CBSE Class 12 2025 Mathematics 65-7-2 Question Paper Set-2: Download Solutions with Answer Key

Study the given graph. It illustrates:

• (1) \( y = \tan^{-1} x \)
• (2) \( y = \cos^{-1} x \)
• (3) \( y = \sec^{-1} x \)
• (4) \( y = \sin^{-1} x \)

If \[ \begin{bmatrix} 2x-1 & 3x\\0 & y^2-1 \end{bmatrix} = \begin{bmatrix} x+3 & 12\\0 & 35 \end{bmatrix} \]
then the value of \( (x - y) \) is :

• (A) 2 or 10
• (B) -2 or 10
• (C) 2 or -10
• (D) -2 or -10

Let \( A \) be a square matrix of order 3. If \( |A| = 5 \), then \( | adj A | \) is :

• (A) 5
• (B) 125
• (C) 25
• (D) 5

If \( A \) and \( B \) are two square matrices each of order 3 with \( |A| = 3 \) and \( |B| = 5 \), then \( |2AB| \) is :

• (A) 30
• (B) 120
• (C) 15
• (D) 225

The matrix \( A = \begin{bmatrix} \sqrt{3} & 0 & 0\\0 & \sqrt{2} & 0\\0 & 0 & \sqrt{5} \end{bmatrix} \) is a/an :

• (A) scalar matrix
• (B) identity matrix
• (C) null matrix
• (D) symmetric matrix

What is the total number of possible matrices of order 3 × 3 with each entry as \( \sqrt{2} \) or \( \sqrt{3} \) ?

• (A) 9
• (B) 512
• (C) 615
• (D) 64

Domain of \( f(x) = \cos^{-1} x + \sin x \) is :

• (A) \( \mathbb{R} \)
• (B) (-1, 1)
• (C) \([-1, 1]\)
• (D) \( \varnothing \)

If \( f(x) = \begin{cases} \frac{\sin^2 ax}{x^2}, & if x \neq 0
1, & if x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of 'a' is :

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

If \( f: \mathbb{R} \to \mathbb{R} \) is defined as \( f(x) = 2x - \sin x \), then \( f \) is :

• (A) a decreasing function
• (B) an increasing function
• (C) maximum at \( x = \frac{\pi}{2} \)
• (D) maximum at \( x = 0 \)

If \( R \) be a relation defined as \( a \, R \, b \) iff \( |a - b| > 0 \), \( a, b \in \mathbb{R} \), then \( R \) is :

• (A) reflexive
• (B) symmetric
• (C) transitive
• (D) symmetric and transitive

If \( f(x) = -2x^8 \), then the correct statement is :

• (A) \( f'( \frac{1}{2} ) = f( - \frac{1}{2} ) \)
• (B) \( f'( \frac{1}{2} ) = - f( - \frac{1}{2} ) \)
• (C) \( -f( \frac{1}{2} ) = f( - \frac{1}{2} ) \)
• (D) \( f'( \frac{1}{2} ) = - f( \frac{1}{2} ) \)

For a function \( f(x) \), which of the following holds true ?

• (A) \( \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \)
• (B) \( \int_a^b f(x) \, dx = 0 \), if \( f \) is an even function
• (C) \( \int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx \), if \( f \) is an odd function
• (D) \( \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a + x) \, dx \)

\[ \int \frac{e^{9 \log x} - e^{8 \log x}}{e^{6 \log x}} \, dx \]
is equal to :

• (A) \( x + C \)
• (B) \( \frac{x^2}{2} + C \)
• (C) \( \frac{x^4}{4} + C \)
• (D) \( \frac{x^3}{3} + C \)

\[ \int \frac{a^x}{\sqrt{1 - a^2 x}} \, dx \]
is equal to :

• (A) \( \sin^{-1} (a^x) + C \)
• (B) \( \log_e (1 - a^2 x) + C \)
• (C) \( \cos^{-1} (a^x) + C \)
• (D) \( \sin^{-1} (a^x) \, \frac{a}{x} + C \)

A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :

• (A) \( \frac{2}{13} \)
• (B) \( \frac{3}{26} \)
• (C) \( \frac{9}{26} \)
• (D) \( \frac{3}{26} \)

A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \( \hat{i} + 15 \hat{j} + 6 \hat{k} \) and the other is along the vector \( 2 \hat{i} + 10 \hat{j} + 6 \hat{k} \), then the value of \( \lambda \) is :

• (A) 6
• (B) 1
• (C) 6
• (D) 4

If \( P(A) = \frac{1}{5}, P(B) = \frac{3}{5} \) and \( P(A \cap B) = \frac{2}{5} \), then \( P(A' \cup B') \) is :

• (A) \( \frac{19}{25} \)
• (B) \( \frac{9}{25} \)
• (C) \( \frac{19}{25} \)
• (D) \( \frac{5}{13} \)

If \( | \vec{a} + \vec{b} | = | \vec{a} - \vec{b} | \) for any two vectors, then vectors \( \vec{a} \) and \( \vec{b} \) are :

• (A) orthogonal vectors
• (B) parallel to each other
• (C) unit vectors
• (D) collinear vectors

Assertion (A): \(f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0
0, & x = 0 \end{cases}\) is continuous at \(x = 0\).

Reason (R): When \(x \to 0\), \(\sin \frac{1}{x}\) is a finite value between -1 and 1.

Assertion (A): The set of values of \(\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)\) is a null set.

Reason (R): \(\sec^{-1} x\) is defined for \(x \in \mathbb{R} - (-1, 1)\).

(a) 10 identical blocks are marked with '0' on two of them, '1' on three of them, and '2' on four of them. If \(X\) denotes the number written on the block, then write the probability distribution of \(X\) and calculate its mean.

If \[ \begin{bmatrix} 3 & -1
0 & 1
2 & -3 \end{bmatrix} A = \begin{bmatrix} 2 & -5
-17 \end{bmatrix} \]
then find matrix \( A \).

Let \(f: A \to B\) be defined by \(f(x) = \frac{x - 2}{x - 3}\), where \(A = \mathbb{R} - \{3\}\) and \(B = \mathbb{R} - \{1\}\). Discuss the bijectivity of the function.

In a Linear Programming Program (LPP) for objective function \( Z = 14x - 10y \)
subject to the constraints: \[ x + y \leq 8, \quad 3x - 2y \geq -6, \quad x, y \geq 0 \]
shade the feasible region and mark the corner points in a neatly drawn graph.

(a) Differentiate \( \left( \frac{5x}{x^5} \right)\) with respect to \(x\).

Let \( 2x + 5y - 1 = 0 \) and \( 3x + 2y - 7 = 0 \) represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each subject book is \( Rs 150 \) (Chemistry), \( Rs 175 \) (Physics) and \( Rs 180 \) (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is \( Rs 35,000 \), what profit did he earn after the sale of two days?

Show that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 4x^3 - 5 \), \( \forall x \in \mathbb{R} \), is one-one and onto.

Let \( R \) be a relation defined on a set \( \mathbb{N} \) of natural numbers such that \( R = \{(x, y) : xy is a square of a natural number, x, y \in \mathbb{N} \} \). Determine if the relation \( R \) is an equivalence relation.

Show that the derivative of \( \tan^{-1} (\sec x + \tan x) \) with respect to \( x \) is equal to \( \frac{1}{2} \) for \( \left( -\frac{\pi}{2} < x < \frac{\pi}{2} \right) \).

Find the dimensions of a rectangle of perimeter 12 cm which will generate maximum volume when swept along a circular rotation keeping the shorter side fixed as the axis.

The scalar product of the vector \( \mathbf{a} = \hat{i} - \hat{j} + 2\hat{k} \) with a unit vector along sum of vectors \( \mathbf{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \) and \( \mathbf{c} = \lambda \hat{i} - 2\hat{j} - 3\hat{k} \) is equal to 1. Find the value of \( \lambda \).

Find the shortest distance between the lines: \[ \mathbf{r}_1 = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda (\hat{i} - 2\hat{j} + 3\hat{k}) \] \[ \mathbf{r}_2 = (\hat{i} + 4\hat{k}) + \mu (3\hat{i} - 6\hat{j} + 9\hat{k}) \]

In the Linear Programming Problem for objective function \( Z = 18x + 10y \) subject to constraints \[ 4x + y \geq 20 \] \[ 2x + 3y \geq 30 \] \[ x, y \geq 0 \]
Find the minimum value of \( Z \).

(a) Find \[ \int \frac{x}{(x - 1)(x^2 + 4)} \, dx \]

(a) Find the point Q on the line \( \frac{2x + 4}{6} = \frac{y + 1}{2} = \frac{-2z + 6}{-4} \) at a distance of \( \frac{\sqrt{5}}{2} \) from the point \( P(1, 2, 3) \).

A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and discovered the scratch passing through the origin inclined at an angle \( \frac{\pi}{4} \) anticlockwise along the positive direction of x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration.

Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.

Let \( A_1 \): People with good health,
\( A_2 \): People with average health,

and \( A_3 \): People with poor health.

During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.

Based upon the above information, answer the following questions:

% Part (i)
(i) A person was tested randomly. What is the probability that he/she has contracted the disease?

Three friends A, B, and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his pre-decided destination, following straight paths from A to C and B to C in such a way that \( \overrightarrow{OA} = \hat{i}, \overrightarrow{OB} = \hat{j} \), and \( \overrightarrow{OC} = 5 \hat{i} - 2 \hat{j} \), respectively.






Based upon the above information, answer the following questions:

% Part (i)
(i) Complete the given figure to explain their entire movement plan along the respective vectors.

Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation if left in the open at room temperature.


(Cylindrical-shaped Camphor tablets)

A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, the differential equation \( \frac{dV}{dt} = -kS \) is the differential equation, where \( V \) is the volume, \( S \) is the surface area, and \( t \) is the time in hours.


Based upon the above information, answer the following questions:

% Part (i)
(i) Write the order and degree of the given differential equation.