Samacheer Kalvi 10th Maths Exercise 1.2 Solutions

10th Maths Exercise 1.2 solutions (Chapter 1 Relations and Functions) with step-by-step answers for all 9 questions, as per the 2026-27 Samacheer Kalvi textbook.

Answers at a glance8 answers · full steps below

Q1 (i) $R_1$ is not a relation from $A$ to $B$
Q1 (ii) $R_2$ is not a relation from $A$ to $B$
Q1 (iii) $R_3$ is a relation from $A$ to $B$
Q1 (iv) $R_4$ is not a relation from $A$ to $B$
Q2 $R=\{(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)\}$ ; Domain $=\{1,2,3,4,5,6\}$ , Range $=\{1,4,9,16,25,36\}$
Q3Domain $=\{0,1,2,3,4,5\}$ , Range $=\{3,4,5,6,7,8\}$
Q4 (i) $R=\{(2,1),(4,2)\}$ (arrow diagram: $2\to1,\;4\to2$ ; graph: points $(2,1),(4,2)$ )
Q4 (ii) $R=\{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)\}$

Q1. Let $A = \{1,2,3,7\}$ and $B = \{3,0,-1,7\}$ , which of the following are relation from …

Let $A = \{1,2,3,7\}$ and $B = \{3,0,-1,7\}$ , which of the following are relation from $A$ to $B$ ?

(i)

$R_1 = \{(2,1), (7,1)\}$

$A\times B$ contains pairs $(x,y)$ with $x\in A$ and $y\in B$ .
Here the second element $1\notin B$ .
So $(2,1)\notin A\times B$ , i.e. $R_1\not\subseteq A\times B$ .

Answer

$R_1$ is not a relation from $A$ to $B$

(ii)

$R_2 = \{(-1,1)\}$

In $(-1,1)$ , the first element $-1\notin A$ and the second element $1\notin B$ .
So $(-1,1)\notin A\times B$ .

Answer

$R_2$ is not a relation from $A$ to $B$

(iii)

$R_3 = \{(2,-1), (7,7), (1,3)\}$

Check each pair: $2\in A,\,-1\in B$ ; $7\in A,\,7\in B$ ; $1\in A,\,3\in B$ .
All the pairs belong to $A\times B$ , so $R_3\subseteq A\times B$ .

Answer

$R_3$ is a relation from $A$ to $B$

(iv)

$R_4 = \{(7,-1), (0,3), (3,3), (0,7)\}$

In $(0,3)$ and $(0,7)$ , the first element $0\notin A$ .
So these pairs are not in $A\times B$ .

Answer

$R_4$ is not a relation from $A$ to $B$

Q2. Let $A = \{1,2,3,4,...,45\}$ and $R$ be the relation defined as “square is of …

Let $A = \{1,2,3,4,...,45\}$ and $R$ be the relation defined as “square is of a number” on $A$ . Write R as a subset of $A \times A$ . Also, find the domain and range of R.

The relation is $x\,\mathrm{R}\,y$ if $y=x^{2}$ , with $x,y\in A=\{1,2,3,\dots,45\}$ .
Since $y\le 45$ , we need $x^{2}\le 45$ , so $x=1,2,3,4,5,6$ (as $7^{2}=49>45$ ).
$R=\{(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)\}\subseteq A\times A$ .

Answer

$R=\{(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)\}$ ; Domain $=\{1,2,3,4,5,6\}$ , Range $=\{1,4,9,16,25,36\}$

Q3. A Relation R is given by the set $\{(x,y) \,/\, y = x + 3,\ x \in \{0,1,2,3,4,5\}\}$ . Determine its …

A Relation R is given by the set $\{(x,y) \,/\, y = x + 3,\ x \in \{0,1,2,3,4,5\}\}$ . Determine its domain and range.

Put each value of $x$ in $y=x+3$ .
$x=0\Rightarrow y=3$ , $x=1\Rightarrow y=4$ , $x=2\Rightarrow y=5$ , $x=3\Rightarrow y=6$ , $x=4\Rightarrow y=7$ , $x=5\Rightarrow y=8$ .
$R=\{(0,3),(1,4),(2,5),(3,6),(4,7),(5,8)\}$ .

Answer

Domain $=\{0,1,2,3,4,5\}$ , Range $=\{3,4,5,6,7,8\}$

Q4. Represent each of the given relations by (a) an arrow diagram, (b) …

Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possible.

(i)

$\{(x,y) \mid x = 2y,\ x \in \{2,3,4,5\},\ y \in \{1,2,3,4\}\}$

Show steps (6)

$x=2y$ means $y=\dfrac{x}{2}$ , so $x$ must be even.
From $x\in\{2,3,4,5\}$ : $x=2\Rightarrow y=1$ and $x=4\Rightarrow y=2$ (both $y$ are in $\{1,2,3,4\}$ ).
$x=3,5$ give $y=1.5,\,2.5$ which are not in the set.
(a) Arrow diagram: from set $\{2,3,4,5\}$ to $\{1,2,3,4\}$ , draw arrows $2\to 1$ and $4\to 2$ only.
(b) Graph: plot the points $(2,1)$ and $(4,2)$ on the $xy$ -plane.
(c) Roster form: $R=\{(2,1),(4,2)\}$ .

Answer

$R=\{(2,1),(4,2)\}$ (arrow diagram: $2\to1,\;4\to2$ ; graph: points $(2,1),(4,2)$ )

(ii)

$\{(x,y) \mid y = x+3,\ x,\ y \text{ are natural numbers} < 10\}$

Show steps (6)

Natural numbers less than $10$ are $1,2,\dots,9$ .
$y=x+3$ and $y<10$ gives $x+3\le 9$ , so $x=1,2,3,4,5,6$ .
$x=1\Rightarrow y=4$ , $x=2\Rightarrow y=5$ , $x=3\Rightarrow y=6$ , $x=4\Rightarrow y=7$ , $x=5\Rightarrow y=8$ , $x=6\Rightarrow y=9$ .
(a) Arrow diagram: arrows $1\to4,\;2\to5,\;3\to6,\;4\to7,\;5\to8,\;6\to9$ .
(b) Graph: plot the points $(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)$ ; they lie on a straight line.
(c) Roster form: $R=\{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)\}$ .

Answer

$R=\{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)\}$

Q5. A company has four categories of employees given by Assistants ( $A$ ), Clerks …

A company has four categories of employees given by Assistants ( $A$ ), Clerks ( $C$ ), Managers ( $M$ ) and an Executive Officer ( $E$ ). The company provide ₹10,000, ₹25,000, ₹50,000 and ₹1,00,000 as salaries to the people who work in the categories $A$ , $C$ , $M$ and $E$ respectively. If $A_1$ , $A_2$ , $A_3$ , $A_4$ and $A_5$ were Assistants; $C_1$ , $C_2$ , $C_3$ , $C_4$ were Clerks; $M_1$ , $M_2$ , $M_3$ were managers and $E_1$ , $E_2$ were Executive officers and if the relation R is defined by $x\mathrm{R}y$ , where $x$ is the salary given to person $y$ , express the relation R through an ordered pair and an arrow diagram.

Show steps (6)

$x$ is the salary and $y$ is the person getting that salary, so the pairs are (salary, person).
Assistants get ₹10,000: $(10000,A_1),(10000,A_2),(10000,A_3),(10000,A_4),(10000,A_5)$ .
Clerks get ₹25,000: $(25000,C_1),(25000,C_2),(25000,C_3),(25000,C_4)$ .
Managers get ₹50,000: $(50000,M_1),(50000,M_2),(50000,M_3)$ .
Executive officers get ₹1,00,000: $(100000,E_1),(100000,E_2)$ .
Arrow diagram: left set $\{10000,25000,50000,100000\}$ , right set of all $14$ persons; from $10000$ draw $5$ arrows to $A_1,\dots,A_5$ , from $25000$ four arrows to $C_1,\dots,C_4$ , from $50000$ three arrows to $M_1,M_2,M_3$ and from $100000$ two arrows to $E_1,E_2$ .

Answer

$R=\{(10000,A_1),(10000,A_2),(10000,A_3),(10000,A_4),(10000,A_5),(25000,C_1),(25000,C_2),(25000,C_3),(25000,C_4),(50000,M_1),(50000,M_2),(50000,M_3),(100000,E_1),(100000,E_2)\}$

Quick check

Let $f$ and $g$ be two functions given by $f = \{(0,1),(2,0),(3,-4),(4,2),(5,7)\}$ , $g = \{(0,2),(1,0),(2,4),(-4,2),(7,0)\}$ then the range of $f \circ g$ is
1. (1) $\{0,2,3,4,5\}$
2. (2) $\{-4,1,0,2,7\}$
3. (3) $\{1,2,3,4,5\}$
4. (4) $\{0,1,2\}$
Show answer
Answer: (4) $\{0,1,2\}$
1. $(f\circ g)(0)=f(2)=0$ , $(f\circ g)(1)=f(0)=1$ .
2. $(f\circ g)(2)=f(4)=2$ , $(f\circ g)(-4)=f(2)=0$ , $(f\circ g)(7)=f(0)=1$ .
3. Range $=\{0,1,2\}$ .
Question 1 of 5
If there are 1024 relations from a set $A = \{1, 2, 3, 4, 5\}$ to a set $B$ , then the number of elements in $B$ is
1. (1) 3
2. (2) 2
3. (3) 4
4. (4) 8
Show answer
Answer: (2) 2
1. Number of relations from $A$ to $B$ is $2^{n(A)\cdot n(B)}$ .
2. $2^{5\,n(B)}=1024=2^{10}$ .
3. $5\,n(B)=10\Rightarrow n(B)=2$ .
Question 2 of 5
Let $A = \{1,2,3,4\}$ and $B = \{4,8,9,10\}$ . A function $f : A \to B$ given by $f = \{(1,4),(2,8),(3,9),(4,10)\}$ is a
1. (1) Many-one function
2. (2) Identity function
3. (3) One-to-one function
4. (4) Into function
Show answer
Answer: (3) One-to-one function
1. Images are $4,8,9,10$ — all different.
2. Different elements of $A$ have different images.
3. So $f$ is one-to-one.
Question 3 of 5
If $f(x) = 2x^2$ and $g(x) = \dfrac{1}{3x}$ , then $f \circ g$ is
1. (1) $\dfrac{3}{2x^2}$
2. (2) $\dfrac{2}{3x^2}$
3. (3) $\dfrac{2}{9x^2}$
4. (4) $\dfrac{1}{6x^2}$
Show answer
Answer: (3) $\dfrac{2}{9x^2}$
1. $(f\circ g)(x)=f\big(g(x)\big)=f\left(\dfrac{1}{3x}\right)$ .
2. $=2\left(\dfrac{1}{3x}\right)^{2}$ .
3. $=\dfrac{2}{9x^{2}}$ .
Question 4 of 5
$A = \{a,b,p\}$ , $B = \{2,3\}$ , $C = \{p,q,r,s\}$ then $n[(A \cup C) \times B]$ is
1. (1) 8
2. (2) 20
3. (3) 12
4. (4) 16
Show answer
Answer: (3) 12
1. $A\cup C=\{a,b,p,q,r,s\}$ , so $n(A\cup C)=6$ .
2. $n(B)=2$ .
3. $n[(A\cup C)\times B]=6\times 2=12$ .
Question 5 of 5

Chapter 1 formulas

Cartesian product of A with B
$A \times B = \{(a,b) \mid \text{for all } a \in A,\ b \in B\}$
A relation R from A to B is a subset of A × B
$R \subseteq A \times B$
Condition for a relation R from X to Y to be a function
$\text{for every } x \in X \text{ there exists only one } y \in Y$
Identity function
$f(x) = x$
Reciprocal function
$f(x) = \dfrac{1}{x}$
Constant function
$f(x) = c$
Linear function
$f(x) = ax + b,\ a \neq 0$
Quadratic function
$f(x) = ax^{2} + bx + c,\ a \neq 0$
Cubic function
$f(x) = ax^{3} + bx^{2} + cx + d,\ a \neq 0$
Composition of two functions f : A → B and g : B → C
$g \circ f : A \to C,\quad (g \circ f)(x) = g(f(x)) \ \text{ for all } x \in A$
Composition is not commutative in general
$f \circ g \neq g \circ f$
Composition is associative
$f \circ (g \circ h) = (f \circ g) \circ h$

Answers checked against the Samacheer Kalvi textbook · Updated 28 September 2026

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Q1. Let A={1,2,3,7}A = \{1,2,3,7\}A={1,2,3,7} and B={3,0,−1,7}B = \{3,0,-1,7\}B={3,0,−1,7} , which of the following are relation from …

(i)

(ii)

(iii)

(iv)

Q2. Let A={1,2,3,4,...,45}A = \{1,2,3,4,...,45\}A={1,2,3,4,...,45} and RRR be the relation defined as “square is of …

Q3. A Relation R is given by the set {(x,y) / y=x+3, x∈{0,1,2,3,4,5}}\{(x,y) \,/\, y = x + 3,\ x \in \{0,1,2,3,4,5\}\}{(x,y)/y=x+3, x∈{0,1,2,3,4,5}} . Determine its …

Q4. Represent each of the given relations by (a) an arrow diagram, (b) …

(i)

(ii)

Q5. A company has four categories of employees given by Assistants (AAA), Clerks …

Quick check

Q1. Let $A = \{1,2,3,7\}$ and $B = \{3,0,-1,7\}$ , which of the following are relation from …

Q2. Let $A = \{1,2,3,4,...,45\}$ and $R$ be the relation defined as “square is of …

Q3. A Relation R is given by the set $\{(x,y) \,/\, y = x + 3,\ x \in \{0,1,2,3,4,5\}\}$ . Determine its …

Q5. A company has four categories of employees given by Assistants ( $A$ ), Clerks …