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Math Class VIII 1 Question Bank

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Section I : Sets

1. Describe the following sets in roster form :

(i) 2{ / , , 2 5}x x n n N n= ∈ ≤ ≤

(ii) {x / x is composite number and 11 < x < 25}

(iii) {x / x ∈ W, x is divisible by 4 and 6, x ≤ 100}

(iv) {x / x is two digit number whose sum of digits is 7}

(v) / , and 5 3

n x x n N n

n

= ∈ ≤

+

(vi) 2 1

/ , and 10 2 3

n x x n W n

n

+ = ∈ ≤

+

(vii) 2{ / , 20}x x W x∈ < (viii) 2{ / 5 , and 400}x x p P I x= ∈ <

(ix) 1

/ , and 5x x n N n n

= ∈ ≤

(x) { /3 – 5 15, }x x x W< ∈

(xi) {First four planets of our solar system}.

Ans. (i) {4, 9, 16, 25}

(ii) {12, 14, 15, 16, 18, 20, 21, 22, 24}

(iii) {0, 12, 24, 36, 48, 60, 72, 84, 96}

(iv) {16, 25, 34, 43, 52, 61, 70}

(v) 1 2 3 4 5

, , , , 4 5 6 7 8

12 SETS AND VENN

DIAGRAMS

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(vi) 1 3 5 7 21

, , , , 3 5 7 9 23

⋅ ⋅ ⋅

(vii) {– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4}

(viii) {– 15, – 10, – 5, 0, 5, 10, 15}

(ix) 1 1 1 1

1, , , , 2 3 4 5

(x) {0, 1, 2, 3, 4, 5, 6}

(xi) {Mercury, Venus, Earth, Mars}

2. Write the following sets in the builder form :

(i) {11, 13, 17, 19, 23, 29, 31, 37}

(ii) 1 1 1

1, , , , 2 3 9

⋅ ⋅⋅

(iii) {21, 23, 25, 27, 29, 31, 33, 35, 37}

(iv) 1 3 5 7

, , , , 3 5 7 9

⋅ ⋅ ⋅

(v) {– 10, – 5, 0, 5, 10, 15, …, 100}

(vi) {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

Ans. (i) {x / x is a prime number, 10 < x < 40}.

(ii) 1

/ , and 10x x n N n n

= ∈

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3. Find, which of the following sets are singleton set :

(i) The set of points of intersection of two non-parallel straight

lines on the same plane.

(ii) A = {x : 7x – 3 = 11}

(iii) B = {y : 2y + 1 < 3 and y ∈ W}

Ans.(i) The set of points of intersection of two non-parallel straight

lines on the same plane is a singleton set.

(ii) A = {x : 7x – 3 = 11}

7x – 3 = 11 ⇒ 7x = 11 + 3

⇒ 7x = 14 ⇒ x = 14

2 7

=

∴ A = {2}

Hence, the given set A has only one element, so it is a singleton set.

(iii) B = {y : 2y + 1< 3 and y W∈ }

2y + 1 < 3

⇒ 2y + 1 – 1 < 3 – 1 (Subtracting 1 from both sides)

⇒ 2y < 2

⇒ y < 2

2 (Dividing both sides by 2)

⇒ y < 1

∴ B = {0}

Hence, it is a singleton set.

4. State whether the following pairs of sets are equivalent or not :

(i) A = {x : x ∈ N and 11 ≥ 2x –1} and

B = {y : y ∈ W and 3 ≤ y ≤ 9}

(ii) Set of whole numbers and set of multiples of 3.

(iii) P = {5, 6, 7, 8} and M = {x : x ∈ W and x ≤ 4}

Ans. (i) A = {x : x ∈ N and 11 ≥ 2x – 1}

11 ≥ 2x – 1

⇒ 11 + 1 ≥ 2x – 1 + 1 ( Adding 1 to both sides)

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⇒ 12 ≥ 2x ⇒ 12

2 ≥ x ⇒ 6 ≥ x

∴ A = {1, 2, 3, 4, 5, 6}

∴ n(A) = 6

B = {y : y W∈ and 3≤ y ≤ 9}

∴ 3 ≤ y ≤ 9

∴ B = {3, 4, 5, 6, 7, 8, 9}

∴ n (B) = 7

∴ Cardinal number of set A = 6 and cardinal number of set B = 7

Hence, set A and set B are not equivalent.

(ii) Set of whole numbers and set of multiples of 3 are equivalent

because both these sets have infinite number of elements.

(iii) P = {5, 6, 7, 8}

n (P) = 4

M = {x : x ∈ W and x ≤ 4}

M = {0, 1, 2, 3, 4}

n(M) = 5

Cardinal number of set P = 4

and Cardinal number of set M = 5. Hence, these sets are not

equivalent.

5. (A) Let A = {all quadrilaterals}, B = {all rectangles}, C = {all

squares} and D = {all rhombuses} in a plane. State, whether the

following statements are true or false.

(i) B C A⊂ ⊂ (ii) C B A⊂ ⊂

(iii) C D A⊂ ⊂ (iv) D C A⊂ ⊂

(v) A B C⊇ ⊇ (vi) A B C⊆ ⊆

Ans.(i) False (ii) True (iii) True (iv) False (v) True (vi) False

(B) Let A = {all triangles}, B = {all isosceles triangles} and C = {all

equilateral triangles}. State whether the following statements are

true or false.

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(i) B C A⊂ ⊂ (ii) C B A⊂ ⊂

Ans. (i) False (ii) True

6. Let A be the set of letters in the word, ‘seed’. Find:

(i) A (ii) n (A)

(iii) Number of subsets of A (iv) Number of proper subsets

Ans. (i) A = {s, e, d}

(ii) n (A) = number of elements = 3

(iii) Number of subsets of A = 23 = 8

{ }, {s}, {e}, {d}, {s, e}, {s, d}, {e, d} {s, e, d}.

(iv) Number of proper subsets of A

= 2n – 1= 23 – 1 = 8 – 1= 7

because { } or φ is not a proper subset.

7. (A) Find the power set of each of the following sets :

(i) A = {0, 5} (ii) B = {7, 9} (iii) C = {2, 4, 6}

Ans. (i) P (A) = {φ , {0}, {5}, {0, 5}}.

(ii) P (B) = {φ , {7}, {9}, {7, 9}}.

(iii) P (C) = {φ , {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}}.

(B) Let A = {1, {2}}. Find the power set of A.

Ans. (i) P (A) = {φ , {1}, {{2}}, {1, {2}}.

8. Let { : , 50},x x N xξ = ∈ < 2{ : },A x x= ∈ξ 2{ : ,B x x n= =

}n N∈ and C = {x : x is a factor of 36}. List the elements of each

of the sets A, B and C. Also, state whether each of the following

statements is true or false :

(i) A B⊆ (ii) A = B (iii) A B↔

(iv) B C↔ (v) n (A) < n (C)

Ans. ξ = { : , 50}x x N x∈ < = {1, 2, 3, 4, 5, …, 49, 50}

A = 2

{ : }x x ∈ξ = {1, 2, 3, 4, 5, 6, 7}

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B = {x : x = n2, n ∈ N} = {1, 4, 9, 16, 25, 36, 49}

C = {x : x is a factor of 36}

= {1, 2, 3, 4, 6, 9, 12, 18, 36}

(i) False (ii) False (iii) True

(iv) False (v) True

9. Let A = {letters of BOMBAY} and B = {letters of CALCUTTA}

(i) Are these sets disjoint or overlapping ?

(ii) Are these sets equal ?

(iii) Are these sets equivalent ?

(iv) Is any of these sets, subset of the other ?

(v) Describe a universal set for this problem.

Ans. A = {letters of BOMBAY} = {A, B, M, O,Y}

B = {letters of CALCUTTA} = {A, C, L, T, U}.

(i) These sets are overlapping.

(ii) These sets are not equal.

(iii) These sets are equivalent.

(iv) Neither A B⊆ nor B A⊆ .

(v) {Letters of English alphabet}.

10. Let { : , 15},x x W xξ = ∈ < A = {multiples of 2}, B = {multiples of 3},

C = {multiples of 5} and D = {multiples of 6}

State whether each of the following statements is true or false :

(i) C and D are disjoint (ii) C D↔ (iii) C = D

(iv) A B↔ (v) A B⊇ (iv) D A⊆

(vii) D B⊆ (viii) C B⊆

Ans. The sets ,ξ A, B, C and D in roster form are

ξ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

A = {2, 4, 6, 8, 10, 12, 14}

B = {3, 6, 9, 12}

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C = {5, 10}

D = {6, 12}

(i) True (ii) True

(iii) False, because C and D are disjoint sets.

(iv) False, because A has more elements than B

(v) False, because B is not contained in A.

(vi) True (vii) True

(viii) False, because C is not contained in B.

11. Given the universal set = { : and 20},x x N x∈ < find :

A = {x : x = 3p ; p ∈ N}

Ans. Universal set U = {x : x ∈ N and x < 20}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

A = {x : x = 3p ; p ∈ N}

x = 3p

If p = 1, then x = 3 × 1 = 3

If p = 2, then x = 3 × 2 = 6

If p = 3, then x = 3 × 3 = 9

If p = 4, then x = 3 × 4 = 12

If p = 5, then x = 3 × 5 = 15

If p = 6, then x = 3 × 6 = 18

∴ A = {3, 6, 9, 12, 15, 18}

12. Find the proper subsets of {x : x2 – 9x – 10 = 0}.

Ans. x2 – 9x – 10 = 0 ⇒ x2 – 10x + x – 10 = 0

⇒ x(x – 10) + 1(x – 10) = 0 ⇒ (x – 10) (x + 1) = 0

If, (x –10) = 0 ⇒ x = 10 and, if (x + 1) = 0 ⇒ x = – 1

∴ Given set = {–1, 10}

∴ Proper subsets of this set = {–1}, {10}, {–1, 10}

13. Let ξ= {1, 2, 3, …, 9}, A = {1, 2, 3, 4, 6, 7} and B = {4, 6, 8}. Find :