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

Question 1

1. If two ordered pairs are equivalent such that $(2x + y, 1) = (5, x - y)$, find the values of $x$ and $y$.

A) $x = 2, y = 1$

B) $x = 1, y = 2$

C) $x = 3, y = -1$

D) $x = 0, y = 5$

Answer

Correct Answer: A

Solution: Two ordered pairs are equal if and only if their corresponding structural components match precisely: 1. $2x + y = 5$ ....(Equation 1) 2. $x - y = 1$ ....(Equation 2) Adding both linear systems together cancels out $y$: $$(2x + y) + (x - y) = 5 + 1 \implies 3x = 6 \implies x = 2$$ Now substitute value $x = 2$ back into Equation 2: $$2 - y = 1 \implies y = 1$$ Final coordinate location matches: $(2, 1)$.

Question 2

2. Let a relation $R$ on the set of natural numbers $\mathbb{N}$ be defined as $R = \{(x, y) : 2x + y = 10\}$. What is the domain of this relation?

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

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

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

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

Answer

Correct Answer: B

Solution: The domain contains every active input value $x$ for which $y$ resolves to a valid element inside the set of natural numbers ($\mathbb{N}$). Isolating the output variable: $y = 10 - 2x$. Since $y \in \mathbb{N}$, it must evaluate to $y \ge 1$. $$10 - 2x \ge 1 \implies 9 \ge 2x \implies x \le 4.5$$ Since input constraints dictate $x \in \mathbb{N}$, the integer parameters can only map to: $1, 2, 3, 4$. $$\text{Domain} = \{1, 2, 3, 4\}.$$

Question 3

3. If set $A$ contains 2 elements and set $B$ contains 3 elements, calculate the total number of distinct relations possible from $A$ to $B$.

Question 4

4. Consider the relation $R = \{(x, x^3) : x \text{ is a prime number less than } 10\}$. Find the range of $R$.

A) $\{1, 8, 27, 64\}$

B) $\{8, 27, 125, 343\}$

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

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

Answer

Correct Answer: B

Solution: The positive prime integers sitting lower than 10 are: $2, 3, 5, 7$. The output tracking elements $(y)$ match the cubic values of these sequential primes: * $2^3 = 8$ * $3^3 = 27$ * $5^3 = 125$ * $7^3 = 343$ $$\text{Range Set} = \{8, 27, 125, 343\}.$$

Question 5

5. If a relation $R$ is defined from $A = \{1, 2, 3\}$ as $R = \{(x, y) : y = 2x + 1\}$, determine the inverse relation $R^{-1}$.

A) $\{(3, 1), (5, 2), (7, 3)\}$

B) $\{(1, 3), (2, 5), (3, 7)\}$

C) $\{(3, 1), (2, 5), (7, 3)\}$

D) $\{(1, 2), (2, 3), (3, 4)\}$

Answer

Correct Answer: A

Solution: First, map the explicit coordinates belonging to forward set $R$: * For $x=1 \implies y=2(1)+1=3 \implies (1, 3)$ * For $x=2 \implies y=2(2)+1=5 \implies (2, 5)$ * For $x=3 \implies y=2(3)+1=7 \implies (3, 7)$ This yields forward mapping $R = \{(1, 3), (2, 5), (3, 7)\}$. The inverse coordinates mapping $R^{-1}$ transposes these element components: $$\text{Inverse Map } R^{-1} = \{(3, 1), (5, 2), (7, 3)\}.$$

Question 6

6. Let $A = \{1, 2, 3\}$. A relation $R$ is defined on $A$ as $R = \{(x, y) : x - y = 10\}$. How many elements are in this relation?

A) $10$

B) $3$

C) $0$

D) $1$

Answer

Correct Answer: C

Solution: The maximum possible distance separating any two items inside the bounded pool $A = \{1, 2, 3\}$ matches $3 - 1 = 2$. There are absolutely no coordinate pairs $(x, y)$ possible such that $x, y \in A$ solves the equation $x - y = 10$. Thus, $R = \emptyset$ represents an empty null relation: $$n(R) = 0.$$

Question 7

7. If the Cartesian product expression evaluates to $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$, what is set $A$?

A) $\{1, 2\}$

B) $\{a, b\}$

C) $\{a, 1\}$

D) $\{b, 2\}$

Answer

Correct Answer: B

Solution: By fundamental definition, the leading set $A$ contains every first input coordinate grouping element situated within the pairs of product space $A \times B$. Extracting these unique first components isolates $a$ and $b$. $$\text{Set } A = \{a, b\}.$$

Question 8

8. A set $A$ has 2 elements. How many non-empty relations can be defined on set $A$?

A) $15$

B) $16$

C) $4$

D) $3$

Answer

Correct Answer: A

Solution: A relation operating on set $A$ is structured as a subset of the self-product matrix $A \times A$. Given $n(A) = 2$, then the total internal elements match $n(A \times A) = 2 \times 2 = 4$. Total complete subsets possible $= 2^4 = 16$. $$\text{Non-Empty Relations} = \text{Total Subsets} - \text{Null Set} = 16 - 1 = 15.$$

Question 9

9. Find the domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{Z}\}$.

A) $\{0, 3, 4, 5\}$

B) $\{0, \pm 3, \pm 4, \pm 5\}$

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

D) $\{-5, 5\}$

Answer

Correct Answer: B

Solution: Evaluating equation $x^2 + y^2 = 25$ for valid integer coordinates ($\mathbb{Z}$): * If $x = 0 \implies y = \pm 5$ (Valid integer) * If $x = \pm 3 \implies y = \pm 4$ (Valid integer) * If $x = \pm 4 \implies y = \pm 3$ (Valid integer) * If $x = \pm 5 \implies y = 0$ (Valid integer) Any intermediate integer assumptions for $x$ (such as $\pm 1$ or $\pm 2$) produce fractional square roots for $y$, which breaks the integer filter constraint. $$\text{Domain (Set of all valid } x \text{ values)} = \{0, \pm 3, \pm 4, \pm 5\}.$$

Question 10

10. Let $A = \{1, 2, 3\}$. The relation $R$ is defined as $R = \{(x, y) : x < y, x, y \in A\}$. Which of the following is the roster form of $R$?

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

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

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

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

Answer

Correct Answer: A

Solution: We search out all possible pairs $(x, y)$ inside product space $A \times A$ where the primary component value strictly scales lower than the second coordinate factor. Evaluating the pairs yields: $(1, 2)$, $(1, 3)$, and $(2, 3)$. Identity sets like $(1, 1)$ drop out since equality fails the strict inequality threshold ($1 < 1$ is false).

Accepted Answer

Correct Answer: C

Solution: The cardinality count within the baseline Cartesian product maps as: $n(A \times B) = n(A) \times n(B) = 2 \times 3 = 6\text{ elements}$. An active mathematical relation scales as any subset configuration of the universal space $A \times B$. The total combinations formula for any power set with $k$ containing items is $2^k$. $$\text{Total Relations possible} = 2^6 = 64.$$

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

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

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4. Consider the relation $R = \{(x, x^3) : x \text{ is a prime number less than } 10\}$. Find the range of $R$.

5. If a relation $R$ is defined from $A = \{1, 2, 3\}$ as $R = \{(x, y) : y = 2x + 1\}$, determine the inverse relation $R^{-1}$.

6. Let $A = \{1, 2, 3\}$. A relation $R$ is defined on $A$ as $R = \{(x, y) : x - y = 10\}$. How many elements are in this relation?

7. If the Cartesian product expression evaluates to $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$, what is set $A$?

8. A set $A$ has 2 elements. How many non-empty relations can be defined on set $A$?

9. Find the domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{Z}\}$.

10. Let $A = \{1, 2, 3\}$. The relation $R$ is defined as $R = \{(x, y) : x < y, x, y \in A\}$. Which of the following is the roster form of $R$?

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

1. If two ordered pairs are equivalent such that $(2x + y, 1) = (5, x - y)$, find the values of $x$ and $y$.

2. Let a relation $R$ on the set of natural numbers $\mathbb{N}$ be defined as $R = \{(x, y) : 2x + y = 10\}$. What is the domain of this relation?

3. If set $A$ contains 2 elements and set $B$ contains 3 elements, calculate the total number of distinct relations possible from $A$ to $B$.

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

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

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4. Consider the relation $R = \{(x, x^3) : x \text{ is a prime number less than } 10\}$. Find the range of $R$.

5. If a relation $R$ is defined from $A = \{1, 2, 3\}$ as $R = \{(x, y) : y = 2x + 1\}$, determine the inverse relation $R^{-1}$.

6. Let $A = \{1, 2, 3\}$. A relation $R$ is defined on $A$ as $R = \{(x, y) : x - y = 10\}$. How many elements are in this relation?

7. If the Cartesian product expression evaluates to $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$, what is set $A$?

8. A set $A$ has 2 elements. How many non-empty relations can be defined on set $A$?

9. Find the domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{Z}\}$.

10. Let $A = \{1, 2, 3\}$. The relation $R$ is defined as $R = \{(x, y) : x < y, x, y \in A\}$. Which of the following is the roster form of $R$?