## Kerala Plus One maths Chapter Wise Questions and Answers Chapter 1 Sets

**Short Answer** **Type Questions**

**(Score 3)**

Question 1.

A = {x: x is a natural number less than 8}

a. Write A in roster form.

b. write a subset of A containing all even numbers in A.

c. Which of the following could not be the number of elements of power set of a set ? [2, 8, 10,16]

Answer:

a. A= {1,2,3,4,5,6,7}

b. {2,4,6} or {2,4,6,7}

c. 10

Question 2.

The number of elements in the power set of the first is 48 more than the total number of elements in the power set of the second. Then find the values of m and n ?

Answer:

P (set 1) = 2^{m }P (set 2) = 2^{n}

2^{m} – 2^{n}= 48

2^{m}-2^{n}= 64-16

2^{m }– 2^{n}= 2^{6}-2^{4} ;It implies m = 6, n=4

Question 3.

a If A is finite, then the number of distinct subset of A is — ——– .

b If nP(A) = 1,then the set A is—————–

c. A set containing a single element is called……………

Answer:

a. 2^{n(}^{A) }n(P(A)) =2^{m}=1, m = 0

c. Singleton set

Question 4.

a Which of the following is equal to (A-B)∪(B-A)

a. A∪B

b. (A∪B) – (A∩B)

c. A∩B

d. A′∩B

b. If A is a set with n (A) = m, then n(P(A)) = …….

Answer:

A-B= A∩B ‘ ; B – A= B ∩ A’

∴ (A∪B)-(A∪B) = (A-B) ∪ (B-A)

If n (A)= m, the number of subsets of A=2^{m }If n (A) = m, then n [P (A)] = 2^{m}

Question 5.

Consider two sets A and B in the same universal set.

i. Represent (A – B) ∪ (B – A) using Venn diagram.

ii. If A= {1,2,3} and B = {3,4,5}, then find (A-B) ∪ (B-A)

Answer:

^{ }ii. A-B = {1,2}, B-A= {4,5},

(A-B) ∪ (B-A)= {1,2,4,5}

Question 6.

Given A ={5,8,4}

i. n(P(A)) =…………

ii. Find the power set of A

iii. If P(A) = P(B), then prove that A= B.

Answer:

i. n(P(A)) = 2^{3} = 8

ii. P(A)={{5,8,4},{5,8},{5,4}, {8, 4}, {5},{8},{4},{}}

iii. P(A) = P(B)

The sets of all subsets of A and B are equal ⇒ A and B are equal ⇒ A = B

Question 7.

a. Let A= {3,6,9,12}, B = {6,12,4,8,7}, C = {9,6,1,2}. Then find the value of A∪ B∪ C.

b. Using the properties of sets, prove that A – (B – C) = (A – B) ∪ (A∩C)

Answer:

Question 8.

a. Represent A={x:x is an integer,x^{2}<4} in roster form.

b. Let U = {1,2,3,4,5,6,7,8}, A = {2,4, 6,8}, B = {2,4, 8}. Show that A – B ≠B-A.

c. Consider the statement: “Integers between -3 and 3”. Write the roster and set builder forms.

Answer:

a. A= {-2,-1,0,1,2}

b. A-B={6} and B-A=φ,A-B ≠ B-A

c. Roster form is {-2,-1,0,1,2}

Set builder form is {x: x is an integer, -3 < x < 3 }

**Short Answer** **Type Questions**

**(Score 4)**

Question 1.

In a certain locality of Delhi, there are 1000 families. A survey indicated that 300 subscribe to ‘The Hindustan Times’ daily newspaper and 250 subscribe to ‘The Statesman’ daily newspaper. Of these two categories, 100 subscribe to both

i. Express the data using Venn diagram.

ii. Find the number of families which subscribe Hindustan Times but not Statesman.

iii. Find the number of families which do not subscribe to any of these newspapers.

Answer:

i.

Question 2.

There are 20 students in a Chemistry class and 30 students in a Physics class. Find the number of students which are either in Physics class or Chemistry class in the following cases.

i. Two classes meet at the same hour.

ii. The two classes meet at different hours and ten students are enrolled in both the courses

Answer:

Question 3.

Out of 20 members in an office, 12 like to take tea and 15 like coffee. Assume that each one likes at least one of the two drinks, how many like

i. Only tea and not coffee

ii. Only coffee and not tea

iii. Both coffee and tea

Answer:

Question 4.

In a group of athletic team in a school 21 are in the basket ball team, 26 in the hockey team and 29 in the football team. If 14 play hockey and basket ball, 12 play football and basket ball, 15 play hockey and football and 8 play all the three games, find the following.

i. How many players are there?

ii. How many play football only?

Answer:

Question 5.

a. Show that sets A={1,2,3}, B={4,5,6} and C={7,8,9} are pair wise disjoint

b. 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}

Answer:

a. (A∩B)=φ,(B∩G)=φ, (A∩C)=φ

b. n(A) = n(E) = n(H). Hence A, E and H are equivalent sets.

n(C) = n(F). Hence C and F are equivalent sets.

n(B) = n(D) = n(G). Hence B, D and G are equivalent sets.

n(B) = n(D) and B, D have the same elements. Hence B and D are equal sets.

n(C) = n(F) and C, F have the same elements. Hence C, F are equal sets.

Question 6.

a. If A= {a, b, c, d}, B={b, c, e}, C={a, e} verify that

An (B – C)= (A∩B) – (A∩C).

b. Using properties of sets, prove that (A∪B) – C = (A – C) ∪ (B – C)

Answer:

Question 7.

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

Find the values of m and n.

b. Prove that A∩(B-C) = (A∩B)-(A∩C)

Answer:

Question 8.

Consider the sets A= {x:x is an integer and -3 < x < 1} and B = {x:x is a letter in the word INDIA}

i. Write A and B in Roster form.

ii. Are A and B equivalent?

iii. How many subsets does A have? Write all possible subsets of A. Hence construct the power set of A.

Answer:

i. A= {-3, -2, -1,0}, B = {I,N, D, A}

ii. Since n(A) = n(B), A and B are equivalent.

iii. Since A contains 4 elements, A will have 2^{4} = 16 subsets.

A= {-3,-2,-1,0}

Subsets are

{-3,-2,-1,0}, {-3}, {-2}, {-1}, {0}, {-3,-2}, {-3,-1}, {-3,0}, {-2,-1}, {-2,0}, {-1,0}, {-3,-2,-1}, {-3,-2,0}, {-2,-1,0}, {-3,-1,0}, φ

Question 9.

a. Write the set of all possible integers whose cube is odd.

b. Let A and B be two sets such that n(A) = 20, n(A∪B) = 42, n(A∩B) = 4. Find

i. n(B)

ii. n(B-A)

iii. n(A-B)

Answer:

a. {2k + 1: k > 0, k ∈ z}, since cube of odd integer is odd.

b. i.n(A∪B) = n(A) + n(B)-n(A∩B) i. e., 42 = 20 + n(B) -4 n(B) = 26

ii. n(A-B) = n(A)-n(A∩B)=20-4= 16

iii. n(B – A) = n(B) – n(A∩B) = 26-4 = 22

Question 20.

In a school with 727 students 600 students are enrolled for mathematics and 173 students are enrolled in both mathematics and physics. How many students are enrolled in

i. Physics

ii. Physics only

Answer:

Let M be the set of students enrolled in Mathematics and P the set of all students enrolled in Physics.

Given n(M ∪ P) = 727, n(M) = 600 n(M∩P)= 17

Using n(M ∪ P) = n(M) +n (P) – n(M∩P)

727 = 600 + n(P) -173

n(P) = 727 -600 +173 = 300 and

n(P ∩M’) = n(P) – n (PM)

= 300-173 = 127

i. Number of students enrolled in Physics = 300

ii. Number of students enrolled in Physics only = 127.

Question 21.

Observe the venn diagram

a. Write A and B in roster form.

b. Verify that (A- B) ∪(A∩B) = A

c. Find (A∩B)’

Answer:

a. A= {1,3,4,8 }

B= {2,3,5}

b. A-B= {1,4,8}

AB= {3}

(A – B) (AB) = A

c. (AB)’ = {1,2,4,5,6,7,8,9}

Question 22.

a. What is a void set ? Give one example

b. Draw a venn diagram to represent (A’-B’)’.

Answer:

a. A set which does not contain any element is called void set. e.g., The set of female students in a school for boys only.

b.

Question 23.

Classify the following sets.

i. Set of lines passing through a point.

ii. Set of lines passing through two given points.

iii. {x: x^{2} – 2 = 0, x is rational}

iv. Set of all concentric circles touching the x-axis at the origin.

v. A = {x: x^{2} – 9x + 20 = 0, x is real} and B = {4,5}

Answer:

i. Infinite set

ii. Empty set

iii. Singleton set

iv. Infinite set

v. Equal sets

Question 24.

A = {1,2}, B = {3,4}

a. Write A× B.

b. Write a relation from A to B in roster form.

c. Represent all possible functions from A to B (Arrow diagram may be used).

Answer:

a. A×B={(1,3) (1,4) (2,3)(2,4)}

b. Any subset of A×B

**Long Answer Type questions **

**(Score 6)**

Question 1.

A survey was conducted among a sample of 10,000 families on the use of mobile phone connections. It was found that 40% families use ‘BSNL’, 20% families use ‘Vodafone’, 10% families use ‘Idea’, 5% families use ‘BSNL’ and ‘Vodafone’, 3% families use ‘Vodafone and ‘Idea’, 4% families use ‘BSNL’ and ‘Idea’ and 2% families use all the three connections.

i. If A, B and C denote the sets of families using ‘BSNL’, ‘Vodafone’ and ‘Idea’ mobile phone connections respectively, match the following columns X and Y in the table.

X | Y |

n(A) | 500 |

n(B) | 1000 |

n(C) | 200 |

n (A ∩ B) | 4000 |

n (B∩C) | 400 |

n(C∩A) | 2000 |

n (A ∩ B ∩C) | 300 |

ii. Determine the number of families that use ‘BSNL’ mobilephone connection only.

iii. Find the number of families that use none of the mobilephone connections.

Answer:

i.

X | Y |

n(A) | 500 |

n(B) | 1000 |

n(C) | 200 |

n (A ∩ B) | 4000 |

n (B∩C) | 400 |

n(C∩A) | 2000 |

n (A ∩ B ∩C) | 300 |

Question 2.

If U = {1,2,3,4,5,6,7,8,9,10,11}, A= {2, 5,9,10} and B = {1,4,7,9}

i. Find A’ and B’

ii. Verify (Au B)’ =A’n B’

iii. (A∩B)’=A’u B’

Answer:

i A’= {1,3,4,6,7,8,11}

B’= {2,3,5,6,8,10,11}

ii. A ∪ B= {1,2,4,5,7,9,10}

(A∪B)’= {3,6,8,11}

A’∩B’= {3,6,8,11}

∴ (A∪B)’= A’∩B’

iii. A∩B = {9}

(A∩B)’= {1,2,3,4,5,6,7,8,10,11}

A’∪B’={l,2,3,4. 5,6, 7,8,10,11}

∴ (A∩B)’=A’∪B’

Question 3.

a Prove that

i. (A∪B)’ ^{=}A’∩B’.

ii. (A∩B)’ = A’∪B’.

for any sets by the help of venn diagrams.

b. Which of the following is equal to

{x/x ∈ R, 3 < x < 4}

a. (3,4)

b. (3,4)

c. (3,4)

d. (3.4)

Answer:

Question 4.

Prove that (A ∪ B) – (A ∩B) = B ∩ A^{1}, h In a survey of 100 students in a music school the number of students learning different music instruments was found to be Guitar 28, Veena 30, Flute 42, Guitar and Veena 8, Guitar and Flute 10, Veena and Flute 5, all musical instruments 3. How many students were learning none of the three musical instruments?

Answer:

Question 5.

A = {1,2,4,5,6, 7, 8,9}, B = {1,3, ,5,6,8} C = {5,6,7,8,10,11}. Find

i. A-B.

ii. A – (B ∪C) and verify

iii. A- (B∪C) =(A – B)∩(A – C)

Answer:

i. A-B= {1,2,4,5,6,7,8,9} – {1,3,5,6,8} = {2,4,7,9}

ii. A -(B ∪ C)

= {1,2,4, 5,6,7, 8,9} – {1, 3, 5,6,7, 8, 10,11} = {2,4,9}

iii. A-(B∪C) = {2,4,9}………………… (1)

A-C = {1,2,4,5,6,7,8,9} – { 5,6,7,8, 10,11} = {1,2,4,9}

A – B = {2,4,7,9} (using (iii) above)

∴ (A-B)∩ (A-C)= {2,4,7,9} ∩ {1,2,4,9} = {2,4,9}………………… (2)

From(1)&(2),

A-(B∪C) = (A-B)∩(A-C)

Question 6.

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

ii. 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 ?

Answer:

i. Let A = Set of people like coffee and

B = Set of people who like tea Then, A∪B = Set of people who like atleast one of the two drinks and

A∩B = Set of people who like both the drinks.

Given, n(A) = 37, n(B) = 52, n(A ∪B) = 70 We know that,

n(A∪B) = n(A) + n(B)-n(A∩B)

70 = 37 + 52-n(A∩B) n(A∩B) = 89 -70 = 19

Hence, 19 people like both coffee and tea.

ii. Let C be the set of the people who like Cricket and T be the set of people who like Tennis.

Then, n (C ∪ T) = 65

n (C) = 40 and n(C∩T)= 10 We know that, n(C∪T) = n(C) + n(T) -n(C∩T)

65=40 + n(T)-10 n(T) = 65 -40+10 = 35

Number of people who like only Tennis = n(T) – n(C ∩T)

= 35-10 = 25

Hence, number of people, who like Tennis only and not cricket is 25 and number of people who like tennis is 35.

Question 7.

a. In a group of400 people, 250 can speak Hindi and 200 can speak English. How many can speak both Hindi and English ?

b. Let F_{1 } be the set of parallelogram, F_{2} be the set of rectangles, F_{3} be the set of rhombus, F_{4}be the set of squares and F_{3 }be the set of trapeziums in a plane. Then, show that F_{t} is equal to union of all sets.

Answer:

a. Let H denote the set of people who speak Hindi and E denote the set of people who speak English.

Given, n(H) = 250, n(E) = 200

n(H∪E) = 400 We know that,

n(H∪E) = n(H) + n(E)-n(H ∩ E)

400 = 250 + 200 – n( H ∩ E)

n(H∩E) = 450-400 = 50

Hence, 50 people can speak Hindi and English.

b. Given, F_{1 }= The set of parallelograms

F_{2} = The set of rectangles

F_{3} = The set of rhombus

F_{4} = The set of squares

F_{5} = The set of trapeziums

By definition of parallelogram, opposite sides are equal and parallel. In rectangles, rhombus and squares, all have opposite sides equal and parallel, therefore

Question 8.

Let A= {x:x ∈ N and x is a multiple of 2} B = {x: x ∈ N and x is a multiple of 5} and C = {x: x e N and x is a multiple of 10} Describe the sets

i. (A∩B)∩C

ii.A∪(B∩C) A ∩( B∪C)

Answer:

A = {x: x ∈ N and x is a multiple of 2 }

= {2,4,6,…}

B = {x: x ∈ N and x is a multiple of 5 }

**NCERT Questions and Answers**

Question 1.

Four different collections are given below.

i. The collection of 12 months of a year.

ii. The collection of novels written by the writer M.T. Vasudevan Nair.

iii. The collection of 11 best cricket batsmen of the world.

iv. The collection of all even integers. Write the collection which is not a set.

Answer:

iii. The collection of 11 best cricket batsmen of the world.

Question 2.

State which of the following sets are finite and which are infinite:

i. {x:x∈ N and (x -1) (x – 2) = 0}

ii. {x: x∈ N and x^{2} = 4}

iii. {x: x ∈ N and 2x -1 = 0}

iv. {x: x ∈ N and x is prime}

v. x {x: x ∈ N and x is odd}

Answer:

(i), (ii) and (iii) are finite sets.

(iv) and (v) are infinite sets.

Question 3.

Verify distributive laws for the following sets: A= {3,5,7,9,11}, B = {7,9,11,13}, C = {H, 13,15}

Answer:

Question 4

Find the union and intersection of the sets

A = {x: x ∈ N and 1 < x < 6}

B = {x : x ∈ N and 6 < x < 10}

ans:

A = {2, 3, 4, 5, 6}, B = {7, 8, 9}

A∪B= {2, 3, 4, 5, 6, 7, 8, 9},

A∩B =φ

Question 5.

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

Answer:

250-x+x + 200-x = 400 ⇒ x=50

Question 6.

In a survey of 400 students in a school, 100 were listed as drinking apple juice, 150 as drinking orange juice and 75 were listed as both drinking apple as well as orange juice. Find how many students were drinking neither apple juice nor orange juice,

Answer:

n(U)=400. n(A)=100, n(B)= 150, n(A ∩ B)=75

Now n(A’∩B’)=n((A∪B) = n(U)-n{A∪B)

= n(U) – n(A) – n(B) + n(A∩B) = 225

Question 7.

Find the intersection of the following pairs of sets and which pairs are disjoint sets?

i. A={1,2,3,4}, B={x:x∈N and 4< x < 6}

ii. C = {a, e, i, o, u}, D = {c, d, e, f}

iii. E = {x: x is an odd natural number}

F = {x: x is an even natural number}

Answer:

i. (A∩B) = {4} ≠φ

ii. (CD)={e}≠φ

iii. (E∩F) = φ. Here E and F are disjoint sets

Question 8.

A market research group conducted a survey of 1000 consumers and reported that 720 consumers liked product A and 450 consumers liked product B.What is the least number that must have liked both products?

Answer:

n(U) = 1000, n(S) = 720, n(T)= 450 n(S∪T) = n(S) + n(T) – n(S∩T)

⇒1170-n(S∩T), n(S ∩ T) is least when n(S u T) is maximum.

But S∪Tc= U⇒n(S∪T)<n(U) =1000.

So minimum value of n(S∪ T) is 1000.

Thus the least value of n(S ∩T) is 170.

Question 9.

Out of 500 car owners investigated, 400 owned car A and 200 owned car B, 50 owned both A and B cars. Is this data correct?

Answer:

n(U) = 500, n(M) = 400, n(S) = 200 and

n(S ∪M) = 50. Then

n(S ∪ M) = n(S)+n(M) – n(S ∩M)= 550

But (S ∪ M) c U ⇒ n(S∪M) < n(U) = 500

This is a contradiction. So the given data is incorrect.

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