CBSE Class 12 Maths 2026 Question Paper 65/4/2 with Solutions

The following graph represents :

1. (A)$y={\sec }^{-1}x$
2. (B)$y={\cot }^{-1}x$
3. (C)$y={\tan }^{-1}x$
4. (D)$y={\text{cosec}}^{-1}x$

Let $A=[a_{ij}]$ be a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{(2i-j{)}^2}{3}$. Then $A^{\prime }$ is :

1. (A)$\left[\begin{array}{cc}\frac{1}{3}& 3\\ 0& \frac{4}{3}\end{array}\right]$
2. (B)$\left[\begin{array}{cc}\frac{1}{3}& 0\\ 3& \frac{4}{3}\end{array}\right]$
3. (C)$\left[\begin{array}{cc}\frac{4}{3}& 3\\ 0& \frac{1}{3}\end{array}\right]$
4. (D)$\left[\begin{array}{cc}\frac{4}{3}& 0\\ 3& \frac{1}{3}\end{array}\right]$

If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :

1. (A)$\frac{4}{7}$
2. (B)$-\frac{3}{7}$
3. (C)4
4. (D)$-4$

Differential of $e^{e^x}$ with respect to x is :

1. (A)$\log x$
2. (B)$e^{e^x}$
3. (C)$e^x{e}^{e^x}$
4. (D)$(e^x{)}^2$

The principal value of ${\sec }^{-1}(\sqrt{2})+2{\text{cosec}}^{-1}(-\sqrt{2})$ is :

1. (A)$-\frac{\pi }{2}$
2. (B)$-\frac{\pi }{4}$
3. (C)$\frac{\pi }{4}$
4. (D)$\frac{\pi }{2}$

If the distance travelled by a particle in t seconds is given by $S=72t+3t^2-t^3$, then time taken by the particle to come to rest is :

1. (A)4 seconds
2. (B)6 seconds
3. (C)3 seconds
4. (D)0 seconds

$\int \frac{dx}{\sqrt{25-16x^2}}$ is equal to :

1. (A)$\frac{1}{5}{\sin }^{-1}4x+C$
2. (B)$\frac{1}{25}{\sin }^{-1}16x+C$
3. (C)$\frac{1}{4}{\sin }^{-1}\frac{4x}{5}+C$
4. (D)$\frac{1}{16}{\sin }^{-1}\frac{4x}{5}+C$

If ${\int }_0^1\frac{dx}{e^x+e^{-x}}={\tan }^{-1}e+k$, then the value of k is :

1. (A)e
2. (B)$\frac{\pi }{4}$
3. (C)0
4. (D)$-\frac{\pi }{4}$

If $f(x)=\{\begin{array}{ll}\frac{\sin x}{x}+\cos x,& x\ne 0\\ k,& x=0\end{array}$ is continuous at x = 0, then the value of k is :

1. (A)0
2. (B)$-2$
3. (C)$-1$
4. (D)2

The area of the region bounded by the curve y = x and x-axis, between x = 0 and x = 2 is :

1. (A)2 sq. units
2. (B)$\frac{1}{2}$ sq. unit
3. (C)1 sq. unit
4. (D)4 sq. units

The greatest integer function, $f(x)=[x]$, $0<x<3$ is not differentiable at how many points ?

1. (A)At only one point
2. (B)At only two points
3. (C)At no point
4. (D)At three points

The sum of the order and the degree of the differential equation $4{\left[\frac{d^{2}y}{d{x}^2}\right]}^2+3\left[1+{\left(\frac{dy}{dx}\right)}^2-y\right]=0$ is :

1. (A)2
2. (B)3
3. (C)not defined
4. (D)4

If $(3\hat{i}-2\hat{j}+5\hat{k})\times (4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}$, then the values of p and q are :

1. (A)$p=-\frac{2}{3},q=\frac{5}{3}$
2. (B)$p=-\frac{8}{3},q=\frac{20}{3}$
3. (C)$p=\frac{20}{3},q=-\frac{8}{3}$
4. (D)$p=0,q=0$

In the graph, the feasible region representing the Linear Programming Problem for maximising objective function $Z=px+qy$, $p,q>0$ is shaded. If all points on segment AB give max (Z), then which of the following is true ?

1. (A)$p=2q$
2. (B)$p=3q$
3. (C)$q=3p$
4. (D)$q=2p$

Three points A(0, 1, 1), B(2, 0, $-1$) and C(1, 0, 3) form $△ABC$. The ar ($△ABC$) is :

1. (A)$\frac{\sqrt{53}}{2}$ sq. units
2. (B)$\sqrt{53}$ sq. units
3. (C)$\frac{\sqrt{11}}{2}$ sq. units
4. (D)$\sqrt{11}$ sq. units

The region represented by the system of inequations $3x+y\ge 3$, $2x-y\ge -5$, $x,y\ge 0$ is :

1. (A)unbounded in 1${}^{st}$ quadrant
2. (B)bounded in 1${}^{st}$ quadrant
3. (C)unbounded in 2${}^{nd}$ quadrant
4. (D)bounded in 2${}^{nd}$ quadrant

Which of the following is not a Linear Differential Equation ?

1. (A)$(1+x^2)dy+2xydx=\cot xdx$
2. (B)$y+\frac{d}{dx}(xy)=x(\sin x+\log x)$
3. (C)$x(1+y^2)dx-y(1+x^2)dy=0$
4. (D)$ydx-(x+3y^2)dy=0$

The probability that it will rain tomorrow in cities A, B and C is 60%, 70% and 80% respectively. The probability that it will rain tomorrow in at least one of the cities is :

1. (A)$\frac{3}{250}$
2. (B)$\frac{244}{250}$
3. (C)1
4. (D)$\frac{9}{10}$

Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

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

Assertion (A) : A function $f:N\to N$ given by $f(x)=x^3+2,\forall x\in N$ is one-one but not onto.
Reason (R) : Since $\forall y\in N$ (Codomain), there does not exist $x=(y-2{)}^{1/3}$ in N (Domain) such that $f(x)=x^3+2=y$.

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

Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi }{4}$ with x-axis, $\frac{\pi }{3}$ with y-axis and an acute angle $\theta$ with z-axis. Find $\theta$ and the components of $\vec{a}$.

Evaluate :
${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}\left(\sin \left(-\frac{\pi }{2}\right)\right)+{\tan }^{-1}\left(\tan \frac{2\pi }{3}\right)$

Find the interval(s) in which the function $f(x)=\frac{x}{\log x}$, where $x\in (0,1)\cup (1,\infty )$, is increasing.

If $(2\hat{i}+3\hat{j}+\hat{k})$ and $(2\hat{i}+\hat{j}-\hat{k})$ represent the sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ respectively of $△ABC$, find the vector representing the median through A.

Show that the function $f(x)=\{\begin{array}{ll}\frac{\cos x}{-x+\frac{\pi }{2}},& x\ne \frac{\pi }{2}\\ 1,& x=\frac{\pi }{2}\end{array}$ is continuous at $x=\frac{\pi }{2}$.

Find whether the function $f(x)=\{\begin{array}{ll}x-1,& x<2\\ 2x-3,& x\ge 2\end{array}$ at x = 2 is differentiable or not.

In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, then compute $A^2-4A-5I$.

Let $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\to B$ is defined by $f(x)=\left(\frac{x-2}{x-3}\right)$. Find whether f is one-one and onto.

Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y)$ is divisible by $n,x,y\in Z\}$. Determine if R is an equivalence relation.

Solve the following Linear Programming Problem graphically :
Minimize $Z=20x+10y$
subject to
$x+2y\le 40$
$3x+y\ge 30$
$4x+3y\ge 60$
$x,y\ge 0$

If $xy=e^{x-y}$, then find $\frac{dy}{dx}$.

Differentiate ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)$ with respect to ${\cos }^{-1}{x}^2$.

Find a point on the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z-3}{2}$ at a distance of $\sqrt{2}$ units from the point (1, 2, 3).

Find the shortest distance between the lines
$\vec{r}=(4+\lambda )\hat{i}+(2\lambda -1)\hat{j}-3\lambda \hat{k}$
$\vec{r}=(1+2\mu )\hat{i}+(2-5\mu )\hat{k}+(4\mu -1)\hat{j}$.

Solve the differential equation $y{e}^{y}dx=(y^3+2x{e}^y)dy$, when $y(0)=1$.

Find the general solution of the differential equation
$(x^3-3x{y}^2)dx=(y^3-3x^{2}y)dy$.

Using integration, find the area of the region bounded by the curve $y=x|x|$, x-axis, x = $-2$ and x = 2.

Find :
$\int \frac{x}{(x-1)(x^2+4)}dx$

Evaluate :
${\int }_0^1\frac{x{\tan }^{-1}x}{(1+x^2{)}^{3/2}}dx$

Find the vector and cartesian equations of the line passing through the point of intersection of the lines $\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})$ and parallel to the line $\frac{x-1}{-2}=\frac{7-y}{-3}=z$.