Grade 11 : Mathematics - Relations (Set 3) | Concept of Maths

Question 1

1. 1. If the ordered pairs $(x+2, 4)$ and $(5, 2y)$ are equal, find the values of $x$ and $y$.

A) $x=3, y=4$

B) $x=7, y=2$

C) $x=3, y=2$

D) $x=5, y=4$

Answer

Correct Answer: C

Question 2

2. 2. Let $A = \{1, 2\}$ and $B = \{3, 4\}$. How many relations are possible from set $A$ to set $B$?

A) 4

B) 16

C) 8

D) 32

Answer

Correct Answer: B

Hint: The number of relations from $A$ to $B$ is $2^{n(A) \times n(B)}$. Here $2^{2 \times 2} = 2^4 = 16$.

Question 3

3. 3. If $n(A) = 3$ and $n(B) = 4$, then the number of elements in $(A \times B) \cap (B \times A)$ is at most:

A) 9

B) 12

C) 16

D) 7

Answer

Correct Answer: A

Hint: The intersection $(A \times B) \cap (B \times A)$ is equal to $(A \cap B) \times (B \cap A)$. The maximum number of common elements in sets of size 3 and 4 is 3. So, $3 \times 3 = 9$.

Question 4

4. 4. Find the range of the relation $R = \{ (x, x^2) : x \text{ is a prime number less than } 10 \}$.

A) $\{1, 4, 9, 25, 49\}$

B) $\{4, 9, 25, 49, 81\}$

C) $\{2, 3, 5, 7\}$

D) $\{4, 9, 25, 49\}$

Question 5

5. 5. If $A \times B = \{ (a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2) \}$, find set $A$.

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

B) $\{a, b\}$

C) $\{a, b, 1, 2, 3\}$

D) $\{a, 1, b, 3\}$

Answer

Correct Answer: B

Hint: Set $A$ consists of all the first elements of the ordered pairs in $A \times B$.

Question 6

6. 6. Let $R$ be a relation on $\mathbb{N}$ defined by $R = \{ (x, y) : x + 2y = 10 \}$. The domain of $R$ is:

A) $\{2, 4, 6, 8, 10\}$

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

C) $\{2, 4, 6, 8\}$

D) $\{1, 3, 5, 7, 9\}$

Answer

Correct Answer: C

Hint: Since $y \in \mathbb{N}$, $y \ge 1$. If $y=1, x=8$; $y=2, x=6$; $y=3, x=4$; $y=4, x=2$. If $y=5, x=0$ (not in $\mathbb{N}$).

Question 7

7. 7. If $R = \{ (x, y) : x, y \in \mathbb{Z}, x^2 + y^2 = 9 \}$, which of the following is NOT an element of $R$?

A) $(1, \sqrt{8})$

B) $(0, 3)$

C) $(-3, 0)$

D) $(0, -3)$

Answer

Correct Answer: A

Question 8

8. 8. Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation $R$ from $A$ to $A$ by $R = \{ (x, y) : y = x + 1 \}$. The range of $R$ is:

A) $\{1, 2, 3, 4, 5, 6\}$

B) $\{1, 2, 3, 4, 5\}$

C) $\{2, 3, 4, 5\}$

D) $\{2, 3, 4, 5, 6\}$

Answer

Correct Answer: D

Hint: For $x \in \{1, 2, 3, 4, 5\}$, $y$ values are $\{2, 3, 4, 5, 6\}$. If $x=6, y=7 \notin A$.

Question 9

9. 9. If $A \times B$ has 6 elements and three of them are $(1, 3), (2, 5), (3, 3)$, then find $A \times B$.

A) $\{(1,3), (1,5), (2,3), (2,5), (3,5)\}$

B) $\{(1,3), (1,5), (2,3), (2,5), (3,3), (3,5)\}$

C) $\{(1,3), (2,3), (3,3), (5,1), (5,2), (5,3)\}$

D) $\{(3,1), (3,2), (3,3), (5,1), (5,2), (5,3)\}$

Answer

Correct Answer: B

Hint: First elements give $A = \{1, 2, 3\}$, second elements give $B = \{3, 5\}$. $n(A \times B) = 3 \times 2 = 6$.

Question 10

10. 10. The Cartesian product $A \times A$ has 9 elements, among which are found $(-1, 0)$ and $(0, 1)$. Find the set $A$.

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

B) $\{0, 1\}$

C) $\{-1, 0, 1\}$

D) $\{-1, 0, 1, 2\}$

Answer

Correct Answer: C

Hint: Since $n(A \times A) = 9$, $n(A) = 3$. Elements gathered from the pairs are $-1, 0, 1$.

Question 11

11. 11. Let $R$ be a relation from $A = \{2, 3, 4, 5\}$ to $B = \{3, 6, 7, 10\}$ defined by $x R y \iff x \text{ divides } y$. Find $R$ in Roaster form.

A) $\{(2,6), (2,10), (3,3), (3,6), (5,10)\}$

B) $\{(2,6), (2,10), (3,6), (5,10)\}$

C) $\{(3,3), (6,2), (10,2), (10,5)\}$

D) $\{(2,3), (2,6), (2,7), (2,10)\}$

Answer

Correct Answer: A

Hint: Check which $x$ from $A$ divides $y$ from $B$. 2 divides 6, 10; 3 divides 3, 6; 5 divides 10.

Question 12

12. 12. If $R^{-1} = \{ (2, 1), (4, 3), (6, 5) \}$, then the domain of the original relation $R$ is:

A) $\{2, 4, 6\}$

B) $\{1, 3, 5\}$

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

D) $\{2, 1, 4, 3, 6, 5\}$

Answer

Correct Answer: B

Question 13

13. 13. Let $A = \{1, 2\}$. The number of non-empty relations on $A$ is:

A) 16

B) 4

C) 15

D) 12

Answer

Correct Answer: C

Hint: Total relations = $2^{n(A) \times n(A)} = 2^4 = 16$. Non-empty = $16 - 1 = 15$.

Question 14

14. 14. If $(x+y, x-y) = (4, 2)$, then the values of $x$ and $y$ are:

A) $x=4, y=2$

B) $x=2, y=1$

C) $x=3, y=0$

D) $x=3, y=1$

Answer

Correct Answer: D

Hint: Solve the linear system: $x+y=4$ and $x-y=2$. Adding them gives $2x=6 \implies x=3$.

Question 15

15. 15. Find the domain of $R = \{ (x, y) : y = |x-1|, x \in \mathbb{Z}, |x| \le 2 \}$.

A) $\{-2, -1, 0, 1, 2\}$

B) $\{0, 1, 2, 3\}$

C) $\{-2, 2\}$

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

Answer

Correct Answer: A

Question 16

16. 16. If $n(A) = p$ and $n(B) = q$, then the total number of non-void relations from $A$ to $B$ is:

A) $pq$

B) $2^{pq} - 1$

C) $2^{pq}$

D) $p^q - 1$

Answer

Correct Answer: B

Question 17

17. 17. Let $A = \{1, 2, 3\}$. Which of the following relations on $A$ is the Identity Relation?

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

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

C) $\{(1,1), (2,2), (3,3)\}$

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

Answer

Correct Answer: C

Question 18

18. 18. If $A = \{x, y, z\}$ and $B = \{1, 2\}$, then $n(A \times B)$ is:

A) 6

B) 5

C) 9

D) 8

Answer

Correct Answer: A

Question 19

19. 19. The relation $R = \{ (x, y) : x, y \in \mathbb{N}, 2x + y = 41 \}$. The range of $R$ is:

A) $\{1, 2, 3, \dots, 20\}$

B) $\{1, 3, 5, \dots, 39\}$

C) $\{2, 4, 6, \dots, 40\}$

D) $\{1, 3, 5, \dots, 39\}$

Answer

Correct Answer: D

Hint: $y = 41 - 2x$. Since $x \ge 1$, $y$ is odd and less than 41. Range is all odd numbers from 1 to 39.

Question 20

20. 20. If $A \subset B$, then $A \times C$ is a subset of:

A) $C \times B$

B) $B \times C$

C) $A \times B$

D) $C \times A$

Answer

Correct Answer: B

Hint: If every element of $A$ is in $B$, then every pair in $A \times C$ must be in $B \times C$.

Accepted Answer

Correct Answer: D

Grade 11 : Mathematics - Relations (Set 3) | Concept of Maths