sets? Justify our answer.

The collection of questions in this Chapter.

Which of the following are sets? Justify our answer.

A collection of most dangerous animals of the world.

If X and Y are two sets such that X ∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩Y have?

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈or ∉ in the blank spaces:

**(1)** 5…A

**(2**) 8…A

**(3)** 0…A

**(4)** 4…A

**(5)** 2…A

**(6)** 10…A

Write the following sets in roster form: A = {*x*: *x* is an integer and –3 < *x *< 7}.

Write the following sets in roster form:

B = {*x*: *x* is a natural number less than 6}.

Write the following sets in roster form:

C = {*x*: *x* is a two-digit natural number such that the sum of its digits is 8}

Write the following sets in roster form:

D = {*x*: *x* is a prime number which is divisor of 60}.

Write the following sets in roster form:

F = The set of all letters in the word BETTER.

Write the following sets in the set-builder form: (3, 6, 9, 12)

Write the following sets in the set-builder form: {2, 4, 8, 16, 32}

Write the following sets in the set-builder form: {5, 25, 125, 625}

Write the following sets in the set-builder form: {2, 4, 6 …}

Write the following sets in the set-builder form: {1, 4, 9 … 100}

List all the elements of the sets: A = {*x*: *x* is an odd natural number}

List all the elements of the following sets:

B = {*x*: *x* is an integer, -1/2 < x < 9/2}

List all the elements of the following sets:

C = {x : x is an integer, x^{2} ≤ 4}

List all the elements of the following sets:

D = {*x*: *x* is a letter in the word “LOYAL”}

List all the elements of the following sets:

E = {*x*: *x* is a month of a year not having 31 days}

List all the elements of the following sets:

F = {*x*: *x* is a consonant in the English alphabet which proceeds *k*}.

Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

1 | {1, 2, 3, 6} | a | {x: x is a prime number and a divisor of 6} |

2 | {2, 3} | b | {x: x is an odd natural number less than 10} |

3 | {M, A,T, H, E, I,C, S} | c | {x: x is natural number and divisor of 6} |

4 | {1, 3, 5, 7, 9} | d | {x: x is a letter of the word MATHEMATICS} |

Write the following sets in roster form:

E = The set of all letters in the word TRIGONOMETRY.

#### Pages 8 - 9

Which of the following are examples of the null set

(i) Set of odd natural numbers divisible by 2

(ii) Set of even prime numbers

(iii) {*x*:*x *is a natural numbers, *x *< 5 and *x *> 7 }

(iv) {*y*:*y *is a point common to any two parallel lines}

Which of the following sets are finite or infinite

The set of months of a year

Which of the following sets are finite or infinite

{1, 2, 3 ...}

Which of the following sets are finite or infinite {1, 2, 3 ... 99, 100}

Which of the following sets are finite or infinite

The set of positive integers greater than 100

Which of the following sets are finite or infinite

The set of prime numbers less than 99

State whether the following set is finite or infinite:

The set of lines which are parallel to the *x*-axis

State whether each of the following set is finite or infinite:

The set of letters in the English alphabet

State whether each of the following set is finite or infinite:

The set of numbers which are multiple of 5

State whether the following set is finite or infinite:

The set of animals living on the earth

State whether each of the following set is finite or infinite:

The set of circles passing through the origin (0, 0)

In the following, state whether A = B or not:

A = {*a*, *b*, *c*, *d*}; B = {*d*, *c*, *b*, *a*}

In the following, state whether A = B or not:

A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

In the following, state whether A = B or not:

A = {2, 4, 6, 8, 10}; B = {*x*: *x *is positive even integer and *x *≤ 10}

In the following, state whether A = B or not:

A = {2, 4, 6, 8, 10}; B = {*x*: *x *is positive even integer and *x *≤ 10}

In the following, state whether A = B or not:

A = {*x*: *x *is a multiple of 10}; B = {10, 15, 20, 25, 30 ...}

Are the following pair of sets equal? Give reasons.

A = {2, 3}; B = {*x*: *x *is solution of *x*^{2} + 5*x *+ 6 = 0}

Are the following pair of sets equal? Give reasons.

A = {*x*: *x *is a letter in the word FOLLOW}; B = {*y*: *y *is a letter in the word WOLF}

From the sets given below, select equal sets:

A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2} E = {–1, 1}, F = {0, *a*}, G = {1, –1}, H = {0, 1}

#### Pages 12 - 13

Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:

**(i)** {2, 3, 4} … {1, 2, 3, 4, 5}

**(ii)** {*a*, *b*, *c*} … {*b*, *c*, *d*}

**(iii)** {*x*: *x* is a student of Class XI of your school} … {*x*: *x* student of your school}

**(iv)** {*x*: *x* is a circle in the plane} … {*x*: *x* is a circle in the same plane with radius 1 unit}

**(v)** {*x*: *x* is a triangle in a plane}…{*x*: *x* is a rectangle in the plane}

**(vi)** {*x*: *x* is an equilateral triangle in a plane}… {*x*: *x* is a triangle in the same plane}

**(vii)** {*x*: *x* is an even natural number} … {*x*: *x* is an integer}

Examine whether the following statements are true or false: {*a*, *b*} ⊄ {*b*, *c*, *a*}

Examine whether the following statements are true or false:

{*a*, *e*} ⊂ {*x*: *x* is a vowel in the English alphabet}

Examine whether the following statements are true or false:

{1, 2, 3} ⊂{1, 3, 5}

Examine whether the following statements are true or false:

{*a*} ⊂ {*a*. *b*, *c*}

Examine whether the following statements are true or false:

{*a*} ∈ (*a*, *b*, *c*)

Examine whether the following statements are true or false:

{*x*: *x* is an even natural number less than 6} ⊂ {*x*: *x* is a natural number which divides 36}

Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?

**(i) **{3, 4}⊂ A

**(ii)** {3, 4}}∈ A

**(iii)** {{3, 4}}⊂ A

**(iv)** 1∈ A

**(v)** 1⊂ A

**(vi) **{1, 2, 5} ⊂ A

**(vii)** {1, 2, 5} ∈ A

**(viii)** {1, 2, 3} ⊂ A

**(ix)** Φ ∈ A

**(x)** Φ ⊂ A

**(xi)** {Φ} ⊂ A

Write down all the subsets of the following sets:

**(i)** {*a*}

**(ii)** {*a*, *b*}

**(iii)** {1, 2, 3}

**(iv)** Φ

How many elements has P(A), if A = Φ?

Write the following as intervals: {*x*: *x* ∈ R, –4 < *x* ≤ 6}

Write the following as intervals: {*x*: *x* ∈ R, –12 < *x* < –10}

Write the following as intervals: {*x*: *x* ∈ R, 0 ≤ *x* < 7}

Write the following as intervals: {*x*: *x* ∈ R, 3 ≤ *x* ≤ 4}

Write the given intervals in set-builder form:

(–3, 0)

Write the given intervals in set-builder form:

[6, 12]

Write the following intervals in set-builder form:

(6, 12]

Write the following intervals in set-builder form:

[–23, 5)

What universal set (s) would you propose for the following:

The set of right triangles

What universal set (s) would you propose for the following:

The set of isosceles triangles

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C

{0, 1, 2, 3, 4, 5, 6}

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C

Φ

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

{1, 2, 3, 4, 5, 6, 7, 8}

#### Pages 17 - 18

Find the union of each of the following pairs of sets:

X = {1, 3, 5} Y = {1, 2, 3}

Find the union of each of the following pairs of sets:

A = {*a*, *e*, *i*, *o*, *u*} B = {*a*, *b*, *c*}

Find the union of each of the following pairs of sets:

A = {*a*, *e*, *i*, *o*, *u*} B = {*a*, *b*, *c*}

Find the union of each of the following pairs of sets:

A = {*x*: *x* is a natural number and multiple of 3} B = {*x*: *x* is a natural number less than 6}

Find the union of each of the following pairs of sets:

A = {*x*: *x* is a natural number and multiple of 3} B = {*x*: *x* is a natural number less than 6}

Find the union of each of the following pairs of sets:

A = {*x*: *x* is a natural number and 1 <* x* ≤ 6} B = {*x*: *x* is a natural number and 6 <* x* < 10}

Find the union of each of the following pairs of sets:

A = {1, 2, 3}, B = Φ

Let A = {*a*, *b*}, B = {*a*, *b*, *c*}. Is A ⊂ B? What is A ∪ B?

If A and B are two sets such that A ⊂ B, then what is A ∪ B?

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

**(i) **A ∪ B

**(ii) **A ∪ C

**(iii) **B ∪ C

**(iv) **B ∪ D

**(v) **A ∪ B ∪ C

**(vi) **A ∪ B ∪ D

**(vii) **B ∪ C ∪ D

Find the intersection of each pair of sets: X = {1, 3, 5} Y = {1, 2, 3}

Find the intersection of each pair of sets:

A = {*a*, *e*, *i*, *o*, *u*} B = {*a*, *b*, *c*}

Find the intersection of each pair of sets:

A = {*x*: *x* is a natural number and multiple of 3} B = {*x*: *x* is a natural number less than 6}

Find the intersection of each pair of sets:

A = {*x*: *x* is a natural number and 1 <* x* ≤ 6} B = {*x*: *x* is a natural number and 6 <* x* < 10}

Find the intersection of each pair of sets: A = {1, 2, 3}, B = Φ

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

**(i) **A ∩ B

**(ii) **B ∩ C

**(iii)** A ∩ C ∩ D

**(iv)** A ∩ C

**(v) **B ∩ D

**(vi)** A ∩ (B ∪ C)

**(vii)** A ∩ D

**(viii)** A ∩ (B ∪ D)

**(ix)** (A ∩ B) ∩ (B ∪ C)

**(x) **(A ∪ D) ∩ (B ∪ C)

If A = {*x*:* x* is a natural number}, B ={*x*:* x* is an even natural number} C = {*x*:* x* is an odd natural number} and D = {*x*:* x *is a prime number}, find

**(i)** A ∩ B

**(ii)** A ∩ C

**(iii)** A ∩ D

**(iv)** B ∩ C

**(v)** B ∩ D

**(vi)** C ∩ D

Which of the following pairs of sets are disjoint

{1, 2, 3, 4} and {*x*:* x *is a natural number and 4 ≤ *x* ≤ 6}

Which of the following pairs of sets are disjoint

{*a*, *e*, *i*, *o*, *u*}and {*c*, *d*, *e*, *f*}

Which of the following pairs of sets are disjoint

{*x*:* x* is an even integer} and {*x: x* is an odd integer}

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find A – B

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20};

find D – B

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find C – D

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find D – C

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find A - C

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20};

find A – D

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find B – A

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20};

find C – A

find D – A

find B – C

find B – D

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find C – B

If X = {*a*, *b*, *c*, *d*} and Y = {*f*, *b*, *d, g*}, find

**(i)** X – Y

**(ii)** Y – X

**(iii)** X ∩ Y

If **R** is the set of real numbers and **Q** is the set of rational numbers, then what is **R** – **Q**?

State whether each of the following statement is true or false. Justify your answer.

{2, 3, 4, 5} and {3, 6} are disjoint sets.

State whether each of the following statement is true or false. Justify your answer.

{*a*, *e*, *i*, *o*, *u* } and {*a*, *b*, *c*, *d*} are disjoint sets.

State whether each of the following statement is true or false. Justify your answer.

{2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

State whether each of the following statement is true or false. Justify your answer.

{2, 6, 10} and {3, 7, 11} are disjoint sets.

#### Pages 20 - 21

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find A'

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find B'

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find (A ∪ C)'

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find (A ∪ B)'

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find (B -C)'

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find (A')'

If U = {*a, b, c, d, e, f, g, h*}, find the complements of the following sets:

A = {*a, b, c*}

If U = {*a, b, c, d, e, f, g, h*}, find the complements of the following sets:

B = {*d, e, f, g*}

If U = {*a, b, c, d, e, f, g, h*}, find the complements of the following sets:

C = {*a, c, e, g*}

If U = {*a, b, c, d, e, f, g, h*}, find the complements of the following sets:

D = {*f*, *g*, *h*, *a*}

Taking the set of natural numbers as the universal set, write down the complements of the following sets:

**(i) **{*x*: *x* is an even natural number}

**(ii) **{*x*: *x* is an odd natural number}

**(iii)** {*x*: *x* is a positive multiple of 3}

**(iv)** {*x*: *x* is a prime number}

**(v) **{*x*: *x* is a natural number divisible by 3 and 5}

**(vi)** {*x*: *x* is a perfect square}

**(vii)** {*x*: *x* is perfect cube}

**(viii)** {*x*: *x* + 5 = 8}

**(ix)** {*x*: 2*x* + 5 = 9}

**(x) **{*x*: *x* ≥ 7}

**(xi)** {*x*: *x* ∈ N and 2*x* + 1 > 10}

If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (A ∪ B)' = A' ∩ B'

If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (A ∩ B)' = A' ∪ B'

Draw appropriate Venn diagram for the following:

(A ∪ B)'

Draw appropriate Venn diagram for the following:

A' ∩ B'

Draw appropriate Venn diagram for the following:

(A ∩ B)'

Draw appropriate Venn diagram for the following:

A' ∪ B'

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A'

Fill in the blanks to make each of the following a true statement:

A ∪ A' = ....

Fill in the blanks to make each of the following a true statement:

Φ′ ∩ A = …

Fill in the blanks to make each of the following a true statement:

A ∩ A' =....

Fill in the blanks to make each of the following a true statement:

U' ∩ A = ...

#### Page 24

If X and Y are two sets such that *n*(X) = 17, *n*(Y) = 23 and *n*(X ∪ Y) = 38, find *n*(X ∩Y).

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

If S and T Are Two Sets Such that S Has 21 Elements, T Has 32 Elements, and S ∩ T Has 11 Elements, How Many Elements Does S ∪ T Have?

If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?

In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?

#### Pages 26 - 27

Decide, among the following sets, which sets are subsets of one and another:

A = {*x*: *x* ∈ R and *x* satisfy *x*^{2} – 8*x* + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If *x* ∈ A and A ∈ B, then *x* ∈ B

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ∈ C, then A ∈ C

determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ⊂ C, then A ⊂ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊄ B and B ⊄ C, then A ⊄ C

determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If *x* ∈ A and A ⊄ B, then *x* ∈ B

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and *x* ∉ B, then *x* ∉ A

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.

Show that the following four conditions are equivalent:

**(i)** A ⊂ B

**(ii)** A – B = Φ

**(iii)** A ∪ B = B

**(iv)** A ∩ B = A

Show that if A ⊂ B, then C – B ⊂ C – A.

Assume that P (A) = P (B). Show that A = B.

Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.

Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)

Using properties of sets show that A ∪ (A ∩ B) = A

Using properties of sets show that A ∩ (A ∪ B) = A.

Show that A ∩ B = A ∩ C need not imply B = C.

Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)

Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = Φ.

In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

In a group of students 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I,11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

(i) the number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.