#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 1: Sets

#### Chapter 1: Sets Exercise 1.1 solutions [Page 2]

What is the difference between a collection and a set? Give reasons to support your answer?

Which of the following collection are sets? Justify your answer:

A collection of all natural numbers less than 50.

Which of the following collection are sets? Justify your answer:

The collection of good hockey players in India.

Which of the following collection are sets? Justify your answer:

The collection of all girls in your class.

Which of the following collection are sets? Justify your answer:

The collection of most talented writers of India.

Which of the following collection are sets? Justify your answer:

The collection of difficult topics in mathematics.

Which of the following collection are sets? Justify your answer:

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

Which of the following collection are sets? Justify your answer:

A collection of novels written by Munshi Prem Chand.

Which of the following collection are sets? Justify your answer:

The collection of all question in this chapter.

Which of the following collection are sets? Justify your answer:

A collection of most dangerous animals of the world.

Which of the following collection are sets? Justify your answer:

The collection of prime integers.

If *A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

4 ...... A

If *A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

−4 ...... A

If *A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

12 ...... A

*A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

9 ...... A

*A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

0 ...... A

*A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

−2 ...... A

#### Chapter 1: Sets Exercise 1.2 solutions [Pages 6 - 7]

Describe the following sets in Roster form:

{*x* : *x* is a letter before *e* in the English alphabet}

Describe the following sets in Roster form:

{*x* ∈ *N* : *x*^{2} < 25};

Describe the following sets in Roster form:

{*x* ∈ *N* : *x* is a prime number, 10 < *x* < 20};

Describe the following sets in Roster form:

{*x* ∈ *N* : *x* = 2*n*, *n* ∈ *N*};

Describe the following sets in Roster form:

{*x* ∈ *R* : *x* > *x*}.

Describe the following sets in Roster form:

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

Describe the following sets in Roster form:

The set of all letters in the word 'Trigonometry'

Describe the following sets in Roster form:

The set of all letters in the word 'Better'.

Describe the following sets in Roster form:

The set of all letters in the word 'Better'.

Describe the following sets in set-builder form:

*A* = {1, 2, 3, 4, 5, 6}

Describe the following sets in set-builder form:

B={1,1/2 ,1/3, 1/4,1/5,...........};

Describe the following sets in set-builder form:

*C* = {0, 3, 6, 9, 12, ...}

Describe the following sets in set-builder form:

*D* = {10, 11, 12, 13, 14, 15};

Describe the following sets in set-builder form:

*E* = {0}

Describe the following sets in set-builder form:

{1, 4, 9, 16, ..., 100}

Describe the following sets in set-builder form:

{2, 4, 6, 8 .....}

Describe the following sets in set-builder form:

{5, 25, 125 625}

List all the elements of the following sets:

\[A = \left\{ x: x^2 \leq 10, x \in Z \right\}\]

List all the elements of the following set:

\[B = \left\{ x: x = \frac{1}{2n - 1}, 1 \leq n \leq 5 \right\}\]

List all the elements of the following set:

\[C = \left\{ x: x \text{ is an integer }, - \frac{1}{2} < x < \frac{9}{2} \right\}\]

List all the elements of the following set:

*D* = {*x* : *x* is a vowel in the word "EQUATION"}

List all the elements of the following set:

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

List all the elements of the following set:

*F* = {*x* : *x* is a letter of the word "MISSISSIPPI"}

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

(i) | {A, P, L, E} |
(i) | x : x + 5 = 5, x ∈ Z |

(ii) | {5, −5} | (ii) | {x : x is a prime natural number and a divisor of 10} |

(iii) | {0} | (iii) | {x : x is a letter of the word "RAJASTHAN"} |

(iv) | {1, 2, 5, 10,} | (iv) | {x: x is a natural number and divisor of 10} |

(v) | {A, H, J, R, S, T, N} |
(v) | x : x^{2} − 25 = 0 |

(vi) | {2, 5} | (vi) | {x : x is a letter of the word "APPLE"} |

Write the set of all vowels in the English alphabet which precede *q*.

Write the set of all positive integers whose cube is odd.

Write the set \[\left\{ \frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50} \right\}\] in the set-builder form.

#### Chapter 1: Sets Exercise 1.3 solutions [Pages 9 - 10]

Which of the following are examples of empty set?

Set of all even natural numbers divisible by 5

Which of the following are examples of empty set?

Set of all even prime numbers

Which of the following are examples of empty set?

{*x* : *x*^{2} −2 = 0 and *x* is rational}

Which of the following are examples of empty set?

{*x* : *x* is a natural number, *x* < 8 and simultaneously *x* > 12};

Which of the following are examples of empty set?

{*x* : *x* is a point common to any two parallel lines}.

Which of the following sets are finite and which are infinite?

Set of concentric circles in a plane

Which of the following sets are finite and which are infinite?

Set of letters of the English Alphabets

Which of the following sets are finite and which are infinite?

{*x* ∈ *N* : *x* > 5}

Which of the following sets are finite and which are infinite?

{*x* = ∈ *N* : *x* < 200}

Which of the following sets are finite and which are infinite?

{*x* ∈ *Z* : *x* < 5};

Which of the following sets are finite and which are infinite?

{*x* ∈ *R* : 0 < *x* < 1}.

Which of the following sets are equal?

(i) \[A = \left\{ 1, 2, 3 \right\};\]

(ii) \[B = \left\{ x \in R : x^2 - 2x + 1 = 0 \right\};\]

(iii) \[C = \left\{ 1, 2, 2, 3 \right\};\]

(iv) \[D = \left\{ x \in R : x^3 - 6 x^2 + 11x - 6 = 0 \right\}\]

Are the following sets equal?*A* = {*x* : *x* is a letter in the word reap}:*B* = {*x* : *x* is a letter in the word paper};*C* = {*x* : *x* is a letter in the word rope}.

From the sets given below, pair the equivalent sets:

\[A = \left\{ 1, 2, 3 \right\}, B = \left\{ t, p, q, r, s \right\}, C = \left\{ \alpha, \beta, \gamma \right\}, D = \left\{ a, e, i, o, u \right\} .\]

Are the following pairs of sets equal? Give reasons.

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

Are the following pairs of sets equal? Give reasons.

*A* = {*x* : *x* is a letter of the word " WOLF"};

*B* = {*x* : *x* is a letter of the word " FOLLOW"}.

From the sets given below, select equal sets and equivalent sets.*A* = {0, *a*}, *B* = {1, 2, 3, 4} *C* = {4, 8, 12}, *D* = {3, 1, 2, 4},*E* = {1, 0}, *F* = {8, 4, 12} *G* = {1, 5, 7, 11}, *H* = {*a*, *b*}.

Which of the following sets are equal?*A* = {*x* : *x* ∈ *N*, *x*, < 3},*B* = {1, 2}*C* = {3, 1}*D* = {*x* : *x* ∈ *N*, *x* is odd, *x* < 5},*E* = {1, 2, 1, 1} *F* = {1, 1, 3}.

Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal.

#### Chapter 1: Sets Exercise 1.4 solutions [Pages 16 - 17]

Which of the following statements are true? Give reason to support your answer.

(i) For any two sets *A* and *B* either \[A \subseteq B o\text{ or } B \subseteq A;\]

Which of the following statements are true? Give reason to support your answer.

Every subset of an infinite set is infinite

Which of the following statements are true? Give reason to support your answer.

Every subset of a finite set is finite

Which of the following statements are true? Give reason to support your answer.

Every set has a proper subset

Which of the following statements are true? Give reason to support your answer.

{*a*, *b*, *a*, *b*, *a*, *b*, ...} is an infinite set

Which of the following statements are true? Give reason to support your answer.

{*a*, *b*, *c*} and {1, 2, 3} are equivalent sets

Which of the following statements are true? Give reason to support your answer.

A set can have infinitely many subsets.

State whether the following statements are true or false:

\[1 \in \left\{ 1, 2, 3 \right\}\]

State whether the following statements are true or false:

\[a \subset {b, c, a}\]

State whether the following statements are true or false:

\[\left\{ a \right\} \in \left\{ a, b, c \right\}\]

State whether the following statements are true or false:

\[\left\{ a, b \right\} = \left\{ a, a, b, b, a \right\}\]

State whether the following statements are true or false:

The set {*x* ; *x* + 8 = 8} is the null set.

Decide among the following sets, which are subsets of which:

\[A = {x : x \text{ satisfies } x^2 - 8x + 12 = 0},\]

\[B = \left\{ 2, 4, 6 \right\}, C = \left\{ 2, 4, 6, 8, . . . \right\}, D = \left\{ 6 \right\} .\]

Write which of the following statements are true? Justify your answer.

The set of all integers is contained in the set of all set of all rational numbers.

Write which of the following statement are true? Justify your answer.

The set of all crows is contained in the set of all birds.

Write which of the following statement are true? Justify your answer.

The set of all rectangle is contained in the set of all squares.

Write which of the following statement are true? Justify your answer.

The set of all real numbers is contained in the set of all complex numbers.

Write which of the following statement are true? Justify your answer.

The sets *P* = {*a*} and *B* = {{*a*}} are equal.

Write which of the following statement are true? Justify your answer.

The sets *A* = {*x* : *x* is a letter of the word "LITTLE"} and,*B* = {*x* : *x* is a letter of the word "TITLE"} are equal.

Which of the following statement are correct?

Write a correct form of each of the incorrect statements.

\[a \subset \left\{ a, b, c \right\}\]

Which of the following statement are correct?

Write a correct form of each of the incorrect statement.

\[\left\{ a \right\} \in \left\{ a, b, c \right\}\]

Which of the following statements are correct?

Write a correct form of each of the incorrect statement.

\[a \in {\left\{ a \right\}, b}\]

Which of the following statement are correct?

Write a correct form of each of the incorrect statement.

\[\left\{ a \right\} \subset \left\{ \left\{ a \right\}, b \right\}\]

Which of the following statement are correct?

Write a correct form of each of the incorrect statement.

\[\left\{ b, c \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}\]

Which of the following statemen are correct?

Write a correct form of each of the incorrect statement.

\[\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}\]

Write a correct form of each of the incorrect statement.

\[\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}\]

Write a correct form of each of the incorrect statement.

\[\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}\]

Write a correct form of each of the incorrect statement.

\[\phi \subset \left\{ a, b, c \right\}\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\left\{ c, d \right\} \subset A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\left\{ c, d \right\} \in A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\left\{ \left\{ c, d \right\} \right\} \subset A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[a \in A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[a \subset A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\left\{ a, b, e \right\} \subset A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statements are false and why?

\[\left\{ a, b, e \right\} \in A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\left\{ a, b, c \right\} \subset A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\phi \in A\]

Let *A* = {*a*, *b*, {*c*, *d*}, *e*}. Which of the following statement are false and why?

\[\phi \in A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[1 \in A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[\left\{ 1, 2, 3 \right\} \subset A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[\left\{ 6, 7, 8 \right\} \in A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[\left\{ \left\{ 4, 5 \right\} \right\} \subset A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[\phi \in A\]

Let *A* = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

\[\phi \in A\]

Let\[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\]Which of the following are true?

Let\[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\] Which of the following are true?

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\] Which of the following are true? \[\left\{ 1 \right\} \in A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\]Which of the following are true? \[\left\{ 1 \right\} \in A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\] Which of the following are true? \[2 \subset A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\]Which of the following are true?\[\left\{ 2 \left\{ 1 \right\} \right\} \not\subset A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\]Which of the following are true? \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\]

Let \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\] Which of the following are true? \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\] Which of the following are true?\[\left\{ \left\{ \phi \right\} \right\} \subset A\]

Write down all possible subsets of each of the following set:

{*a*}

Write down all possible subsets of each of the following set:

{0, 1},

Write down all possible subsets of each of the following set:

{*a*, *b*, *c*},

Write down all possible subsets of each of the following set:

{1, {1}},

Write down all possible subsets of each of the following set:

\[\left\{ \phi \right\}\]

Write down all possible proper subsets each of the following set:

{1, 2},

Write down all possible proper subsets each of the following set:

{1, 2, 3}

Write down all possible proper subsets each of the following set:

{1}.

What is the total number of proper subsets of a set consisting of *n* elements?

If *A* is any set, prove that: \[A \subseteq \phi \Leftrightarrow A = \phi .\]

Prove that:

\[A \subseteq B, B \subseteq C \text{ and } C \subseteq A \Rightarrow A = C .\]

How many elements has \[P \left( A \right), \text{ if } A = \phi\]

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

The set of right triangles.

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

The set of isosceles triangles.

If \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]

#### Chapter 1: Sets Exercise 1.5 solutions [Page 21]

If *A* and *B* are two set such that \[A \subset B\]then find:

\[A \cap B\]

If *A* and *B* are two sets such that \[A \subset B\] then find:

\[A \cup B\]

If *A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[A \cup B\]

If *A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[A \cup C\]

If *A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[B \cup C\]

If *A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find\[B \cup D\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[A \cup B \cup C\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[A \cup B \cup D\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[B \cup C \cup D\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[A \cap \left( B \cup C \right)\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[\left( A \cap B \right) \cap \left( B \cap C \right)\]

*A* = {1, 2, 3, 4, 5}, *B* = {4, 5, 6, 7, 8}, *C* = {7, 8, 9, 10, 11} and *D* = {10, 11, 12, 13, 14}, find:

\[\left( A \cup D \right) \cap \left( B \cup C \right)\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\]and *D* = {*x* : *x* is a prime natural number}. Find: \[A \cap B\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\] and *D* = {*x* : *x* is a prime natural number}. Find: \[A \cap C\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\] and *D* = {*x* : *x* is a prime natural number}. Find: \[A \cap D\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\] and *D* = {*x* : *x* is a prime natural number}. Find: \[B \cap C\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\] and *D* = {*x* : *x* is a prime natural number}. Find: \[B \cap D\]

Let \[A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}\] and *D* = {*x* : *x* is a prime natural number}. Find: \[C \cap D\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}. Find: \[A - B\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}. Find: \[A - C\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}. Find: \[A - D\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}. Find:

\[B - A\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}. Find: \[C - A\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}.

Find: \[D - A\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}.

Find: \[B - C\]

Let *A* = {3, 6, 12, 15, 18, 21}, *B* = {4, 8, 12, 16, 20}, *C* = {2, 4, 6, 8, 10, 12, 14, 16} and *D* = {5, 10, 15, 20}.

Find: \[B - D\]

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

Find \[\left( A \cap C \right)'\]

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

Find \[\left( A \cup B \right)'\]

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, *A* = {1, 2, 3, 4}, = {2, 4, 6, 8} and *C* = {3, 4, 5, 6}. Find \[\left( A' \right)'\]

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

Find \[\left( B - C \right)'\]

Let *U* = {1, 2, 3, 4, 5, 6, 7, 8, 9}, *A* = {2, 4, 6, 8} and *B* = {2, 3, 5, 7}. Verify that \[\left( A \cup B \right)' = A' \cap B'\]

Let *U* = {1, 2, 3, 4, 5, 6, 7, 8, 9}, *A* = {2, 4, 6, 8} and *B* = {2, 3, 5, 7}. Verify that \[\left( A \cap B \right)' = A' \cup B'\]

#### Chapter 1: Sets Exercise 1.6 solutions [Page 27]

We have to find the smallest set *A* such that\[A \cup \left\{ 1, 2 \right\} = \left\{ 1, 2, 3, 5, 9 \right\}\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identities:

\[A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup C \right)\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left( A \cap C \right)\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A \cap \left( B - C \right) = \left( A \cap B \right) - \left( A \cap C \right)\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A - \left( B \cup C \right) = A\left( A - B \right) \cap \left( A - C \right)\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A - \left( B \cap C \right) = \left( A - B \right) \cup \left( A - C \right)\]

Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]

If *U* = {2, 3, 5, 7, 9} is the universal set and *A* = {3, 7}, *B* = {2, 5, 7, 9}, then prove that:

\[\left( A \cup B \right)' = A' \cap B'\]

If *U* = {2, 3, 5, 7, 9} is the universal set and *A* = {3, 7}, *B* = {2, 5, 7, 9}, then prove that:

\[\left( A \cap B \right)' = A'B' .\]

For any two sets *A* and *B*, prove that

*B* ⊂ *A* ∪ B

For any two sets *A* and *B*, prove that

*A* ∩ *B *⊂ *A*

For any two sets *A* and *B*, prove that *A* ⊂ *B *⇒ *A* ∩ *B *= *A *

For any two sets *A* and *B*, show that the following statements are equivalent:

(i) \[A \subset B\]

(ii) \[A \subset B\]=ϕ

(iii) \[A \cup B = B\]

(iv) \[A \cap B = A .\]

For three sets *A*, *B* and *C*, show that \[A \cap B = A \cap C\]

For three sets *A*, *B* and *C*, show that \[A \subset B \Rightarrow C - B \subset C - A\]

For any two sets, prove that:

\[A \cup \left( A \cap B \right) = A\]

For any two sets, prove that:

\[A \cap \left( A \cup B \right) = A\]

Find sets *A*, *B* and *C* such that \[A \cap B, A \cap C \text{ and } B \cap C\]are non-empty sets and\[A \cap B \cap C = \phi\]

For any two sets *A* and *B*, prove that: \[A \cap B = \phi \Rightarrow A \subseteq B'\]

If *A* and *B* are sets, then prove that \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint.

Using properties of sets, show that for any two sets *A* and *B*,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]

For any two sets of *A* and *B*, prove that:

\[A' \cup B = U \Rightarrow A \subset B\]

For any two sets of *A* and *B*, prove that:

\[B' \subset A' \Rightarrow A \subset B\]

Is it true that for any sets *A* and \[B, P \left( A \right) \cup P \left( B \right) = P \left( A \cup B \right)\]? Justify your answer.

Show that for any sets *A* and *B*, \[A = \left( A \cap B \right) \cup \left( A - B \right)\]

Show that for any sets *A* and *B*, \[A \cup \left( B - A \right) = A \cup B\]

Each set *X*, contains 5 elements and each set *Y*, contains 2 elements and \[\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r\] If each element of *S* belong to exactly 10 of the *X _{r}*'s and to eactly 4 of

*Y*'s, then find the value of

_{r}*n*.

#### Chapter 1: Sets Exercise 1.7 solutions [Pages 34 - 35]

For any two sets *A* and *B*, prove that :

\[A' - B' = B - A\]

For any two sets *A* and *B*, prove the following:

\[A \cap \left( A' \cup B \right) = A \cap B\]

For any two sets *A* and *B*, prove the following:

\[A - \left( A - B \right) = A \cap B\]

For any two sets *A* and *B*, prove the following:

\[A \cap \left( A \cup B \right)' = \phi\]

For any two sets *A* and *B*, prove the following:

\[A - B = A \Delta\left( A \cap B \right)\]

If *A*, *B*, *C* are three sets such that \[A \subset B\]then prove that \[C - B \subset C - A\]

For any two sets *A* and *B*, prove that \[\left( A \cup B \right) - B = A - B\]

For any two sets *A* and *B*, prove that \[A - \left( A \cap B \right) = A - B\]

For any two sets *A* and *B*, prove that \[A - \left( A - B \right) = A \cap B\]

For any two sets *A* and *B*, prove that

\[A \cup \left( B - A \right) = A \cup B\]

For any two sets *A* and *B*, prove that \[\left( A - B \right) \cup \left( A \cap B \right) = A\]

#### Chapter 1: Sets Exercise 1.8 solutions [Pages 46 - 47]

If *A* and *B* are two sets such that \[n \left( A \cup B \right) = 50, n \left( A \right) = 28 \text{ and } n \left( B \right) = 32\]\[n \left( A \cap B \right)\]

If *P* and *Q* are two sets such that *P* has 40 elements, \[P \cup Q\]has 60 elements and\[P \cap Q\]has 10 elements, how many elements does *Q* have?

In a school there are 20 teachers who teach athematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics?

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 like both coffee and tea?

Let *A* and *B* be two sets such that :\[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] Find\[n\left( B \right)\]

Let *A* and *B* be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( A - B \right)\]

Let *A* and *B* be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( B - A \right)\]

A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:

how many can speak both Hindi and English:

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: how many can speak Hindi only

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:

how many can speak English only.

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:

(i) how may drink tea and coffee both;

(ii) how many drink coffee but not tea.

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 numbers of people who read at least one of the newspapers.

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

Of the members of three athletic teams in a certain school, 21 are in the basketball team, 26 in hockey team and 29 in the football team, 14 play hockey and basket ball 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all?

In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?

\[\cap\]A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?

In a survey of 100 persons it was found that 28 read magazine *A*, 30 read magazine *B*, 42 read magazine *C*, 8 read magazines *A* and *B*, 10 read magazines *A* and *C*, 5 read magazines *B* and *C* and 3 read all the three magazines. Find:

(i) How many read none of three magazines?

(ii) How many read magazine *C* only?

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:

(i) How many students were studying Hindi?

(ii) How many students were studying English and Hindi?

In a survey it was found that 21 persons liked product *P*_{1}, 26 liked product *P*_{2} and 29 liked product *P*_{3}. If 14 persons liked products *P*_{1} and *P*_{2}; 12 persons liked product *P*_{3} and *P*_{1} ; 14 persons liked products *P*_{2} and *P*_{3} and 8 liked all the three products. Find how many liked product *P*_{3} only.

#### Chapter 1: Sets Exercise 1.9 solutions [Page 49]

If a set contains *n* elements, then write the number of elements in its power set.

Write the number of elements in the power set of null set.

Let *A* = {*x* : *x* ∈ *N*, *x* is a multiple of 3} and *B* = {*x* : *x* ∈ *N* and *x* is a multiple of 5}. Write \[A \cap B\]

Let *A* and *B* be two sets having 3 and 6 elements respectively. Write the minimum number of elements that \[A \cup B\]

If *A* = {*x* ∈ *C* : *x*^{2} = 1} and *B* = {*x* ∈ *C* : *x*^{4} = 1}, then write *A* − *B* and *B* − *A*.

If *A* and *B* are two sets such that \[A \subset B\], then write *B*' − *A*' in terms of *A* and *B*.

Let *A* and *B* be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that \[A \cup B\] can have.

If \[A = \left\{ \left( x, y \right) : y = \frac{1}{x}, 0 \neq x \in R \right\}\]and\[B = \left\{ \left( x, y \right) : y = - x, x \in R \right\}\] then write\[A \cap B\]

If \[A = \left\{ \left( x, y \right) : y = e^x , x \in R \right\} and B = \left\{ \left( x, y \right) : y = e^{- x} , x \in R \right\}\]write\[A \cap B\]

If *A* and *B* are two sets such that \[n \left( A \right) = 20, n \left( B \right) = 25\]\text{ and } \[n \left( A \cup B \right) = 40\], then write \[n \left( A \cap B \right)\]

If *A* and *B* are two sets such that \[n \left( A \right) = 115, n \left( B \right) = 326, n \left( A - B \right) = 47,\] then write \[n \left( A \cup B \right)\]

#### Chapter 1: Sets solutions [Pages 49 - 51]

For any set *A*, (*A*')' is equal to

(a)

*A*'(b)

*A*(c) ϕ

(d) none of these.

Let *A* and *B* be two sets in the same universal set. Then,\[A - B =\]

(a) \[A \cap B\]

(b)\[A' \cap B\]

(c)\[A \cap B'\]

(d) none of these.

The number of subsets of a set containing *n* elements is

(a)

*n*(b)

*2*^{n}− 1(c)

*n*^{2}(d) 2

^{n}

For any two sets *A* and *B*,\[A \cap \left( A \cup B \right) =\]

(a)

*A*(b)

*B*(c) ϕ

(d) none of these.

If *A* = {1, 3, 5, *B*} and *B* = {2, 4}, then

(a)\[4 \in A\]

(b)\[\left\{ 4 \right\} \subset A\]

(c)\[B \subset A\]

(d) none of these.

The symmetric difference of *A* and *B* is

(a)\[\left( A - B \right) \cap \left( B - A \right)\]

(b)\[\left( A - B \right) \cup \left( B - A \right)\]

(c) \[\left( A \cup B \right) - \left( A \cap B \right)\]

(d) \[\left\{ \left( A \cup B \right) - A \right\} \cup \left\{ \left( A \cup B \right) - B \right\}\]

The symmetric difference of *A* = {1, 2, 3} and *B* = {3, 4, 5} is

(a) {1, 2}

(b) {1, 2, 4, 5}

(c) {4, 3}

(d) {2, 5, 1, 4, 3}

For any two sets *A* and *B*,\[\left( A - B \right) \cup \left( B - A \right) =\]

(a) \[\left( A - B \right) \cup A\]

(b)\[\left( B - A \right) \cup B\]

(c)\[\left( A \cup B \right) - \left( A \cap B \right)\]

(d)\[\left( A \cup B \right) \cap \left( A \cap B \right)\]

Which of the following statements is false:

\[A - B = A \cap B'\]

\[A - B = A - \left( A \cap B \right)\]

\[A - B = A - B'\]

\[A - B = \left( A \cup B \right) - B .\]

For any three sets A, B and C

(a) \[A \cap \left( B - C \right) = \left( A \cap B \right) - \left( A \cap C \right)\]

(b) \[A \cap \left( B - C \right) = \left( A \cap B \right) - C\]

(c) \[A \cup \left( B - C \right) = \left( A \cup B \right) \cap \left( A \cup C' \right)\]

(d) \[A \cup \left( B - C \right) = \left( A \cup B \right) - \left( A \cup C \right) .\]

Let \[A = \left\{ x : x \in R, x \geq 4 \right\} \text{ and } B = \left\{ x \in R : x < 5 \right\}\] Then, \[n \left( A' \cap B' \right) =\]

(a) (4, 5]

(b) (4, 5)

(c) [4, 5)

(d) [4, 5]

Let *U* be the universal set containing 700 elements. If *A*, *B* are sub-sets of *U* such that \[n \left( A \right) = 200, n \left( B \right) = 300 \text{ and } \left( A \cap B \right) = 100\].Then \[n \left( A' \cap B' \right) =\]

(a) 400

(b) 600

(c) 300

(d) none of these.

Let *A* and *B* be two sets that \[n \left( A \right) = 16, n \left( B \right) = 14, n \left( A \cup B \right) = 25\] Then, \[n \left( A \cap B \right)\]

(a) 30

(b) 50

(c) 5

(d) none of these

If *A* = |1, 2, 3, 4, 5|, then the number of proper subsets of *A* is

(a) 120

(b) 30

(c) 31

(d) 32

In set-builder method the null set is represented by

(a) { }

(b) Φ

(c) \[\left| x : x \neq x \right|\]

(d) \[\left| x : x = x \right|\]

\[\cap\] If *A* and *B* are two disjoint sets, then \[n \left( A \cup B \right)\]is equal to

(a) \[n \left( A \right) + n\left( B \right)\]

(b) \[n \left( A \right) + n\left( B \right) - n\left( A \cap B \right)\]

(c)\[n \left( A \right) + n \left( B \right) + n \left( A \cap B \right)\]

(d) \[n \left( A \right) n \left( B \right)\]

(e) \[n \left( A \right) - n \left( B \right)\]

For two sets [A \cup B = A\] iff

(a) \[B \subseteq A\]

(b) \[A \subseteq B\]

(c) \[A \neq B\]

(d) \[A = B\]

If *A* and *B* are two sets such that \[n \left( A \right) = 70, n \left( B \right) = 60, n \left( A \cup B \right) = 110\] then \[n \left( A \cap B \right)\]

(a) 240

(b) 50

(c) 40

(d) 20

If *A* and *B* are two given sets, then \[A \cap \left( A \cap B \right)^c\]

(a)

*A*(b)

*B*(c) Φ

(d)\[A \cap B^c\]

If *A* = {*x* : *x* is a multiple of 3} and , *B* = {*x* : *x* is a multiple of 5}, then *A* − *B* is

(a) \[A \cap B\]

(b) \[A \cap B\]

(c) \[A \cap B\]

(d) \[A \cap B\]

In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus is

(a) 80%

(b) 40%

(c) 60%

(d) 70%

If \[A \cap B - B\]

(a) \[A \subset B\]

(b) \[B \subset A\]

(c) \[A = \Phi\]

(d) \[B = \Phi\]

An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee and tea. The investigator reported that 10 students take all three drinks milk, coffee and tea; 20 students take milk and coffee; 25 students take milk and tea; 12 students take milk only; 5 students take coffee only and 8 students take tea only. Then the number of students who did not take any of three drinks is

10

20

25

30

N/A

Two finite sets have *m* and *n* elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set. Then, the values of *m* and *n* are:

7, 6

6, 3

7, 4

3, 7

In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?

(a) 35

(b) 48

(c) 60

(d) 22

(e) 30

Suppose \[A_1 , A_2 , . . . , A_{30}\] are thirty sets each having 5 elements and \[B_1 , B_2 , . . . , B_n\] are *n* sets each with 3 elements. Let \[\cup^{30}_{i = 1} A_i = \cup^n_{j = 1} B_j = S\] and each element of *S* belong to exactly 10 of the \[A_i 's\]and exactly 9 of the\[B_j 's\] then *n* is equal to

(a) 15

(b) 3

(c) 45

(d) 35

Two finite sets have *m* and *n* elements. The number of subsets of the first set is 112 more than that of the second. The values of *m* and *n* are respectively

(a) 4, 7

(b) 7, 4

(c) 4, 4

(c) 4, 4

For any two sets *A* and *B*, \[A \cap \left( A \cup B \right)'\]is equal to

(a)

*A*(b)

*B*(c)\[\phi\]

(d)\[A \cap B\]

The set \[\left( A \cup B' \right)' \cup \left( B \cap C \right)\] is equal to

(a) \[A' \cup B \cup C\]

(b) \[A' \cup B\]

(c) \[A' \cup c\]

(d) \[A' \cap B\]

(e) n/a

Let *F*_{1} be the set of all parallelograms, *F*_{2 }the set of all rectangles, *F*_{3 }the set of all rhombuses, *F*_{4}_{ }the set of all squares and *F*_{5 }the set of trapeziums in a plane. Then *F*_{1}may be equal to

(a) \[F_2 \cap F_3\]

(b) \[F_3 \cap F_4\]

(c) \[F_2 \cup F_3\]

(d) \[F_2 \cup F_3 \cup F_4 \cup F_1\]

## Chapter 1: Sets

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 1 - Sets

RD Sharma solutions for Class 11 Maths chapter 1 (Sets) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 1 Sets are Practical Problems on Union and Intersection of Two Sets, Power Set, Subsets, Equal Sets, Finite and Infinite Sets, The Empty Set, Sets and Their Representations, Complement of a Set, Union Set, Venn Diagrams, Universal Set, Proper and Improper Subset, Open and Close Intervals, Operation on Set - Disjoint Sets, Intersection of Sets, Difference of Sets, Element Count Set, Intrdouction of Operations on Sets.

Using RD Sharma Class 11 solutions Sets exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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