Solve for [math] and [math]: [math].

[math] Solution: Equating corresponding coordinates: $[math][math][math][math]\boxed{x = 9\text{, }y = 5}$.

Class 9 · Advanced MathematicsNCERT 2026 Revised Syllabus

Chapter 3: Relations and Functions

Class IX CBSE Advanced Mathematics

Explores Cartesian products of sets, relations between two sets, domain and range, and the definition of a function with vertical-line reasoning and graphical shifts of modulus and quadratic curves.

Find complete NCERT solutions for Class 9 Advanced Mathematics, Class IX CBSE Advanced Mathematics Chapter 3. Every question has a clear, step-by-step answer.

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Exercise 3.1(6 questions)

Solve for $x$ and $y$ : $(x - 5, y + 1) = (4, 6)$ .

School Level Solution

Sol.

Solution:Equating corresponding coordinates: $$x - 5 = 4 \Rightarrow x = 9$$ $$y + 1 = 6 \Rightarrow y = 5$$Answer: $\boxed{x = 9\text{, }y = 5}$ .

Given $A = \{1, 2\}$ and $B = \{2, 3, 5\}$ , find $A \times B$ and $B \times A$ .

School Level Solution

Sol.

Solution:Cartesian product pairs every element of the first set with every element of the second.$$A \times B = \{(1,2),(1,3),(1,5),(2,2),(2,3),(2,5)\}$$$$B \times A = \{(2,1),(2,2),(3,1),(3,2),(5,1),(5,2)\}$$Answer:

$\boxed{A \times B = \{(1,2),(1,3),(1,5),(2,2),(2,3),(2,5)\}\text{, }B \times A = \{(2,1),(2,2),(3,1),(3,2),(5,1),(5,2)\}}$

If $n(A \times B) = 20$ and $n(A) = 4$ , find $n(B)$ .

School Level Solution

Sol.

Solution:$$n(A \times B) = n(A) \cdot n(B)$$ $$20 = 4 \cdot n(B) \Rightarrow n(B) = 5$$Answer: $\boxed{n(B) = 5}$ .

Given $A = \{1, 2, 3\}$ and $B = \{x, y\}$ , find $A \times B$ , $B \times A$ , $A \times A$ and $B \times B$ .

School Level Solution

Sol.

Solution:$$A \times B = \{(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)\}$$$$B \times A = \{(x,1),(x,2),(x,3),(y,1),(y,2),(y,3)\}$$$$A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$$$B \times B = \{(x,x),(x,y),(y,x),(y,y)\}$$Answer:

$\boxed{A \times B = \{(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)\}\text{, }B \times A = \{(x,1),(x,2),(x,3),(y,1),(y,2),(y,3)\}\text{, }A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}\text{, }B \times B = \{(x,x),(x,y),(y,x),(y,y)\}}$

Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 7\}$ , find $A \times B$ , $B \times A$ and $(A \times B) \cap (B \times A)$ .

School Level Solution

Sol.

Solution:$$A \times B = \{(1,2),(1,3),(1,7),(2,2),(2,3),(2,7),(3,2),(3,3),(3,7)\}$$$$B \times A = \{(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(7,1),(7,2),(7,3)\}$$The common ordered pairs: $$(A \times B) \cap (B \times A) = \{(2,2),(2,3),(3,2),(3,3)\}$$Answer: $\boxed{(A \times B) \cap (B \times A) = \{(2,2),(2,3),(3,2),(3,3)\}}$ .

Hard

Given $A = \{1, 2\}$ , $B = \{2, 3\}$ , $C = \{4, 5\}$ , verify that $A \times (B \cup C) = (A \times B) \cup (A \times C)$ .

School Level Solution

Sol.

Solution: $B \cup C = \{2, 3, 4, 5\}$ , so: $$A \times (B \cup C) = \{(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5)\}$$ $A \times B = \{(1,2),(1,3),(2,2),(2,3)\}$ and $A \times C = \{(1,4),(1,5),(2,4),(2,5)\}$ , so: $$(A \times B) \cup (A \times C) = \{(1,2),(1,3),(2,2),(2,3),(1,4),(1,5),(2,4),(2,5)\}$$Both sides are equal. Verified.Answer: $\boxed{\text{Verified: }A \times (B \cup C) = (A \times B) \cup (A \times C)}$ .

Exercise 3.2(6 questions)

Given $A = \{1, 2, 3\}$ , $B = \{4, 5, 6, 7\}$ and $R = \{(a, b) : a + b = 7, a \in A, b \in B\}$ , write $R$ and find its domain and range.

School Level Solution

Sol.

Solution:For $a = 1$ , $b = 6$ ; for $a = 2$ , $b = 5$ ; for $a = 3$ , $b = 4$ : $$R = \{(1,6),(2,5),(3,4)\}$$Domain $= \{1, 2, 3\}$ , Range $= \{4, 5, 6\}$ .Answer: $\boxed{R = \{(1,6),(2,5),(3,4)\}\text{, Domain }= \{1, 2, 3\}\text{, Range }= \{4, 5, 6\}}$ .

Given $A = \{2, 3, 4, 5\}$ , $B = \{3, 6, 7, 10\}$ and $R = \{(a, b) : a \text{ divides } b\}$ , write $R$ and find its domain and range.

School Level Solution

Sol.

Solution: $2 \mid 6$ , $2 \mid 10$ ; $3 \mid 3$ , $3 \mid 6$ ; $5 \mid 10$ : $$R = \{(2,6),(2,10),(3,3),(3,6),(5,10)\}$$Domain $= \{2, 3, 5\}$ , Range $= \{3, 6, 10\}$ .Answer: $\boxed{R = \{(2,6),(2,10),(3,3),(3,6),(5,10)\}\text{, Domain }= \{2, 3, 5\}\text{, Range }= \{3, 6, 10\}}$ .

Hard

Let $R = \{(a, b) : a + 2b = 12, a, b \in \mathbb{N}\}$ . Write $R$ and find its domain and range.

School Level Solution

Sol.

Solution:From $a + 2b = 12$ : for $b = 1, 2, 3, 4, 5$ , we get $a = 10, 8, 6, 4, 2$ : $$R = \{(10,1),(8,2),(6,3),(4,4),(2,5)\}$$Domain $= \{2, 4, 6, 8, 10\}$ , Range $= \{1, 2, 3, 4, 5\}$ .Answer: $\boxed{R = \{(10,1),(8,2),(6,3),(4,4),(2,5)\}\text{, Domain }= \{2, 4, 6, 8, 10\}\text{, Range }= \{1, 2, 3, 4, 5\}}$ .

Let $R = \{(x, x^2) : x \text{ is prime and } x < 10\}$ . Write $R$ and find its range.

School Level Solution

Sol.

Solution:Primes less than 10 are $2, 3, 5, 7$ : $$R = \{(2,4),(3,9),(5,25),(7,49)\}$$Range $= \{4, 9, 25, 49\}$ .Answer: $\boxed{R = \{(2,4),(3,9),(5,25),(7,49)\}\text{, Range }= \{4, 9, 25, 49\}}$ .

Hard

Given $A = \{p, q, r, s\}$ and $B = \{1, 2\}$ , how many relations can be formed from $A$ to $B$ ? Give four examples.

School Level Solution

Sol.

Solution:$$n(A \times B) = 4 \times 2 = 8$$Number of relations $=$ number of subsets of $A \times B$ : $$2^8 = 256$$Four examples: $\{(p,1)\}$ , $\{(p,1),(q,2)\}$ , $\{(p,1),(q,1),(r,2)\}$ , and $A \times B$ itself.Answer: $\boxed{2^8 = 256\text{ relations}}$ .

Given $A = \{1, 2, 3, 4, 5\}$ and $R = \{(a, b) : |a - b| = 2\}$ , write $R$ and find its domain and range.

School Level Solution

Sol.

Solution:Pairs differing by 2: $$R = \{(1,3),(3,1),(2,4),(4,2),(3,5),(5,3)\}$$Domain $= \{1, 2, 3, 4, 5\}$ , Range $= \{1, 2, 3, 4, 5\}$ .Answer:

$\boxed{R = \{(1,3),(3,1),(2,4),(4,2),(3,5),(5,3)\}\text{, Domain }= \{1, 2, 3, 4, 5\}\text{, Range }= \{1, 2, 3, 4, 5\}}$

Exercise 3.3(2 questions)

Hard

Which of the following relations are functions. Justify your answer.

a.Unique image for every element
b.One element has two images
c.One element has two images
d.One element has two images

School Level Solution

Sol.

Solution:- (a) Not a function: one element of the domain has two images. - (b) Not a function: one element of the domain has no image. - (c) Function: every element of the domain has exactly one image. - (d) Not a function: one element of the domain has multiple images.Answer: (a) not a function; (b) not a function; (c) a function; (d) not a function.

Hard

Which of the following relations from $A = \{3, 5, 7, 9\}$ to $B = \{1, 2, 3, 4, 5\}$ are functions? (a) $R_1 = \{(3,2),(5,4),(7,5),(9,5)\}$ (b) $R_2 = \{(1,3),(3,5),(5,7)\}$ (c) $R_3 = \{(2,3),(2,5),(2,7),(3,5),(3,7),(5,7)\}$ (d) $R_4 = \{(3,3),(5,5)\}$ .

a. $R_1 = \{(3,2),(5,4),(7,5),(9,5)\}$
b. $R_2 = \{(1,3),(3,5),(5,7)\}$
c. $R_3 = \{(2,3),(2,5),(2,7),(3,5),(3,7),(5,7)\}$
d. $R_4 = \{(3,3),(5,5)\}$

School Level Solution

Sol.

Solution:(a) $R_1$ — every element of $A$ has a unique image. Function. (b) $R_2$ — domain is $\{1, 3, 5\} \neq A$ ; also $7, 9$ have no image. Not a function. (c) $R_3$ — domain $\{2, 3, 5\} \neq A$ and element 2 has three images. Not a function. (d) $R_4$ — $7, 9$ have no image. Not a function.Answer: $\boxed{\text{Only }R_1\text{ is a function}}$ .

Exercise 3.4(2 questions)

Hard

For each of the following relations, find the domain and range and state whether it is a function: (a) $R = \{(5,1),(4,1),(3,1),(2,0)\}$ (b) $R = \{(1,-1),(2,-2),(3,-3),(4,-4),(5,-5)\}$ (c) $R = \{(3,-1),(3,0),(3,1),(3,2)\}$ .

a. $R = \{(5,1),(4,1),(3,1),(2,0)\}$
b. $R = \{(1,-1),(2,-2),(3,-3),(4,-4),(5,-5)\}$
c. $R = \{(3,-1),(3,0),(3,1),(3,2)\}$

School Level Solution

Sol.

Solution:(a) Each first element has a unique image. Domain $= \{2,3,4,5\}$ , Range $= \{0,1\}$ . Function.(b) Domain $= \{1,2,3,4,5\}$ , Range $= \{-1,-2,-3,-4,-5\}$ . Function.(c) Element 3 has four images. Domain $= \{3\}$ , Range $= \{-1,0,1,2\}$ . Not a function.Answer:

$\boxed{\text{(a) Domain }= \{2,3,4,5\}\text{, Range }= \{0,1\}\text{, Function; (b) Domain }= \{1,2,3,4,5\}\text{, Range }= \{-1,-2,-3,-4,-5\}\text{, Function; (c) Domain }= \{3\}\text{, Range }= \{-1,0,1,2\}\text{, Not a function}}$

Hard

For each relation, find the domain and range: (a) $R = \{(x, y) : xy = 8, x, y \in \mathbb{Z}\}$ (b) $R = \{(x, y) : x = |y|, x \in \mathbb{Z}, 0 \leq x \leq 5\}$ (c) $R = \{(x, y) : y = -\sqrt{x}, x \in (0, \infty)\}$ .

a. $xy = 8, x, y \in \mathbb{Z}$
b. $x = |y|, x \in \mathbb{Z}, 0 \leq x \leq 5$
c. $y = -\sqrt{x}, x \in (0, \infty)$

School Level Solution

Sol.

Solution:(a) $xy = 8$ pairs: $(1,8),(2,4),(4,2),(8,1),(-1,-8),(-2,-4),(-4,-2),(-8,-1)$ . Domain $= \{\pm1, \pm2, \pm4, \pm8\} =$ Range.(b) For each $x$ , $y = \pm x$ (except $x = 0$ ). Domain $= \{0,1,2,3,4,5\}$ , Range $= \{-5,-4,\dots,0,\dots,4,5\}$ .(c) $y = -\sqrt{x}$ . Domain $= (0, \infty)$ , Range $= (-\infty, 0)$ .Answer:

$\boxed{\text{(a) Domain }= \{\pm1, \pm2, \pm4, \pm8\}\text{ = Range; (b) Domain }= \{0,1,2,3,4,5\}\text{, Range }= \{-5,-4,\dots,4,5\}\text{; (c) Domain }= (0, \infty)\text{, Range }= (-\infty, 0)}$

Other Chapters

1 2 3 4 5 6

Chapter 3: Relations and Functions

Exercise 3.1(6 questions)

Solve for $x$ and $y$: $(x - 5, y + 1) = (4, 6)$.

Given $A = \{1, 2\}$ and $B = \{2, 3, 5\}$, find $A \times B$ and $B \times A$.

If $n(A \times B) = 20$ and $n(A) = 4$, find $n(B)$.

Given $A = \{1, 2, 3\}$ and $B = \{x, y\}$, find $A \times B$, $B \times A$, $A \times A$ and $B \times B$.

Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 7\}$, find $A \times B$, $B \times A$ and $(A \times B) \cap (B \times A)$.

Given $A = \{1, 2\}$, $B = \{2, 3\}$, $C = \{4, 5\}$, verify that $A \times (B \cup C) = (A \times B) \cup (A \times C)$.

Exercise 3.2(6 questions)

Given $A = \{1, 2, 3\}$, $B = \{4, 5, 6, 7\}$ and $R = \{(a, b) : a + b = 7, a \in A, b \in B\}$, write $R$ and find its domain and range.

Given $A = \{2, 3, 4, 5\}$, $B = \{3, 6, 7, 10\}$ and $R = \{(a, b) : a \text{ divides } b\}$, write $R$ and find its domain and range.

Let $R = \{(a, b) : a + 2b = 12, a, b \in \mathbb{N}\}$. Write $R$ and find its domain and range.

Let $R = \{(x, x^2) : x \text{ is prime and } x < 10\}$. Write $R$ and find its range.

Given $A = \{p, q, r, s\}$ and $B = \{1, 2\}$, how many relations can be formed from $A$ to $B$? Give four examples.

Given $A = \{1, 2, 3, 4, 5\}$ and $R = \{(a, b) : |a - b| = 2\}$, write $R$ and find its domain and range.

Exercise 3.3(2 questions)

Which of the following relations are functions. Justify your answer.

Which of the following relations from $A = \{3, 5, 7, 9\}$ to $B = \{1, 2, 3, 4, 5\}$ are functions? (a) $R_1 = \{(3,2),(5,4),(7,5),(9,5)\}$ (b) $R_2 = \{(1,3),(3,5),(5,7)\}$ (c) $R_3 = \{(2,3),(2,5),(2,7),(3,5),(3,7),(5,7)\}$ (d) $R_4 = \{(3,3),(5,5)\}$.

Exercise 3.4(2 questions)

For each of the following relations, find the domain and range and state whether it is a function: (a) $R = \{(5,1),(4,1),(3,1),(2,0)\}$ (b) $R = \{(1,-1),(2,-2),(3,-3),(4,-4),(5,-5)\}$ (c) $R = \{(3,-1),(3,0),(3,1),(3,2)\}$.

For each relation, find the domain and range: (a) $R = \{(x, y) : xy = 8, x, y \in \mathbb{Z}\}$ (b) $R = \{(x, y) : x = |y|, x \in \mathbb{Z}, 0 \leq x \leq 5\}$ (c) $R = \{(x, y) : y = -\sqrt{x}, x \in (0, \infty)\}$.

Solve for $x$ and $y$ : $(x - 5, y + 1) = (4, 6)$ .

Given $A = \{1, 2\}$ and $B = \{2, 3, 5\}$ , find $A \times B$ and $B \times A$ .

If $n(A \times B) = 20$ and $n(A) = 4$ , find $n(B)$ .

Given $A = \{1, 2, 3\}$ and $B = \{x, y\}$ , find $A \times B$ , $B \times A$ , $A \times A$ and $B \times B$ .

Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 7\}$ , find $A \times B$ , $B \times A$ and $(A \times B) \cap (B \times A)$ .

Given $A = \{1, 2\}$ , $B = \{2, 3\}$ , $C = \{4, 5\}$ , verify that $A \times (B \cup C) = (A \times B) \cup (A \times C)$ .

Given $A = \{1, 2, 3\}$ , $B = \{4, 5, 6, 7\}$ and $R = \{(a, b) : a + b = 7, a \in A, b \in B\}$ , write $R$ and find its domain and range.

Given $A = \{2, 3, 4, 5\}$ , $B = \{3, 6, 7, 10\}$ and $R = \{(a, b) : a \text{ divides } b\}$ , write $R$ and find its domain and range.

Let $R = \{(a, b) : a + 2b = 12, a, b \in \mathbb{N}\}$ . Write $R$ and find its domain and range.

Let $R = \{(x, x^2) : x \text{ is prime and } x < 10\}$ . Write $R$ and find its range.

Given $A = \{p, q, r, s\}$ and $B = \{1, 2\}$ , how many relations can be formed from $A$ to $B$ ? Give four examples.

Given $A = \{1, 2, 3, 4, 5\}$ and $R = \{(a, b) : |a - b| = 2\}$ , write $R$ and find its domain and range.

Which of the following relations from $A = \{3, 5, 7, 9\}$ to $B = \{1, 2, 3, 4, 5\}$ are functions? (a) $R_1 = \{(3,2),(5,4),(7,5),(9,5)\}$ (b) $R_2 = \{(1,3),(3,5),(5,7)\}$ (c) $R_3 = \{(2,3),(2,5),(2,7),(3,5),(3,7),(5,7)\}$ (d) $R_4 = \{(3,3),(5,5)\}$ .

For each of the following relations, find the domain and range and state whether it is a function: (a) $R = \{(5,1),(4,1),(3,1),(2,0)\}$ (b) $R = \{(1,-1),(2,-2),(3,-3),(4,-4),(5,-5)\}$ (c) $R = \{(3,-1),(3,0),(3,1),(3,2)\}$ .

For each relation, find the domain and range: (a) $R = \{(x, y) : xy = 8, x, y \in \mathbb{Z}\}$ (b) $R = \{(x, y) : x = |y|, x \in \mathbb{Z}, 0 \leq x \leq 5\}$ (c) $R = \{(x, y) : y = -\sqrt{x}, x \in (0, \infty)\}$ .