CBSE Class 12 2025 Mathematics 65-6-1 Question Paper Set-1: Download Solutions with Answer Key

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 \[ A = \begin{pmatrix} -1 & 0 & 0
0 & 3 & 0
0 & 0 & 5 \end{pmatrix} \]
then \(A\) is a/an:

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

The following graph is a combination of:

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

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

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

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

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

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

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

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) \(\frac{a}{x}\)

If \(y = a \cos(\log x) + b \sin(\log x)\), then \(x^2y'' + xy'1\) is:

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

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

• (1) \(f\) has a minimum value at \(x = 0\)
• (2) \(f\) has no maximum value in \(\mathbb{R}\)
• (3) \(f\) is continuous at \(x = 0\)
• (4) \(f\) is differentiable at \(x = 0\)

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

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

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

The order and degree of the following differential equation are, respectively: \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y^2 = 0. \]

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

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

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) \((Value of Z at Q) < (Value of Z at R)\)

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.

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 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

A box has 4 green, 8 blue, and 3 red pens. A student picks up a pen at random, checks its color and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:

• (1) 124/125
• (2) 125/125
• (3) 64/125
• (4) 125/125

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.

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

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

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

Determine the values of \(x\) for which the function \(f(x) = \frac{x-4}{x+1}\) is an increasing or a decreasing function.

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 if the lines \(\mathbf{r}_1 = ( \hat{i} + \hat{j} - \hat{k} ) + \lambda ( 3\hat{i} - \hat{j} )\) and \(\mathbf{r}_2 = ( 4\hat{i} - \hat{k} ) + \mu ( 2\hat{i} + 3\hat{k} )\) intersect with each other.

Let \[ A = \begin{pmatrix} 1 & 4
-2 & 1 \end{pmatrix} \quad and \quad C = \begin{pmatrix} 3 & 4 & 2
12 & 16 & 8
-6 & -8 & -4 \end{pmatrix}. \]
Then, find the matrix \(B\) if \(AB = C\).

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.

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 \( BC \).

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and in committee II is 0.01, then find the probability that the person makes the correct decision of selection:
(i) in both committees
(ii) in only one committee

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

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

Draw a rough sketch for the curve \(y = 2 + |x + 1|\). Using integration, find the area of the region bounded by the curve \(y = 2 + |x + 1|\), \(x = -4\), \(x = 3\), and \(y = 0\).

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 \( y(0) = 0 \).

Let the polished side of the mirror be along the line \[ \frac{x}{1} = \frac{1 - y}{2} = \frac{2z - 4}{6}. \]
A point \( P(1, 6, 3) \), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point \( P \) and its image.

(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?

(i) Find the probability that it was defective.

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