CBSE Class 12 Maths 2026 Question Paper 65/4/3 with Solutions

The following graph represents :

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

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

1. (A)$\left[\begin{array}{cc}12& \frac{25}{3}\\ \frac{16}{3}& 3\end{array}\right]$
2. (B)$\left[\begin{array}{cc}12& \frac{16}{3}\\ \frac{25}{3}& 3\end{array}\right]$
3. (C)$\left[\begin{array}{cc}3& \frac{25}{3}\\ \frac{16}{3}& 12\end{array}\right]$
4. (D)$\left[\begin{array}{cc}3& \frac{16}{3}\\ \frac{25}{3}& 12\end{array}\right]$

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 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

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

A cylindrical tank is being filled with sand at a rate of 314 m${}^3$/h. If the radius of the tank is 10 m, then the height of sand in the tank increases at the rate of :

1. (A)1.1 m/h
2. (B)1 m/h
3. (C)$\pi$ m/h
4. (D)$\frac{\pi }{2}$ m/h

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)$(e^x{)}^2$

$\int \frac{dx}{\sqrt{25-16x^2}}$ 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 sum of the order and the degree of the differential equation $y=x\frac{dy}{dx}+3\left[2-{\left(\frac{dy}{dx}\right)}^2\right]$ is :

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

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

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)$p=0,q=0$

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

The probability that a particular item is available in three shops A, B and C is $\frac{4}{5}$, $\frac{3}{4}$ and $\frac{2}{3}$ respectively. If a person visits all the three shops to buy the item, then what is the probability that it will be available in at least one of the shops ?

1. (A)$\frac{59}{60}$
2. (B)1
3. (C)$\frac{1}{60}$
4. (D)$\frac{9}{60}$

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.

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.

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 $\vec{a}$.

Find the interval(s) for which the function $f(x)=\frac{x}{3}+\frac{3}{x}$, $x\ne 0$ is increasing.

If in a parallelogram PQRS, $\overrightarrow{PQ}=2\hat{i}+3\hat{j}+\hat{k}$ and $\overrightarrow{QR}=\hat{i}-2\hat{j}+\hat{k}$, then find the unit vectors parallel to the diagonals $\overrightarrow{PR}$ and $\overrightarrow{SQ}$.

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

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

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.

Solve the following Linear Programming Problem graphically :
Maximize $Z=20x+10y$
subject to constraints
$x+2y\le 28$
$3x+y\le 24$
$x\ge 2$
$x,y\ge 0$

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 $xy=e^{x-y}$, 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 :
$\int \frac{x}{(x-1)(x^2+4)}dx$

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

Using integration, find the area of the region bounded by $y=5x+4$, $y=0$, $x=-1$ and $x=1$.

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 length of the perpendicular drawn from the point P(1, 2, 3) to the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Also, find the equation of the perpendicular line joining P and the foot of the perpendicular.