CBSE Class 12 Mathematics Compartment Question Paper 2025 (Available):Download Solution with answer Key

The domain of the function \( f(x) = \cos^{-1}(2x) \) is:

• (A) \([-1, 1]\)
• (B) \(\left[0, \frac{1}{2}\right]\)
• (C) \([-2, 2]\)
• (D) \(\left[-\frac{1}{2}, \frac{1}{2}\right]\)

If \[ \begin{vmatrix} 2x & 3
x & -8
\end{vmatrix} = 0, \textbf{ then the value of } \(x\) \textbf{ is:} \]

• (A) zero
• (B) 3
• (C) \(2\sqrt{3}\)
• (D) \(\pm 2\sqrt{3}\)

For a non-singular matrix \(X\), if \(X^2 = I\), then \(X^{-1}\) is equal to:

• (A) \(X\)
• (B) \(X^2\)
• (C) \(I\)
• (D) \(O\)

The area of a triangle with vertices \((3, 0), (0, k), (-3, 0)\) is \(9\) sq units. The value of \(k\) is:

• (A) \(9\)
• (B) \(-9\)
• (C) \(3\)
• (D) \(6\)

The value of the determinant \[ \begin{vmatrix} \cos 75^\circ & \sin 75^\circ
\sin 15^\circ & \cos 15^\circ \end{vmatrix} \]
is:

• (A) \(1\)
• (B) zero
• (C) \(\dfrac{1}{2}\)
• (D) \(\dfrac{\sqrt{3}}{2}\)

The derivative of \(\sin^{-1}(2x^2 - 1)\) with respect to \(\sin^{-1}x\) is:

• (A) \(\dfrac{2}{x}\)
• (B) \(2\)
• (C) \(\dfrac{\sqrt{1 - x^2}}{\sqrt{1 - 4x^2}}\)
• (D) \(1 - x^2\)

If \(A\) is an identity matrix of order \(n\), then \(A (Adj A)\) is a/an:

• (A) identity matrix
• (B) row matrix
• (C) zero matrix
• (D) skew symmetric matrix

The area bounded by the curve \(x = y^2\), the y-axis, and the lines \(y = 3\) and \(y = 4\) is:

• (A) \(\dfrac{74}{3} sq units\)
• (B) \(\dfrac{37}{3} sq units\)
• (C) \(74 sq units\)
• (D) \(37 sq units\)

In an LPP, corner points of the feasible region determined by the system of linear constraints are \((1,1), (3,0), (0,3)\).
If \(Z = ax + by\), where \(a > 0, b > 0\) is to be minimized, the condition on \(a\) and \(b\) so that the minimum of \(Z\) occurs at \((3, 0)\) and \((1, 1)\) will be:

• (A) \(a = 2b\)
• (B) \(a = \dfrac{b}{2}\)
• (C) \(a = 3b\)
• (D) \(a = b\)

If \(\dfrac{d}{dx}f(x) = 3x^2 - \dfrac{3}{x^4}\), and \(f(1) = 0\), then \(f(x)\) is:

• (A) \(6x + \dfrac{12}{x^5}\)
• (B) \(x^4 - \dfrac{1}{x^3} + 2\)
• (C) \(x^3 + \dfrac{1}{x^3} - 2\)
• (D) \(x^3 + \dfrac{1}{x^3} + 2\)

The maximum value of \(Z = 3x + 4y\) subject to the constraints \(x + y \leq 1\), \(x \geq 0\), \(y \geq 0\) is:

• (A) 3
• (B) 4
• (C) 7
• (D) 0

\[ \int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \, dx is equal to: \]

• (A) \(\sec \sqrt{x} + C\)
• (B) \(2\sqrt{x} \tan x - x + C\)
• (C) \(2\left( \tan \sqrt{x} - \sqrt{x} \right) + C\)
• (D) \(2 \tan \sqrt{x} - x + C\)

A coin is tossed three times. The probability of getting at least two heads is:

• (A) \(\frac{1}{2}\)
• (B) \(\frac{3}{8}\)
• (C) \(\frac{1}{8}\)
• (D) \(\frac{1}{4}\)

If \( |\vec{a}|^2 = 1 \), \( |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 2 \), then the value of \( |\vec{a} + \vec{b}| \) is:

• (A) \( 9 \)
• (B) \( 3 \)
• (C) \( -3 \)
• (D) \( 2 \)

If the rate of change of volume of a sphere is twice the rate of change of its radius, then the surface area of the sphere is:

• (A) 1 sq unit
• (B) 2 sq units
• (C) 3 sq units
• (D) 4 sq units

The general solution of the differential equation \( \frac{dy}{dx} = 2x \cdot e^{x^2 + y} \) is:

• (A) \( e^{x^2 + y} = C \)
• (B) \( e^{x^2} + e^{-y} = C \)
• (C) \( e^{x^2} = e^y + C \)
• (D) \( e^{x^2 - y} = C \)

If \( m' \) and \( n' \) are the degree and order respectively of the differential equation \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y}{dx^2} \), then the value of \( (m + n) \) is:

• (A) \( 4 \)
• (B) \( 3 \)
• (C) \( 2 \)
• (D) \( 5 \)

Two vectors \( \vec{a} \) and \( \vec{b} \) are such that \( |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \). The angle between the two vectors is:

• (A) \( 30^\circ \)
• (B) \( 60^\circ \)
• (C) \( 45^\circ \)
• (D) \( 90^\circ \)

Consider the function \( f : \mathbb{R} \rightarrow \mathbb{R} \), defined as \( f(x) = x^3 \).
Assertion (A): \( f(x) \) is a one-one function.
Reason (R): \( f(x) \) is a one-one function, if co-domain = range.

• (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): \( f(x) = [x] \), \( x \in \mathbb{R} \), the greatest integer function is not differentiable at \( x = 2 \).
Reason (R): The greatest integer function is not continuous at any integral value.

• (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.

For the curve \( \sqrt{x} + \sqrt{y} = 1 \), find the value of \( \frac{dy}{dx} \) at the point \( \left( \frac{1}{9}, \frac{1}{9} \right) \).

(a) Find the principal value of \[ \cos^{-1} \left(-\frac{1}{2}\right) + 2\sin^{-1} \left(\frac{1}{2}\right) \]

(b) Prove that: \[ \tan^{-1}(\sqrt{x}) = \frac{1}{2} \cos^{-1}\left( \frac{1 - x}{1 + x} \right), \quad x \in [0, 1] \]

\quad (a) Find the value of \( \lambda \), if the points \( A(-1, -1, 2),\ B(2, 8, \lambda),\ C(3, 11, 6) \) are collinear.

(b)\quad \( \vec{a} \) and \( \vec{b} \) are two co-initial vectors forming the adjacent sides of a parallelogram such that: \[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \]
Find the area of the parallelogram.

A ladder 13 m long is leaning against the wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 2 m/s.
How fast is the height on the wall decreasing when the foot of the ladder is 12 m away from the wall?

Determine the vector equation of a line passing through the point \( (1, 2, -3) \) and perpendicular to both the given lines: \[ \frac{x - 8}{3} = \frac{y + 16}{-16} = \frac{z - 10}{7} \quad and \quad \frac{x - 15}{3} = \frac{y - 29}{-8} = \frac{z - 5}{-5} \]

(a) Evaluate: \[ I = \int_{2}^{4} \left( |x - 2| + |x - 3| + |x - 4| \right) dx \]

(b) Find: \[ \int \frac{dx}{(x + 2)(x^2 + 1)} \]

Find the maximum slope of the curve \( y = x^3 + 3x^2 + 9x - 30 \).

(a) Find the general solution of the differential equation \[ x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
OR

(b) Find the particular solution of the differential equation \[ \frac{dy}{dx} - y \cot x = \sin 2x, \quad given that y = 2 when x = \frac{\pi}{2}. \]

If \( \hat{a}, \hat{b} \) and \( \hat{c} \) are unit vectors such that \[ \hat{a} \cdot \hat{b} = \hat{a} \cdot \hat{c} = 0 \]
and the angle between \( \hat{b} \) and \( \hat{c} \) is \( \frac{\pi}{6} \), then prove that: \[ \hat{a} = \pm 2 (\hat{b} \times \hat{c}) \]

(a) Four students of class XII are given a problem to solve independently.
Their chances of solving the problem respectively are: \[ \frac{1}{2},\quad \frac{1}{3},\quad \frac{2}{3},\quad \frac{1}{5} \]
Find the probability that at most one of them will solve the problem.

(b)The probability distribution of a random variable \( X \) is given below:


 

Solve the following LPP graphically:


Maximize: \[ Z = 2x + 3y \]

Subject to:
\begin{align*
x + 4y &\leq 8 \quad \text{(1)

2x + 3y &\leq 12 \quad \text{(2)

3x + y &\leq 9 \quad \text{(3)

x &\geq 0, \quad y \geq 0 \quad \text{(non-negativity)
\end{align*

If \[ A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}, \]
find \( A^{-1} \). Using \( A^{-1} \), solve the given system of equations:
\begin{align*
2x - 3y + 5z &= 11 \quad \text{(1)

3x + 2y - 4z &= -5 \quad \text{(2)

x + y - 2z &= -3 \quad \text{(3)
\end{align*

\[ l_1: \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(4\hat{i} + 6\hat{j} + 12\hat{k}) \] \[ l_2: \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(6\hat{i} + 9\hat{j} + 18\hat{k}) \]

(b) Show that the lines \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \quad and \quad \frac{x - 4}{5} = \frac{y - 1}{2} = z \]
intersect. Also, find their point of intersection.

If \[ y = \cos x^2 + \cos^2 x + \cos^2(x^2) + \cos(x^x), \]
find \( \frac{dy}{dx} \).

Find the intervals in which the function \[ f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 \]
is:

[(i)] strictly increasing
[(ii)] strictly decreasing

Using integration, find the area of the region \[ \{(x, y) : 0 \leq y \leq x^2, \, 0 \leq y \leq x, \, 0 \leq x \leq 3\}. \]

A window is in the form of a rectangle surmounted by an equilateral triangle on its length. Let the rectangular part have length and breadth \( x \) and \( y \) metres respectively.



(i) If the perimeter of the window is 12 m, find the relation between \( x \) and \( y \).

(iii) (b) If the area of the window is 50 m\(^2\), find an expression for its perimeter in terms of \( x \).

During the festival season, a mela was organized by the Resident Welfare Association at a park near the society. The main attraction of the mela was a huge swing, which traced the path of a parabola given by: \[ x^2 = y \quad or \quad f(x) = x^2 \]



(i) Let \( f: \mathbb{N} \to \mathbb{R} \) be defined by \( f(x) = x^2 \). What will be the range?

Two persons are competing for a position on the Managing Committee of an organisation. The probabilities that the first and the second person will be appointed are \( 0.5 \) and \( 0.6 \), respectively. Also, if the first person gets appointed, then the probability of introducing a waste treatment plant is \( 0.7 \), and the corresponding probability is \( 0.4 \) if the second person gets appointed.



Based on the above information, answer the following:


(i) What is the probability that the waste treatment plant is introduced?