CBSE Class 12 Mathematics Set 2- (65/4/2) Question Paper 2026 with Solution PDF

The following graph represents :

• (A) \(y = \cos^{-1} x\)
  
• (B) \(y = \sec^{-1} x\)
  
• (C) \(y = \tan^{-1} x\)
  
• (D) \(y = \text{cosec}^{-1} x\)

Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

• (A) \(\begin{bmatrix} \frac{1}{3} & 3 \\ 0 & \frac{4}{3} \end{bmatrix}\)
  
• (B) \(\begin{bmatrix} \frac{1}{3} & 0 \\ 3 & \frac{4}{3} \end{bmatrix}\)
  
• (C) \(\begin{bmatrix} \frac{4}{3} & 3 \\ 0 & \frac{1}{3} \end{bmatrix}\)
  
• (D) \(\begin{bmatrix} \frac{4}{3} & 0 \\ 3 & \frac{1}{3} \end{bmatrix}\)

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

• (A) \(\frac{4}{7}\)
  
• (B) \(-\frac{3}{7}\)
  
• (C) \(4\)
  
• (D) \(-4\)

Differential of \(e^{e^x}\) with respect to x is :

• (A) \(\log x\)
  
• (B) \(e^{e^x}\)
  
• (C) \(e^x e^{e^x}\)
  
• (D) \((e^x)^2\)

The principal value of \[ \sec^{-1}(\sqrt{2})+2\operatorname{cosec}^{-1}(-\sqrt{2}) \] is: 

• (A) \(-\frac{\pi}{2}\)
  
• (B) \(-\frac{\pi}{4}\)
  
• (C) \(\frac{\pi}{4}\)
  
• (D) \(\frac{\pi}{2}\)

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

• (A) 4 seconds
  
• (B) 6 seconds
  
• (C) 3 seconds
  
• (D) 0 seconds

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

• (A) \(\frac{1}{5} \sin^{-1} 4x + C\)
  
• (B) \(\frac{1}{25} \sin^{-1} 16x + C\)
  
• (C) \(\frac{1}{4} \sin^{-1} \frac{4x}{5} + C\)
  
• (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 :

• (A) \(e\)
  
• (B) \(\frac{\pi}{4}\)
  
• (C) \(0\)
  
• (D) \(-\frac{\pi}{4}\)

If \[ f(x)= \begin{cases} \dfrac{\sin x}{x}+\cos x, & x\neq 0,\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then the value of \(k\) is: 

• (A) \(0\)
  
• (B) \(-2\)
  
• (C) \(-1\)
  
• (D) \(2\)

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

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

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

• (A) At only one point
  
• (B) At only two points
  
• (C) At no point
  
• (D) At three points

The sum of the order and the degree of the differential equation

\[ 4\left(\dfrac{d^2y}{dx^2}\right)^2 + 3\left[1 + \left(\dfrac{dy}{dx}\right)^2 - y\right] = 0 \]

is :

• (A) 2
• (B) 3
• (C) not defined
• (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: 

• (A) \(p = -\frac{2}{3}, q = \frac{5}{3}\)
  
• (B) \(p = -\frac{8}{3}, q = \frac{20}{3}\)
  
• (C) \(p = \frac{20}{3}, q = -\frac{8}{3}\)
  
• (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 ?

• (A) \(p = 2q\)
  
• (B) \(p = 3q\)
  
• (C) \(q = 3p\)
  
• (D) \(q = 2p\)

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

• (A) \(\frac{\sqrt{53}}{2} \text{ sq. units}\)
  
• (B) \(\sqrt{53} \text{ sq. units}\)
  
• (C) \(\frac{\sqrt{11}}{2} \text{ sq. units}\)
  
• (D) \(\sqrt{11} \text{ sq. units}\)

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

• (A) unbounded in \(1^{st}\) quadrant
  
• (B) bounded in \(1^{st}\) quadrant
  
• (C) unbounded in \(2^{nd}\) quadrant
  
• (D) bounded in \(2^{nd}\) quadrant

Which of the following is not a Linear Differential Equation ?

• (A) \((1 + x^2) dy + 2xy dx = \cot x dx\)
  
• (B) \(y + \frac{d}{dx}(xy) = x (\sin x + \log x)\)
  
• (C) \(x (1 + y^2) dx - y (1 + x^2) dy = 0\)
  
• (D) \(y dx - (x + 3y^2) dy = 0\)

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:

• (A) \(\frac{3}{250}\)
  
• (B) \(\frac{244}{250}\)
  
• (C) \(1\)
  
• (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.

• (A) Both Assertion (A) and Reason (R) are true and 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.

Assertion (A): A function \[ f:\mathbb{N}\to\mathbb{N} \] given by \[ f(x)=x^3+2,\quad \forall x\in\mathbb{N} \] is one-one but not onto. 

Reason (R): Since, for every \(y\in\mathbb{N}\) (codomain), there does not exist \[ x=(y-2)^{1/3}\in\mathbb{N} \] (domain) such that \[ f(x)=x^3+2=y. \] 

• (A) Both Assertion (A) and Reason (R) are true and 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.

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 \(\vec{AB}\) and \(\vec{AC}\) respectively of \(\Delta ABC\), find the vector representing the median through A.

Show that the function \[ f(x)= \begin{cases} \dfrac{\cos x}{\frac{\pi}{2}-x}, & x\neq \frac{\pi}{2},\\ 1, & x=\frac{\pi}{2} \end{cases} \] is continuous at \[ x=\frac{\pi}{2}. \] 

Show that the function \(f(x) = \begin{cases} \frac{\cos x}{\frac{\pi}{2} - x}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}\) is continuous at \(x = \frac{\pi}{2}\).

Find whether the function \(f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \ge 2 \end{cases}\) 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 \(A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}\), 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 \[ f(x)=\frac{x-2}{x-3}. \] Find whether \(f\) is one-one and onto. 

Let \(n\) be a fixed positive integer. A relation \(R\) is defined on the set \(\mathbb{Z}\) by \[ R=\{(x,y):(x-y)\text{ is divisible by }n,\; x,y\in\mathbb{Z}\}. \] Determine whether \(R\) is an equivalence relation. 

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

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

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 \( (x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy \).

Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) 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 \( \frac{x - 1}{-2} = \frac{7 - y}{-3} = z \).

An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box \(i\) is selected and a cap picked out is \(i/6\), where \(i = 1, 2, 3\).

At a birthday party, children are being served orange juice in conical cups. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0.1 cm\(^3\)/s.

A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.