SETS AND RELATIONS + FACTORIZATION AND ALGEBRAIC MANIPULATION

1

In a class of 45 students, 25 like math, 20 like physics, and 10 like both. How many like only physics?

A 10 B 15 C 20 D 5

2

The commutative property for intersection of sets states that:

A A ∩ B = B ∩ A B (A ∩ B) ∩ C = A ∩ (B ∩ C) C A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) D A ∩ U = A

3

Zara is verifying the associative property for sets. She has A = {1}, B = {2}, C = {3}. Which of the following is true?

A (A ∪ B) ∪ C = A ∪ (B ∪ C) B (A ∩ B) ∪ C = A ∩ (B ∪ C) C A ∪ (B ∩ C) = (A ∪ B) ∩ C D A ∩ (B ∪ C) = (A ∩ B) ∪ C

4

Which property is used in (A ∪ B) ∪ C = A ∪ (B ∪ C)?

A Commutative B Associative C Distributive D Identity

5

Simplify: (2x + 6)/(x² + 3x)

A 2/x B 2/(x + 3) C 2x/(x + 3) D 2/(x)

6

Nadia is factorizing x² - 7x + 12. She needs two numbers with product 12 and sum -7. What is the factorization?

A (x - 3)(x - 4) B (x + 3)(x + 4) C (x - 2)(x - 6) D (x - 1)(x - 12)

7

Which of the following is a subset of A = {1, 2, 3}?

A {1, 2, 3, 4} B {1, 2} C {0, 1} D {3, 4}

8

If R = {(x, y) | x ∈ {1, 2}, y ∈ {3, 4}} is a binary relation, which is a valid ordered pair in R?

A (1, 3) B (3, 1) C (1, 1) D (3, 3)

9

Simplify: (x + 1)/(x² - 1) + (x - 1)/(x² - 1)

A 2/(x - 1) B 2x/(x² - 1) C 2/(x + 1) D 2x/(x - 1)

10

What is the square root of (x² + 6x + 9)?

A x + 3 B x - 3 C (x + 3)² D x² + 6x + 9

11

In a group of 40 students, 25 play hockey, 20 play cricket, and 10 play both. How many play neither hockey nor cricket?

A 5 B 10 C 15 D 20

12

Which property states that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)?

A Associative property B Commutative property C Distributive property D Identity property

13

Junaid is verifying De Morgan's law for sets. If U = {1,2,3,4,5}, A = {1,2}, B = {3,4}, which of the following is true?

A (A ∪ B)' = A' ∩ B' B (A ∩ B)' = A' ∪ B' C Both A and B D Neither A nor B

14

Simplify: (x² - 4)/(x - 2)

A x + 2 B x - 2 C x + 4 D x - 4

15

Komal is finding the square root of 49a⁶/64b². She takes the square root of each part. What is the result?

A 7a³/8b B 7a³/8b² C 7a²/8b D 49a³/64b

16

Sohail is factorizing x² - x - 6. He needs two numbers with product -6 and sum -1. What is the factorization?

A (x - 3)(x + 2) B (x + 3)(x - 2) C (x - 6)(x + 1) D (x + 6)(x - 1)

17

What is the domain of the relation R = {(1,2), (2,3), (3,4)}?

A {1, 2, 3} B {2, 3, 4} C {1, 2, 3, 4} D {1, 3}

18

What is the complement of the universal set U?

A U B ∅ C U' = U D Undefined

19

Maryam is multiplying the fractions (x/3) × (6/x²). She multiplies numerators and denominators. What is the product in simplest form?

A 2/x B 2x C x/2 D 6/x

20

Sadia has two sets: P = {1,2,3} and Q = {2,3,4}. She finds P ∩ Q. What is the result?

A {1} B {2,3} C {4} D {1,2,3,4}

21

Rizwan defines a relation R = {(1,2), (2,3), (3,4)}. What is the range of R?

A {1,2,3} B {2,3,4} C {1,2,3,4} D {2,3}

22

In a class of 30 students, 18 like Math, 15 like Science, and 8 like both. A student is chosen at random. How many like neither Math nor Science?

A 5 B 7 C 8 D 10

23

If A = {x | x is a factor of 12} and B = {x | x is a factor of 18}, what is A ∩ B?

A {1, 2, 3, 6} B {1, 2, 4, 6, 12} C {1, 2, 3, 6, 9, 18} D {1, 2, 3}

24

Which of the following is a binary relation from A = {1, 2} to B = {3, 4}?

A {(1,3), (2,4)} B {(1,1), (2,2)} C {(3,1), (4,2)} D {(1,2,3)}

25

Which of the following is the identity element for union of sets?

A Universal set B Empty set C The set itself D No identity exists