Class 12 Mathematics 2025 Board Exam

Assertion (A): Let $\mathbb{Z}$ be the set of integers. A function $f:\mathbb{Z}\to \mathbb{Z}$ defined as $f(x)=3x-5$, $\forall x\in \mathbb{Z}$ is a bijective.

Reason (R): A function is a bijective if it is both surjective and injective.

(A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.

If $A$ denotes the set of continuous functions and $B$ denotes set of differentiable functions, then which of the following depicts the correct relation between set $A$ and $B$ ?

(A) Venn Diagram $A$ inside $B$
(B) Venn Diagram $B$ inside $A$
(C) Venn Diagram $A\cap B$
(D) Venn Diagram Disjoint

Let $R$ be a relation defined over $\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, defined as '$mRn$ if and only if $m$ is a multiple of $n$, $m,n\in \mathbb{N}$.' Find whether $R$ is reflexive, symmetric and transitive or not.

Prove that $f:\mathbb{N}\to \mathbb{N}$ defined as $f(x)=ax+b$ $(a,b\in \mathbb{N})$ is one-one but not onto.

Let $R$ be a relation on set of real numbers $\mathbb{R}$ defined as $\{(x,y):x-y+\sqrt{3}$ is an irrational number, $x,y\in \mathbb{R}\}$. Verify $R$ for reflexivity, symmetry and transitivity.

A class-room teacher is keen to assess the learning of her students the concept of 'relations' taught to them. She writes the following five relations each defined on the set $A=\{1,2,3\}$ :

$R_1=\{(2,3),(3,2)\}$

$R_2=\{(1,2),(1,3),(3,2)\}$

$R_3=\{(1,2),(2,1),(1,1)\}$

$R_4=\{(1,1),(1,2),(3,3),(2,2)\}$

$R_5=\{(1,1),(1,2),(3,3),(2,2),(2,1),(2,3),(3,2)\}$

The students are asked to answer the following questions about the above relations :

(i) Identify the relation which is reflexive, transitive but not symmetric.

(ii) Identify the relation which is reflexive and symmetric but not transitive.

(iii) (a) Identify the relations which are symmetric but neither reflexive nor transitive.

OR

(iii) (b) What pairs should be added to the relation $R_2$ to make it an equivalence relation ?

A school is organizing a debate competition with participants as speakers $S=\{S_1,S_2,S_3,S_4\}$ and these are judged by judges $J=\{J_1,J_2,J_3\}$. Each speaker can be assigned one judge. Let $R$ be a relation from set $S$ to $J$ defined as $R=\{(x,y):\text{speaker }x\text{ is judged by judge }y,x\in S,y\in J\}$.

Based on the above, answer the following :

(i) How many relations can be there from $S$ to $J$ ?

(ii) A student identifies a function from $S$ to $J$ as $f=\{(S_1,J_1),(S_2,J_2),(S_3,J_2),(S_4,J_3)\}$. Check if it is bijective.

(iii) (a) How many one-one functions can be there from set $S$ to set $J$ ?

OR

(iii) (b) Another student considers a relation $R_1=\{(S_1,S_2),(S_2,S_4)\}$ in set $S$. Write minimum ordered pairs to be included in $R_1$ so that $R_1$ is reflexive but not symmetric.

The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?

(A) Graph A
(B) Graph B
(C) Graph C
(D) Graph D

(This question references a diagram.)

If $y={\sin }^{-1}x$, $-1\le x\le 0$, then the range of $y$ is

(A) $\left[-\frac{\pi }{2},0\right]$
(B) $\left[\frac{\pi }{2},0\right]$
(C) $\left[-\frac{\pi }{2},0\right)$
(D) $\left[-\frac{\pi }{2},0\right]$

The principal value of ${\sin }^{-1}\left(\cos \frac{43\pi }{5}\right)$ is

(A) $-\frac{7\pi }{5}$
(B) $-\frac{\pi }{10}$
(C) $\frac{\pi }{10}$
(D) $\frac{3\pi }{5}$