Grade 11 : Mathematics - Relations (Set 1) | Concept of Maths

Question 1

1. If \( (x + 2, 4) = (5, y - 1) \), find the values of \( x \) and \( y \).

A) \( x=7, y=3 \)

B) \( x=3, y=5 \)

C) \( x=3, y=3 \)

D) \( x=5, y=5 \)

Answer

Correct Answer: B

Question 2

2. If set \( A \) has 3 elements and set \( B \) has 4 elements, how many elements are in \( A \times B \)?

A) 12

B) 7

C) \( 2^{12} \)

D) 1

Answer

Correct Answer: A

Question 3

3. A relation \( R \) from set \( A \) to set \( B \) is mathematically defined as:

A) \( R = A \cup B \)

B) \( R = A \cap B \)

C) \( R \subseteq A \times B \)

D) \( R \supseteq A \times B \)

Answer

Correct Answer: C

Question 4

4. Find the domain of the relation \( R = \{(1, 2), (3, 4), (5, 6)\} \).

A) \( \{1, 3, 5\} \)

B) \( \{2, 4, 6\} \)

C) \( \{1, 2, 3, 4, 5, 6\} \)

D) \( \emptyset \)

Answer

Correct Answer: A

Question 5

5. Find the range of the relation \( R = \{(x, y) : y = x + 5; x \in \{1, 2, 3\}\} \).

A) \( \{1, 2, 3\} \)

B) \( \{6, 7, 8\} \)

C) \( \{5, 10, 15\} \)

D) \( \{0, 5, 10\} \)

Answer

Correct Answer: B

Hint: Substitute \( x \) values into \( y = x+5 \) to get the set of second elements.

Question 6

6. If \( n(A) = p \) and \( n(B) = q \), what is the total number of relations that can be defined from \( A \) to \( B \)?

A) \( pq \)

B) \( p^q \)

C) \( q^p \)

D) \( 2^{pq} \)

Answer

Correct Answer: D

Question 7

7. Let \( A = \{1, 2\} \). The identity relation \( I_A \) on set \( A \) is:

A) \( \{(1, 2), (2, 1)\} \)

B) \( \{(1, 1), (2, 2), (1, 2)\} \)

C) \( \{(1, 1), (2, 2)\} \)

D) \( \emptyset \)

Answer

Correct Answer: C

Hint: Identity relation contains only pairs where the first and second elements are the same for every element in \( A \).

Question 8

8. If \( R \) is a relation from \( A \) to \( B \), then the inverse relation \( R^{-1} \) is a relation from:

A) \( B \) to \( A \)

B) \( A \) to \( A \)

C) \( B \) to \( B \)

D) \( A \times B \) to \( \emptyset \)

Answer

Correct Answer: A

Question 9

9. If \( R = \{(1, 2), (2, 3)\} \), then \( R^{-1} \) is:

A) \( \{(1, 2), (2, 3)\} \)

B) \( \{(2, 1), (3, 2)\} \)

C) \( \{(1, 1), (2, 2)\} \)

D) \( \{(3, 2), (2, 1), (1, 1)\} \)

Answer

Correct Answer: B

Question 10

10. The set of all second elements of the ordered pairs in a relation \( R \) from \( A \) to \( B \) is called the:

A) Domain

B) Codomain

C) Range

D) Cartesian Product

Answer

Correct Answer: C

Question 11

11. If \( A = \{a, b\} \), how many relations can be defined on set \( A \)?

A) 4

B) 16

C) 8

D) 2

Answer

Correct Answer: B

Hint: Number of relations on set \( A \) is \( 2^{n(A) \times n(A)} = 2^{2 \times 2} \).

Question 12

12. If \( (x, 1) \in \{(2, 1), (3, 4)\} \), then the value of \( x \) is:

A) 1

B) 3

C) 4

D) 2

Answer

Correct Answer: D

Question 13

13. The smallest relation on any non-empty set \( A \) is the:

A) Empty relation \( \emptyset \)

B) Identity relation

C) Universal relation

D) Singleton relation

Answer

Correct Answer: A

Question 14

14. If \( R \) is a relation from \( A \) to \( B \), and \( B = \{1, 2, 3\} \), then the Codomain of \( R \) is:

A) The Range of \( R \)

B) The Domain of \( R \)

C) The set \( B \)

D) The set \( A \)

Answer

Correct Answer: C

Hint: For a relation from \( A \) to \( B \), the entire set \( B \) is the codomain.

Question 15

15. Let \( A = \{1, 2\} \) and \( B = \{3, 4\} \). Which of the following is NOT a relation from \( A \) to \( B \)?

A) \( \{(1, 3), (2, 4)\} \)

B) \( \{(3, 1), (4, 2)\} \)

C) \( \{(1, 4)\} \)

D) \( \emptyset \)

Answer

Correct Answer: B

Hint: Elements of a relation from \( A \) to \( B \) must be of form \( (a, b) \) where \( a \in A \) and \( b \in B \).

Question 16

16. If \( n(A \times B) = 15 \) and \( n(A) = 3 \), find \( n(B) \).

A) 5

B) 12

C) 45

D) 18

Answer

Correct Answer: A

Question 17

17. If \( R = \{(x, y) : x^2 + y^2 = 0\} \) on the set of real numbers, then \( R \) is:

A) \( \{(1, -1)\} \)

B) \( \{(1, 1)\} \)

C) \( \{(0, 0)\} \)

D) \( \emptyset \)

Answer

Correct Answer: C

Hint: For real numbers, \( x^2 + y^2 = 0 \) only when both \( x \) and \( y \) are zero.

Question 18

18. If \( A \times B = \emptyset \), it implies:

A) \( A \) is not empty

B) \( B \) is not empty

C) Both \( A \) and \( B \) are infinite

D) At least one of \( A \) or \( B \) is an empty set

Answer

Correct Answer: D

Question 19

19. Let \( R = \{(x, y) : y = x^2\} \). If \( (3, k) \in R \), what is the value of \( k \)?

A) 6

B) 9

C) 3

D) 27

Answer

Correct Answer: B

Question 20

20. Range of a relation is always a subset of the:

A) Codomain

B) Domain

C) Set \( A \)

D) None of these

Answer

Correct Answer: A

Hint: The Range consists of actual outputs, while Codomain is the set of all possible outputs.

Grade 11 : Mathematics - Relations (Set 1) | Concept of Maths