#### 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

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 1 SetsSolved Examples [Pages 4 - 12]

#### Short Answer

Write the following sets in the roaster form.

A = {x | x is a positive integer less than 10 and 2^{x} – 1 is an odd number}

Write the following sets in the roaster form.

C = {x : x^{2} + 7x – 8 = 0, x ∈ R}

#### State whether the following is True or False:

37 ∉ {x | x has exactly two positive factors}

False

True

28 ∈ {y | the sum of the all positive factors of y is 2y}

True

False

7,747 ∈ {t | t is a multiple of 37}

True

False

If X and Y are subsets of the universal set U, then show that Y ⊂ X ∪ Y

If X and Y are subsets of the universal set U, then show that X ∩ Y ⊂ X

If X and Y are subsets of the universal set U, then show that X ⊂ Y ⇒ X ∩ Y = X

Given that N = {1, 2, 3, ..., 100}, then write the subset A of N, whose element are odd numbers.

Given that N = {1, 2, 3, ..., 100}, then write the subset B of N, whose element are represented by x + 2, where x ∈ N.

Given that E = {2, 4, 6, 8, 10}. If n represents any member of E, then, write the following sets containing all numbers represented by n + 1

Given that E = {2, 4, 6, 8, 10}. If n represents any member of E, then, write the following sets containing all numbers represented by n^{2}

Let X = {1, 2, 3, 4, 5, 6}. If n represent any member of X, express the following as sets:

n ∈ X but 2n ∉ X

Let X = {1, 2, 3, 4, 5, 6}. If n represent any member of X, express the following as sets:

n + 5 = 8

Let X = {1, 2, 3, 4, 5, 6}. If n represent any member of X, express the following as sets:

n is greater than 4

Draw the Venn diagrams to illustrate the following relationship among sets E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.

All the students who study Mathematics study English, but some students who study English do not study Mathematics.

Draw the Venn diagrams to illustrate the following relationship among sets E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.

There is no student who studies both Mathematics and English.

Draw the Venn diagrams to illustrate the following relationship among sets E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.

Some of the students study Mathematics but do not study English, some study English but do not study Mathematics, and some study both.

Not all students study Mathematics, but every students studying English studies Mathematics.

For all sets A, B and C is (A ∩ B) ∪ C = A ∩ (B ∪ C)? Justify your statement.

Use the properties of sets to prove that for all the sets A and B

#### Long Answer

For all sets A, B and C is (A – B) ∩ (C – B) = (A ∩ C) – B? Justify your answer.

Let A, B and C be sets. Then show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Let P be the set of prime numbers and let S = {t | 2^{t} – 1 is a prime}. Prove that S ⊂ P.

From 50 students taking examinations in Mathematics, Physics and Chemistry, each of the student has passed in at least one of the subject, 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, at most 29 Mathematics and Chemistry and at most 20 Physics and Chemistry. What is the largest possible number that could have passed all three examination?

#### Objective Type Questions from 14 to 16

Each set X_{r} contains 5 elements and each set Y_{r} contains 2 elements and \[\bigcup\limits_{r=1}^{20} X_{r} = S = \bigcup\limits_{r=1}^{n} Y_{r}\] If each element of S belong to exactly 10 of the X_{r}’s and to exactly 4 of the Y_{r}’s, then n is ______.

10

20

100

50

Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are ______.

7, 6

5, 1

6, 3

8, 7

The set (A ∪ B ∪ C) ∩ (A ∩ B′ ∩ C′)′ ∩ C′ is equal to ______.

B ∩ C′

A ∩ C

B ∪ C′

A ∩ C′

#### Fill in the blanks 17 and 18:

If A and B are two finite sets, then n(A) + n(B) is equal to ______.

If A is a finite set containing n element, then number of subsets of A is ______.

#### State whether the following is True or False: 19 and 20

Let R and S be the sets defined as follows:

R = {x ∈ Z | x is divisible by 2}

S = {y ∈ Z | y is divisible by 3}

then R ∩ S = φ

True

False

Q ∩ R = Q, where Q is the set of rational numbers and R is the set of real numbers.

True

False

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 1 SetsExercise [Pages 12 - 18]

#### Short Answer

Write the following sets in the roaster from:

A = {x : x ∈ R, 2x + 11 = 15}

Write the following sets in the roaster from:

B = {x | x^{2} = x, x ∈ R}

Write the following sets in the roaster from:

C = {x | x is a positive factor of a prime number p}

Write the following sets in the roaster form:

D = {t | t^{3} = t, t ∈ R}

Write the following sets in the roaster form:

E = `{w | (w - 2)/(w + 3) = 3, w ∈ R}`

Write the following sets in the roaster form:

F = {x | x 4 – 5x^{2} + 6 = 0, x ∈ R}

If Y = {x | x is a positive factor of the number 2^{p – 1} (2^{p} – 1), where 2^{p} – 1 is a prime number}.Write Y in the roaster form.

#### State whether the following is True or False:

35 ∈ {x | x has exactly four positive factors}.

True

False

128 ∈ {y | the sum of all the positive factors of y is 2y}

True

False

3 ∉ {x | x^{4} – 5x^{3} + 2x^{2} – 112x + 6 = 0}

True

False

496 ∉ {y | the sum of all the positive factors of y is 2y}.

True

False

Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}. Verify that L – (M ∪ N) = (L – M) ∩ (L – N)

If A and B are subsets of the universal set U, then show that A ⊂ A ∪ B

If A and B are subsets of the universal set U, then show that A ⊂ B ⇔ A ∪ B = B

If A and B are subsets of the universal set U, then show that (A ∩ B) ⊂ A

Given that N = {1, 2, 3, ... , 100}. Then write the subset of N whose elements are even numbers.

Given that N = {1, 2, 3, ... , 100}. Then write the subset of N whose element are perfect square numbers.

If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by 4n

If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by n + 6

If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by `n/2`

If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by n – 1

If Y = {1, 2, 3, ... 10}, and a represents any element of Y, write the following sets, containing all the elements satisfying the given conditions.

a ∈ Y but a^{2} ∉ Y

If Y = {1, 2, 3, ... 10}, and a represents any element of Y, write the following sets, containing all the elements satisfying the given conditions.

a + 1 = 6, a ∈ Y

If Y = {1, 2, 3, ... 10}, and a represents any element of Y, write the following sets, containing all the elements satisfying the given conditions.

a is less than 6 and a ∈ Y

A, B and C are subsets of Universal Set U. If A = {2, 4, 6, 8, 12, 20} B = {3, 6, 9, 12, 15}, C = {5, 10, 15, 20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.

Let U be the set of all boys and girls in a school, G be the set of all girls in the school, B be the set of all boys in the school, and S be the set of all students in the school who take swimming. Some, but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets U, G, B and S.

For all sets A, B and C, show that (A – B) ∩ (C – B) = A – (B ∪ C)

#### State whether the following statement is True or False: 13 – 17 Justify your answer.

For all sets A and B, (A – B) ∪ (A ∩ B) = A

True

False

For all sets A, B and C, A – (B – C) = (A – B) – C

True

False

For all sets A, B and C, if A ⊂ B, then A ∩ C ⊂ B ∩ C

True

False

For all sets A, B and C, if A ⊂ B, then A ∪ C ⊂ B ∪ C

True

False

For all sets A, B and C, if A ⊂ C and B ⊂ C, then A ∪ B ⊂ C

True

False

#### Using properties of sets prove the statements given in 18 to 22

For all sets A and B, A ∪ (B – A) = A ∪ B

For all sets A and B, A – (A – B) = A ∩ B

For all sets A and B, A – (A ∩ B) = A – B

For all sets A and B, (A ∪ B) – B = A – B

Let T = `{x | (x + 5)/(x - 7) - 5 = (4x - 40)/(13 - x)}`. Is T an empty set? Justify your answer.

#### Long Answer

Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed in English and Mathematics but not in Science.

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed in Mathematics and Science but not in English

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed in Mathematics only

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed in more than one subject only

In a class of 60 students, 25 students play cricket and 20 students play tennis, and 10 students play both the games. Find the number of students who play neither?

In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects.

In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. Find the number of families which buy newspaper A only.

In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. Find the number of families which buy none of A, B and C

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study French only

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study English only

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study Sanskrit only

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study English and Sanskrit but not French

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study French and Sanskrit but not English

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study French and English but not Sanskrit

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study at least one of the three languages

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study none of the three languages

#### Objective Type Questions from 29 to 43

Suppose A_{1}, A_{2}, ..., A_{30} are thirty sets each having 5 elements and B_{1}, B_{2}, ..., Bn are n sets each with 3 elements, let \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] = and each element of S belongs to exactly 10 of the A_{i}’s and exactly 9 of the B,’S. then n is equal to ______.

15

3

45

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 set. The values of m and n are, respectively ______.

4, 7

7, 4

4, 4

7, 7

The set (A ∩ B′)′ ∪ (B ∩ C) is equal to ______.

A′ ∪ B ∪ C

A′ ∪ B

A′ ∪ C′

A′ ∩ B

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

F

_{2}∩ F_{3}F

_{3}∩ F_{4 }F

_{2}∪ F_{5}F

_{2}∪ F_{3}∪ F_{4}∪ F_{1}

Let S = set of points inside the square, T = the set of points inside the triangle and C = the set of points inside the circle. If the triangle and circle intersect each other and are contained in a square. Then ______.

S ∩ T ∩ C = Φ

S ∪ T ∪ C = C

S ∪ T ∪ C = S

S ∪ T = S ∩ C

Let R be set of points inside a rectangle of sides a and b (a, b > 1) with two sides along the positive direction of x-axis and y-axis. Then ______.

R = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b}

R = {(x, y) : 0 ≤ x < a, 0 ≤ y ≤ b}

R = {(x, y) : 0 ≤ x ≤ a, 0 < y < b}

R = {(x, y) : 0 < x < a, 0 < y < b}

In a class of 60 students, 25 students play cricket and 20 students play tennis, and 10 students play both the games. Then, the number of students who play neither is ______.

0

25

35

45

In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then the number of persons who read neither is ______.

210

290

180

260

If X = {8^{n} – 7n – 1 | n ∈ N} and Y = {49n – 49 | n ∈ N}. Then ______.

X ⊂ Y

Y ⊂ X

X = Y

X ∩ Y = Φ

A survey shows that 63% of the people watch a News Channel whereas 76% watch another channel. If x% of the people watch both channel, then ______.

x = 35

x = 63

39 ≤ x ≤ 63

x = 39

If sets A and B are defined as A = `{(x, y) | y = 1/x, 0 ≠ x ∈ "R"}` B = {(x, y) | y = – x, x ∈ R}, then ______.

A ∩ B = A

A ∩ B = B

A ∩ B = Φ

A ∪ B = A

If A and B are two sets, then A ∩ (A ∪ B) equals ______.

A

B

Φ

A ∩ B

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {2, 4, ..., 18} and N the set of natural numbers is the universal set, then A′ ∪ (A ∪ B) ∩ B′) is ______.

Φ

N

A

B

Let S = {x | x is a positive multiple of 3 less than 100}

P = {x | x is a prime number less than 20}. Then n(S) + n(P) is ______.

34

41

33

30

If X and Y are two sets and X′ denotes the complement of X, then X ∩ (X ∪ Y)′ is equal to ______.

X

Y

Φ

X ∩ Y

#### Fill in the blanks 44 to 51:

The set {x ∈ R : 1 ≤ x < 2} can be written as ______.

When A = Φ, then number of elements in P(A) is ______.

If A and B are finite sets such that A ⊂ B, then n (A ∪ B) = ______.

If A and B are any two sets, then A – B is equal to ______.

Power set of the set A = {1, 2} is ______.

Given the sets A = {1, 3, 5}. B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then the universal set of all the three sets A, B and C can be ______.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 5}, B = {2, 4, 6, 7} and C = {2, 3, 4, 8}. Then (B ∪ C)′ is ______.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 5}, B = {2, 4, 6, 7} and C = {2, 3, 4, 8}. Then (C – A)′ is ______.

For all sets A and B, A – (A ∩ B) is equal to ______.

#### Match the following sets for all sets A, B and C

Column A |
Column B |

(i) ((A′ ∪ B′) – A)′ | (a) A – B |

(ii) [B′ ∪ (B′ – A)]′ | (b) A |

(iii) (A – B) – (B – C) | (c) B |

(iv) (A – B) ∩ (C – B) | (d) (A × B) ∩ (A × C) |

(v) A × (B ∩ C) | (e) (A × B) ∪ (A × C) |

(vi) A × (B ∪ C) | (f) (A ∩ C) – B |

#### State whether the following is True or False: 53 to 58

If A is any set, then A ⊂ A.

True

False

Given that M = {1, 2, 3, 4, 5, 6, 7, 8, 9} and if B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, then B ⊄ M.

True

False

The sets {1, 2, 3, 4} and {3, 4, 5, 6} are equal.

True

False

Q ∪ Z = Q, where Q is the set of rational numbers and Z is the set of integers.

True

False

Let sets R and T be defined as

R = {x ∈ Z | x is divisible by 2}

T = {x ∈ Z | x is divisible by 6}. Then T ⊂ R

True

False

Given A = {0, 1, 2}, B = {x ∈ R | 0 ≤ x ≤ 2}. Then A = B.

True

False

## Chapter 1: Sets

## NCERT solutions for Mathematics Exemplar Class 11 chapter 1 - Sets

NCERT solutions for Mathematics Exemplar Class 11 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 Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

Using NCERT 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 NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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