CBSE Class 12 2025 Mathematics Set-3 (Available): Download Answer Key and Solution PDF

If \( \mathbf{p} \) and \( \mathbf{q} \) are unit vectors, then which of the following values of \( \mathbf{p} \cdot \mathbf{q} \) is not possible?

• (1) \( -\frac{1}{2} \)
• (2) \( \frac{1}{\sqrt{2}} \)
• (3) \( \frac{\sqrt{3}}{2} \)
• (4) \( \sqrt{3} \)

Which of the following can be both a symmetric and skew-symmetric matrix?

• (1) Unit Matrix
• (2) Diagonal Matrix
• (3) Null Matrix
• (4) Row Matrix

If \( \int_0^a x \, dx \leq \frac{a}{2} + 6 \), then which of the following holds for \( a \)?

• (1) \( -4 \leq a \leq 3 \)
• (2) \( a \geq 4, a \leq -3 \)
• (3) \( -3 \leq a \leq 4 \)
• (4) \( -3 \leq a \leq 0 \)

If \( A \) and \( B \) are square matrices of the same order, then \( (A B^T - B A^T) \) is a:

• (1) Symmetric matrix
• (2) Skew-symmetric matrix
• (3) Null matrix
• (4) Unit matrix

The value of \( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \) is:

• (1) \( -1 \)
• (2) \( \frac{-\sqrt{3}}{2} \)
• (3) \( 0 \)
• (4) \( 1 \)

If \( p \) and \( q \) are respectively the order and degree of the differential equation \( \frac{d}{dx} \left( \frac{dy}{dx} \right)^3 = 0 \), then \( (p - q) \) is:

• (1) \( 0 \)
• (2) \( 1 \)
• (3) \( 2 \)
• (4) \( 3 \)

The function \( f(x) = x^2 - 4x + 6 \) is increasing in the interval:

• (1) \( (0, 2) \)
• (2) \( (-\infty, 2] \)
• (3) \( [1, 2] \)
• (4) \( [2, \infty) \)

The line \( x = 1 + 5\mu, y = -5 + \mu, z = -6 - 3\mu \) passes through which of the following points?

• (1) \( (1, -5, 6) \)
• (2) \( (1, 5, 6) \)
• (3) \( (1, -5, -6) \)
• (4) \( (-1, -5, 6) \)

The area of the shaded region (figure) represented by the curves \( y = x^2 \), \( 0 \leq x \leq 2 \), and the y-axis is given by:

• (1) \( \int_0^2 x^2 \, dx \)
• (2) \( \int_0^2 \sqrt{y} \, dy \)
• (3) \( \int_0^4 x^2 \, dx \)
• (4) \( \int_0^4 \sqrt{y} \, dy \)

If \( E \) and \( F \) are two events such that \( P(E) > 0 \) and \( P(F) \neq 1 \), then \( P(E \,|\, F) \) is:

• (1) \( \frac{P(\bar{E})}{P(\bar{F})} \)
• (2) \( 1 - P(\bar{E} \,|\, F) \)
• (3) \( 1 - P(E \,|\, F) \)
• (4) \( \frac{1 - P(E \cup F)}{P(\bar{F})} \)

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





Then \( E(X) \) of distribution is:

• (1) \( -1.8 \)
• (2) \( -1 \)
• (3) \( 1 \)
• (4) \( 1.8 \)

If the projection of \( \mathbf{a} = \alpha \hat{i} + \hat{j} + 4 \hat{k} \) on \( \mathbf{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \) is 4 units, then \( \alpha \) is:

• (1) \( -13 \)
• (2) \( -5 \)
• (3) \( 13 \)
• (4) \( 5 \)

The equation of a line parallel to the vector \( 3 \hat{i} + \hat{j} + 2 \hat{k} \) and passing through the point \( (4, -3, 7) \) is:

• (1) \( x = 4t + 3, y = -3t + 1, z = 7t + 2 \)
• (2) \( x = 3t + 4, y = t + 3, z = 2t + 7 \)
• (3) \( x = 3t + 4, y = -3, z = 2t + 7 \)
• (4) \( x = 3t + 4, y = -3t + 1, z = 2t + 7 \)

If a line makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive directions of \( x \), \( y \), and \( z \)-axes respectively, then \( \theta \) is:

• (1) \( -\frac{\pi}{3} \)
• (2) \( \frac{\pi}{3} \) only
• (3) \( \frac{\pi}{6} \)
• (4) \( \pm \frac{\pi}{3} \)

A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function \( Z = 5x + 7y \), where \( x \) and \( y \) are the number of units of X and Y respectively sold. Which of the following statements is correct?

• (1) The objective function maximizes the difference of the profit earned from products X and Y.
• (2) The objective function measures the total production of products X and Y.
• (3) The objective function maximizes the combined profit earned from selling X and Y.
• (4) The objective function ensures the company produces more of product X than product Y.

If \( A \) denotes the set of continuous functions and \( B \) denotes the set of differentiable functions, then which of the following depicts the correct relation between set \( A \) and \( B \)?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (1)
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Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:

• (1) Abhay: \( 6 AB \)
  (2) Bina: \( 7 AB - BA \)
  (3) Chhaya: \( 8 AB \)
  (4) Devesh: \( 7 BA - AB \)

If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?

• (1) \( A = B \)
• (2) \( AB = BA \)
• (3) \( A = 0 \) or \( B = 0 \)
• (4) \( A = I \) or \( B = I \)

Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.


Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.

• (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): \( A = diag [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).


Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

• (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 values of \( a \) for which \( f(x) = \sin x - ax + b \) is increasing on \( \mathbb{R} \).

Find: \( \int 2x^3 e^{x^2} \,dx \).

If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).

If \( f(x) = \begin{cases} 2x - 3, & -3 \leq x \leq -2
x + 1, & -2 < x \leq 0 \end{cases} \), check the differentiability of \( f(x) \) at \( x = -2 \).

If \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \) and \( \mathbf{a} \cdot \mathbf{b} = 4 \), then evaluate \( |\mathbf{a} + 2\mathbf{b}| \).

A vector \( \mathbf{a} \) makes equal angles with all the three axes. If the magnitude of the vector is \( 5\sqrt{3} \) units, then find \( \mathbf{a} \).

If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

If \( y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 \), then show that \( x(x + 1)^2 y_2 + (x + 1)^2 y_1 = 2 \).

If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \), for \( -1 < x < 1, x \neq y \), then prove that \[ \frac{dy}{dx} = \frac{-1}{(1 + x)^2}. \]

Let \( R \) be a relation on the set of real numbers \( \mathbb{R} \) defined as \[ R = \{(x, y) : x - y + \sqrt{3} is an irrational number, x, y \in \mathbb{R} \}. \]
Verify \( R \) for reflexivity, symmetry, and transitivity.

Solve the following linear programming problem graphically: \[ Minimise Z = 2x + y \]
subject to the constraints: \[ 3x + y \geq 9, \] \[ x + y \geq 7, \] \[ x + 2y \geq 8, \] \[ x, y \geq 0. \]

A die with numbers 1 to 6 is biased such that \( P(2) = \frac{3}{10} \) and the probability of other numbers is equal. Find the mean of the number of times number 2 appears on the die, if the die is thrown twice.

Two dice are thrown. Defined are the following two events A and B: \[ A = \{(x, y) : x + y = 9\}, \quad B = \{(x, y) : x \neq 3\}, \]
where \( (x, y) \) denote a point in the sample space.

Check if events \( A \) and \( B \) are independent or mutually exclusive.

Solve the differential equation \( 2(y + 3) - xy \frac{dy}{dx} = 0 \); given \( y(1) = -2 \).

Solve the differential equation: \[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2. \]

If \( \int_a^b x^3 dx = 0 \) and \( \int_a^b x^2 dx = \frac{2}{3} \), then find the values of \( a \) and \( b \).

Find the shortest distance between the lines: \[ \frac{x+1}{2} = \frac{y-1}{1} = \frac{z-9}{-3} \]
and \[ \frac{x-3}{2} = \frac{y+15}{-7} = \frac{z-9}{5}. \]

Find the image \( A' \) of the point \( A(2,1,2) \) in the line \[ l: \mathbf{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda (\hat{i} - \hat{j} - \hat{k}). \]
Also, find the equation of the line joining \( A A' \). Find the foot of the perpendicular from point \( A \) on the line \( l \).

Find: \[ I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx. \]

Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \]
the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).

Given



find \( AB \). Hence, solve the system of linear equations: \[ x - y + z = 4, \] \[ x - 2y - 2z = 9, \] \[ 2x + y + 3z = 1. \]

If



then find \( A^{-1} \). Hence, solve the system of linear equations: \[ x - 2y = 10, \] \[ 2x - y - z = 8, \] \[ -2y + z = 7. \]

(i) How many relations can be there from \( S \) to \( J \)?

(ii) Check if the function \( f \) is bijective, given: \[ f = \{ (S_1, J_1), (S_2, J_2), (S_3, J_2), (S_4, J_3) \}. \]

(a) How many one-one functions can be there from \( S \) to \( J \)?

(b) Minimum ordered pairs required to make \( R_1 \) reflexive but not symmetric, given: \[ R_1 = \{(S_1, S_2), (S_2, S_4) \} \]

(a) What is the probability that a randomly selected car is electric?

(b) What is the probability that a randomly selected car is a petrol car?

Given that a car is electric, what is the probability that it was manufactured by Comet?

Given that a car is electric, what is the probability that it was manufactured by Amber or Bonzi?

(i) Find the intervals on which \( f(x) \) is increasing or decreasing for \( x \in [0, \pi] \).

(ii)Verify whether each critical point in \( [0, \pi] \) is a local maximum, local minimum, or a point of inflection.