CBSE Class 12 2025 Mathematics 65-6-3 Question Paper Set-3: Download Solutions with Answer Key

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 \[ A = \begin{bmatrix} 0 & 0 & -5
0 & 3 & 0
4.3 & 0 & 0 \end{bmatrix}, then A is a: \]

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

The graph shown below depicts:

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

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

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

Let both \(AB'\) and \(B'A\) be defined for matrices A and B. If 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) \(m \times n\)

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 \(y = a \cos(\log x) + b \sin(\log x)\), then \(x^2 y_2 + x y_1\) is:

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

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

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

If \(f(x) = x^x\), find the critical point:

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

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

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

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

The order and degree of the differential equation \[ \left[ \left( \frac{d^2 y}{dx^2} \right)^2 - 1 \right]^2 = \frac{dy}{dx} are, respectively: \]

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

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

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

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.

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

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

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

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

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

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

A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up is 0.10, member B does not turn up is 0.20 and member C does not turn up is 0.05. The probability of the meeting being cancelled is:

• (1) 0.35
• (2) 0.316
• (3) 0.001
• (4) 0.65

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.

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 \(\mathbf{r}\).

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

Find the interval in which \(f(x) = x + \frac{1}{x}\) is always increasing, \(x \neq 0\).

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

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

Find the angle at which the given lines are inclined to each other:
\[ l_1: \frac{x - 5}{2} = \frac{y + 3}{1} = \frac{z - 1}{-3} \] \[ l_2: \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 5}{-1} \]

Find the value of \(x\), if \[ \begin{bmatrix} 1 & 3 & 2
2 & 5 & 1
15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1
x
2 \end{bmatrix} = \begin{bmatrix} 0
0
0 \end{bmatrix} \]

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 \( BC \).

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 \(Z\).

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.

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 \(x \in (0, 1)\).

Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly from Bag I is transferred to Bag II and then a ball is drawn randomly from Bag II. Find the probability that the ball drawn is white.

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 \( y(0) = 0 \).

Using integration, find the area of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad bounded between the lines \quad x = -\frac{a}{2} \quad to \quad x = \frac{a}{2}. \]

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

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

Show that the line passing through the points A \((0, -1, -1)\) and B \((4, 5, 1)\) intersects the line joining points C \((3, 9, 4)\) and D \((-4, 4, 4)\).

(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?

(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.