CBSE Class 12 Maths 2026 Question Paper 65/4/1 with Solutions

The following graph represents :

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

If $A=[a_{ij}]$ is a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{|i-3j|}{2}$, then $A^{\prime }$ is :

1. (A)$\left[\begin{array}{cc}1& \frac{5}{2}\\ \frac{1}{2}& 2\end{array}\right]$
2. (B)$\left[\begin{array}{cc}1& \frac{1}{2}\\ \frac{5}{2}& 2\end{array}\right]$
3. (C)$\left[\begin{array}{cc}2& \frac{5}{2}\\ \frac{1}{2}& 1\end{array}\right]$
4. (D)$\left[\begin{array}{cc}2& \frac{1}{2}\\ \frac{5}{2}& 1\end{array}\right]$

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 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 surface area of a sphere when its volume changes at the same rate as its radius is :

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

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 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

$\int \frac{dx}{\sqrt{25-16x^2}}$ 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)$-\frac{\pi }{4}$

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

Product of the order and degree of differential equation $1+{\left(\frac{dy}{dx}\right)}^3=\lambda {\left(\frac{d^{3}y}{d{x}^3}\right)}^2$ is :

1. (A)5
2. (B)6
3. (C)2
4. (D)3

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 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

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)$q=2p$

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)$p=0,q=0$

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

A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?

1. (A)$\frac{27}{125}$
2. (B)$\frac{8}{125}$
3. (C)$\frac{2}{125}$
4. (D)$\frac{98}{125}$

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.

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

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.

Find the values of x for which $f(x)=x^x$, $x>0$ is increasing.

If the position vectors of three points A, B and C are $3\hat{i}+\hat{j}$, $5\hat{i}+6\hat{j}-3\hat{k}$ and $4\hat{j}$ respectively, then show that they form an isosceles triangle.

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 $\vec{a}$.

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.

If $A=\left[\begin{array}{ccc}3& 2& 0\\ 1& 4& 0\\ 0& 0& 5\end{array}\right]$, then compute $A^2-7A+10I$.

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 ${\cos }^{-1}{x}^2$.

Solve the following Linear Programming Problem graphically :
Maximise $Z=200x+120y$
subject to the constraints
$x+y\le 300$
$3x+y\le 600$
$x-y\ge -100$
$x,y\ge 0$

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
$\vec{r}=(4+\lambda )\hat{i}+(2\lambda -1)\hat{j}-3\lambda \hat{k}$
$\vec{r}=(1+2\mu )\hat{i}+(2-5\mu )\hat{k}+(4\mu -1)\hat{j}$.

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.

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

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

Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between x = 4 and x = 8.

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
$(x^3-3x{y}^2)dx=(y^3-3x^{2}y)dy$.

Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ and is parallel to the vector $3\hat{i}+2\hat{j}-8\hat{k}$.