CBSE Class 12 2025 Mathematics 65-6-2 Question Paper Set-2: Download Solutions with Answer Key

Sum of two skew-symmetric matrices of same order is always a/an :

• (1) skew-symmetric matrix
• (2) symmetric matrix
• (3) null matrix
• (4) identity matrix

If A = \[ A = \begin{pmatrix} 0 & -3 & 8
3 & 0 & 5
-8 & -5 & 0 \end{pmatrix} \]
then A is a :

• (1) null matrix
• (2) symmetric matrix
• (3) skew-symmetric matrix
• (4) diagonal matrix

The graph shown below depicts :

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

Let both AB' and B'A be defined for matrices A and B. If the order of A is \(n \times m\), then the order of B is :

• (1) \(n \times n\)
• (2) \(n \times m\)
• (3) \(m \times m\)
• (4) \(m \times n\)

If \(f(x) = \frac{\log(1 + ax) + \log(1 - bx)}{x}\) for \(x \neq 0\) and \(f(x) = k\) for \(x = 0\), is continuous at \(x = 0\), then the value of \(k\) is :

• (1) \(a\)
• (2) \(a + b\)
• (3) \(a - b\)
• (4) \(b\)

If \(y = a \cos(\log x) + b \sin(\log x)\), then \(x^2 y^2 + x y\) is :

• (1) \(\cot(\log x)\)
• (2) \(y\)
• (3) \(-y\)
• (4) \(\tan(\log x)\)

\[ \sec^{-1} \left( -\sqrt{2} \right) - \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \]
is equal to:

• (1) \(\frac{11\pi}{12}\)
• (2) \(\frac{5\pi}{12}\)
• (3) \(-\frac{5\pi}{12}\)
• (4) \(\frac{7\pi}{12}\)

If \(\tan^{-1} (x^2 - y^2) = a\), where \(a\) is a constant, then \(\frac{dy}{dx}\) is:

• (1) \(\frac{x}{y}\)
• (2) \(-\frac{x}{y}\)
• (3) \(\frac{a}{y}\)
• (4) \(\frac{a}{x}\)

Let \(f(x) = x^2\), \(x \in \mathbb{R}\). Then, which of the following statements is incorrect ?

• (1) Minimum value of \(f\) does not exist.
• (2) There is no point of maximum value of \(f\) in \(\mathbb{R}\).
• (3) \(f\) is continuous at \(x = 0\).
• (4) \(f\) is differentiable at \(x = 0\).

\[ \int \frac{x + 5}{(x + 6)^2} e^x \, dx \]
is equal to:

• (1) \(\log(x + 6) + C\)
• (2) \(e^x + C\)
• (3) \(\frac{e^x}{x + 6} + C\)
• (4) \(-\frac{1}{(x + 6)^2} e^x + C\)

Let \(f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5\), \(f(1) = 0\). Then, \(f(x)\) is:

• (1) \(x^3 + 3x^2 + \frac{2}{x^2} + 5x + 11\)
• (2) \(x^3 + 3x^2 + \frac{2}{x^2} + 5x - 11\)
• (3) \(x^3 + 3x^2 - \frac{2}{x^2} + 5x - 11\)
• (4) \(x^3 - 3x^2 - \frac{2}{x^2} + 5x - 11\)

The order and degree of the differential equation \[ \frac{d^2y}{dx^2} + 4 \left(\frac{dy}{dx}\right) = x \log \left(\frac{d^2y}{dx^2}\right) are respectively: \]

• (1) 0, 3
• (2) 2, 1
• (3) 2, not defined
• (4) 1, not defined

For a Linear Programming Problem (LPP), the given objective function is \(Z = x + 2y\). The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph.






The point \(P = ( \frac{3}{13}, \frac{24}{13} )\), \(Q = ( \frac{3}{15}, \frac{15}{4} )\), \(R = ( \frac{7}{3}, \frac{3}{2} )\), \(S = ( \frac{18}{7}, \frac{7}{7} )\).

Which of the following statements is correct?

• (1) \(Z\) is minimum at \(S \left( \frac{18}{7}, \frac{7}{7} \right)\)
• (2) \(Z\) is maximum at \(R \left( \frac{7}{3}, \frac{3}{2} \right)\)
• (3) \((Value of Z at P) > (Value of Z at Q)\)
• (4) \((Value of Z at Q) < (Value of Z at R)\)

The area of the region bounded by the curve \(y^2 = x\) between \(x = 0\) and \(x = 1\) is:

• (1) \(3\) sq units
• (2) \(2\) sq units
• (3) \(4\) sq units
• (4) \(3 \frac{1}{2}\) sq units

Let \(\mathbf{| \mathbf{a} |} = 5\) and \(-2 \leq z \leq 1\). Then, the range of \(|\mathbf{a}|\) is:

• (1) \([5, 10]\)
• (2) \([-2, 5]\)
• (3) \([2, 1]\)
• (4) \([-10, 5]\)

The solution for the differential equation \(\log \left( \frac{dy}{dx} \right) = 3x + 4y\) is:

• (1) \(3e^{3y} + 4e^{-3x} + C = 0\)
• (2) \(e^{3x} + 4y + C = 0\)
• (3) \(3e^{-y} + 4e^{x} + 12C = 0\)
• (4) \(3e^{-y} + 4e^{3x} + 12C = 0\)

In a Linear Programming Problem (LPP), the objective function \(Z = 2x + 5y\) is to be maximized under the following constraints:




\[ x + y \leq 4, \quad 3x + 3y \geq 18, \quad x, y \geq 0. \]
Study the graph and select the correct option.

• (1) The solution of the given LPP lies in the shaded unbounded region.
• (2) The solution lies in the shaded region \(\triangle AOB\).
• (3) The solution does not exist.
• (4) The solution lies in the combined region of \(\triangle AOB\) and unbounded shaded region.

Chances that three persons A, B, and C go to the market are 30%, 60% and 50% respectively. The probability that at least one will go to the market is :

• (1) \(\frac{14}{10}\)
• (2) \(\frac{43}{50}\)
• (3) \(\frac{9}{100}\)
• (4) \(\frac{7}{50}\)

Assertion (A): If \(| \mathbf{a} \times \mathbf{b} |^2 + | \mathbf{a} \cdot \mathbf{b} |^2 = 256\) and \(| \mathbf{b} | = 8\), then \(| \mathbf{a} | = 2\).

Reason (R): \(\sin^2 \theta + \cos^2 \theta = 1\) and \(| \mathbf{a} \times \mathbf{b} | = | \mathbf{a} | | \mathbf{b} | \sin \theta\) and \( \mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | | \mathbf{b} | \cos \theta\).

(A) Both Assertion (A) and Reason (R) are true and the 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): Let \(f(x) = e^x\) and \(g(x) = \log x\). Then \((f + g)(x) = e^x + \log x\) where the domain of \((f + g)\) is \(\mathbb{R}\).

Reason (R): \(Dom(f + g) = Dom(f) \cap Dom(g)\).

(A) Both Assertion (A) and Reason (R) are true and the 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

(a) Differentiate \(\sqrt{e^{\sqrt{2x}}}\) with respect to \(e^{\sqrt{2x}}\) for \(x > 0\).

If \((x)^y = (y)^x\), then find \(\frac{dy}{dx}\).

If \(\mathbf{a}\) and \(\mathbf{b}\) are position vectors of point A and point B, respectively, find the position vector of point C on \(\overrightarrow{BA}\) such that \(BC = 3BA\).

Vector \(\mathbf{r}\) is inclined at equal angles to the three axes \(x\), \(y\), and \(z\). If the magnitude of \(\mathbf{r}\) is \(5\sqrt{3}\) units, then find \(\mathbf{r}\).

Determine those values of \(x\) for which \(f(x) = \frac{2}{x} - 5\), \(x \ne 0\) is increasing or decreasing.

Find the domain of the function \(f(x) = \sin^{-1}(-x^2)\).

Find the value of \(\lambda\) if the following lines are perpendicular to each other:
\[ l_1: \frac{1 - x}{-3} = \frac{3y - 2}{2\lambda} = \frac{z - 3}{3}, \quad l_2: \frac{x - 1}{3\lambda} = \frac{1 - y}{1} = \frac{2z - 5}{3} \]

If \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1
-1 & 3 & 4
0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2
3
4 \end{bmatrix} \]
are three matrices, then find \(ABC\).

Consider the Linear Programming Problem, where the objective function \[ Z = x + 4y \]
needs to be minimized subject to the following constraints: \[ 2x + y \geq 1000, \] \[ x + 2y \geq 800, \] \[ x \geq 0, \quad y \geq 0. \]
Draw a neat graph of the feasible region and find the minimum value of \(Z\).

Find the distance of the point \(P(2, 4, -1)\) from the line \[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}. \]

Let the position vectors of points A, B and C be \( \mathbf{a} = 3\hat{i} - \hat{j} - 2\hat{k} \), \( \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k} \), and \( \mathbf{c} = \hat{i} + 5\hat{j} + 3\hat{k} \), respectively. Find the vector and Cartesian equations of the line passing through \( A \) and parallel to line \( BC \).

Differentiate \(y = \sin^{-1}(3x - 4x^3)\) with respect to \(x\), for \(x \in \left[ -\frac{1}{2}, \frac{1}{2} \right]\).

Differentiate \(y = \cos^{-1}\left( \frac{1 - x^2}{1 + x^2} \right)\) with respect to \(x\), when \(x \in (0, 1)\).

A student wants to pair up natural numbers such that they satisfy the equation \(2x + y = 41\), where \(x, y \in \mathbb{N}\). Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric, and transitive. Hence, state whether it is an equivalence relation or not.

Show that the function \(f: \mathbb{N} \to \mathbb{N}\), where \(\mathbb{N}\) is the set of natural numbers, given by \[ f(n) = \begin{cases} n - 1, & if n is even
n + 1, & if n is odd \end{cases} \]
is a bijection.

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the probability distribution of number of tails. Hence, find the mean of the distribution.

Solve the differential equation: \[ x^2y \, dx - (x^3 + y^3) \, dy = 0. \]

Solve the differential equation \( (1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0 \) subject to initial condition \( y(0) = 0 \).

Use integration to find the area of the region enclosed by the curve \(y = -x^2\) and the straight lines \(x = -3\), \(x = 2\) and \(y = 0\). Sketch a rough figure to illustrate the bounded region.

Find \[ \int \frac{x^2 + 1}{(x - 1)^2 (x + 3)} \, dx. \]

Evaluate \[ \int_0^{\frac{\pi}{2}} \frac{x}{\sin x + \cos x} \, dx. \]

Find the foot of the perpendicular drawn from point \((2, -1, 5)\) to the line \[ \frac{x - 11}{10} = \frac{y + 2}{-4} = \frac{z + 8}{-11} \]
Also, find the length of the perpendicular.

(i) Find the probability that it was defective.

(ii) What is the probability that this defective smartphone was manufactured by company B?

(i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form \( A \mathbf{X} = B \).

(ii) Find \( |A| \) and confirm if it is possible to find \( A^{-1} \).

(iii) Find \( A^{-1} \), if possible, and write the formula to find \( \mathbf{X} \).

(iii) (b) Find \( A^2 - I \), where \( I \) is the identity matrix.

(i) Express the distance \( y \) between the wall and foot of the ladder in terms of \( h \) and height \( x \) on the wall at a certain instant. Also, write an expression in terms of \( h \) and \( x \) for the area \( A \) of the right triangle, as seen from the side by an observer.

(ii) Find the derivative of the area \( A \) with respect to the height on the wall \( x \), and find its critical point.

(iii) (a) Show that the area \( A \) of the right triangle is maximum at the critical point.

(iii) (b) If the foot of the ladder, whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance \( y \) is \( 2 \, m/s \), then at what rate is the height on the wall \( x \) increasing when the foot of the ladder is 3 m away from the wall?