CBSE Class 12 2025 Mathematics 65-1-2 Question Paper (Available):download Solutions with Answer Key

If \( E \) and \( F \) are two independent events such that \( P(E) = \frac{2}{3} \), \( P(F) = \frac{3}{7} \), then \( P(E \,|\, F) \) is equal to:

• (A) \( \frac{1}{6} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{2}{3} \)
• (D) \( \frac{7}{9} \)

If \( \vec{\alpha} = \hat{i} - 4 \hat{j} + 9 \hat{k} \) and \( \vec{\beta} = 2\hat{i} - 8 \hat{j} + \lambda \hat{k} \) are two mutually parallel vectors, then \( \lambda \) is equal to:

• (A) \( \frac{-18}{9} \)
• (B) \( 18 \)
• (C) \( \frac{-34}{9} \)
• (D) \( \frac{34}{9} \)

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

• (A) \( \tan x - 2 \sec x + C \)
• (B) \( -\tan x + 2 \sec x + C \)
• (C) \( -\tan x - 2 \sec x + C \)
• (D) \( \tan x + 2 \sec x + C \)

If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

• (A) \( \frac{\pi}{6} \)
• (B) \( \frac{\pi}{4} \)
• (C) \( \frac{\pi}{3} \)
• (D) \( \frac{\pi}{2} \)

The graph of a trigonometric function is as shown. Which of the following will represent the graph of its inverse?

If \( A \) is a square matrix of order 3 such that \( \det(A) = 9 \), then \( \det(9A^{-1}) \) is equal to:

• (A) \( 9 \)
• (B) \( 9^2 \)
• (C) \( 9^3 \)
• (D) \( 9^4 \)

If \( f(x) = |x| + |x - 1| \), then which of the following is correct?

• (A) \( f(x) \) is both continuous and differentiable, at \( x = 0 \) and \( x = 1 \)
• (B) \( f(x) \) is differentiable but not continuous, at \( x = 0 \) and \( x = 1 \)
• (C) \( f(x) \) is continuous but not differentiable, at \( x = 0 \) and \( x = 1 \)
• (D) \( f(x) \) is neither continuous nor differentiable, at \( x = 0 \) and \( x = 1 \)

Which of the following is not a homogeneous function of \( x \) \text{ and \( y \)?

• (A) \( y^2 - xy \)
• (B) \( x - 3y \)
• (C) \( \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \)
• (D) \( \tan x - \sec y \)

Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:

• (A) \( m \times m \)
• (B) \( n \times n \)
• (C) \( m \times n \)
• (D) \( n \times m \)

The feasible region of a linear programming problem with objective function \( Z = ax + by \), is bounded, then which of the following is correct?

• (A) It will only have a maximum value.
• (B) It will only have a minimum value.
• (C) It will have both maximum and minimum values.
• (D) It will have neither maximum nor minimum value.

If \[ A = \begin{bmatrix} -1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{bmatrix}, then A^{-1} is: \]

• (A) \( \begin{bmatrix} -1 & 0 & 0
  0 & -1 & 0
  0 & 0 & -1 \end{bmatrix} \)
• (B) \( \begin{bmatrix} 1 & 0 & 0
  0 & -1 & 0
  0 & 0 & -1 \end{bmatrix} \)
• (C) \( \begin{bmatrix} -1 & 0 & 0
  0 & -1 & 0
  0 & 0 & 1 \end{bmatrix} \)
• (D) \( \begin{bmatrix} -1 & 0 & 0
  0 & 1 & 0
  0 & 0 & 1 \end{bmatrix} \)

The integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1 + y}{x} \) is:

• (A) \( x e^x \)
• (B) \( \frac{e^x}{x} \)
• (C) \( \frac{x}{e^x} \)
• (D) \( \frac{1}{x e^x} \)

Let \( A = [a_{ij}] \) be a square matrix of order 3 such that \( a_{ij} = \hat{i} - 2 \hat{j} \). Then, which of the following is true?

• (A) \( a_{12} > 0 \)
• (B) \( all \ a_{ij} < 0 \)
• (C) \( a_{13} + a_{31} = -6 \)
• (D) \( a_{23} > a_{32} \)

The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in \( [0, 2] \) is:

• (A) 0
• (B) 2
• (C) 4
• (D) 5

If \( \int \frac{1}{2x^2} \, dx = k \cdot 2x + C \), then \( k \) is equal to:

• (A) \( -1 \)
• (B) \( \log 2 \)
• (C) \( -\log 2 \)
• (D) \( 1/2 \)

If \( A \) and \( B \) are invertible matrices, then which of the following is not correct?

• (A) \( (A + B)^{-1} = B^{-1} + A^{-1} \)
• (B) \( (AB)^{-1} = B^{-1} A^{-1} \)
• (C) \( adj(A) = |A| A^{-1} \)
• (D) \( |A|^{-1} = |A^{-1}| \)

The corner points of the feasible region in graphical representation of a L.P.P. are \( (2, 72), (15, 20) \) and \( (40, 15) \). If \( Z = 18x + 9y \) be the objective function, then:

• (A) \( Z \) is maximum at \( (2, 72), \) minimum at \( (15, 20) \)
• (B) \( Z \) is maximum at \( (15, 20), \) minimum at \( (40, 15) \)
• (C) \( Z \) is maximum at \( (40, 15), \) minimum at \( (15, 20) \)
• (D) \( Z \) is maximum at \( (40, 15), \) minimum at \( (2, 72) \)

The area of the shaded region bounded by the curves \( y^2 = x, x = 4 \) and the x-axis is given by:

• (A) \( \int_0^4 x \, dx \)
• (B) \( 2 \int_0^4 \sqrt{x} \, dx \)
• (C) \( 4 \int_0^4 \sqrt{x} \, dx \)
• (D) \( 4 \int_0^4 \frac{1}{\sqrt{x}} \, dx \)
• (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.

Assertion (A): Let \( Z \) be the set of integers. A function \( f : Z \to Z \) defined as \( f(x) = 3x - 5 \), \( \forall x \in Z \), is a bijective.


Reason (R): A function is bijective if it is both surjective and injective.

 Assertion (A)
Assertion (A): \(f(x) = \begin{cases} 3x - 8, & x \leq 5
2k, & x > 5 \end{cases}\)
is continuous at \(x = 5\) for \(k = \frac{5}{2}\).
 Reason (R)
Reason (R):
For a function \(f\) to be continuous at \(x = a\), \(\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = f(a)\).

(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors \( \mathbf{a} = 3 \hat{i} + \hat{j} + 2 \hat{k} \) and \( \mathbf{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k} \).
Determine the angle formed between the kite strings. Assume there is no slack in the strings.

Find the values of \( a \) for which \( f(x) = x^2 - 2ax + b \) is an increasing function for \( x > 0 \).

(a) Differentiate \(2\cos^2 x\) w.r.t. \(\cos^2 x\).

(b) If \(\tan^{-1}(x^2 + y^2) = a^2\), then find \(\frac{dy}{dx}\).

Evaluate: \( \sin^{-1} \left( \sin \frac{3\pi}{5} \right) \).

The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.

(a) Verify that lines given by \( \vec{r} = (1 - \lambda) \hat{i} + (\lambda - 2) \hat{j} + (3 - 2\lambda) \hat{k} \) and \( \vec{r} = (\mu + 1) \hat{i} + (2\mu - 1) \hat{j} - (2\mu + 1) \hat{k} \) are skew lines. Hence, find shortest distance between the lines.

(b) During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by \( \vec{B} = 2\hat{i} + 8\hat{j} \), \( \vec{W} = 6\hat{i} + 12\hat{j} \) and \( \vec{F} = 12\hat{i} + 18\hat{j} \) respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

Solve the following linear programming problem graphically:
Maximise \( Z = 20x + 30y \)
Subject to the constraints: \[ x + y \leq 80, \quad 2x + 3y \geq 100, \quad x \geq 14, \quad y \geq 14. \]

The area of an expanding rectangle is increasing at the rate of \( 48 \, cm^2/s \). The length of the rectangle is always square of its breadth. At what rate is the length of the rectangle increasing at an instant, when breadth \( = 4.5 \, cm \)?

(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows: \[ \begin{array}{|c|c|} \hline X & P(X)
\hline 0 & p
2 & 2p
4 & 3p
5 & p
\hline \end{array} \]
Where \( X \) is the number of students absent.

• (i) Calculate \( p \).
• (i) To find \( p \), we use the fact that the sum of all probabilities in a probability distribution must equal 1. Therefore: \[ p + 2p + 3p + p = 1 \] \[ 7p = 1 \implies p = \frac{1}{7} \]

(b) For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data, it was revealed that two-thirds of the total applicants were females and the other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in the written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.

(a) Find: \[ \int \frac{\cos 2x}{(\sin x + \cos x)^2} \, dx \]

(b) Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]

Sketch the graph of \( y = |x + 3| \) and find the area of the region enclosed by the curve, x-axis, between \( x = -6 \) and \( x = 0 \), using integration.

(a) Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad w.r.t. \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]

(b) Find: \[ \frac{dy}{dx}, \quad if \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]

Find the absolute maximum and absolute minimum of the function \( f(x) = 2x^3 - 15x^2 + 36x + 1 \) on \( [1, 5] \).

A school wants to allocate students into three clubs: Sports, Music, and Drama, under the following conditions:

- The number of students in the Sports club should be equal to the sum of the number of students in the Music and Drama clubs.

- The number of students in the Music club should be 20 more than half the number of students in the Sports club.

- The total number of students to be allocated in all three clubs is 180.

Find the number of students allocated to different clubs, using the matrix method.

(a) Find the image \( A' \) of the point \( A(1, 6, 3) \) in the line \( \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \). Also, find the equation of the line joining \( A \) and \( A' \).

(b) Find a point \( P \) on the line \( \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9} \) such that its distance from point \( Q(2, 4, -1) \) is 7 units. Also, find the equation of the line joining \( P \) and \( Q \).

A bank offers loans to its customers on different types of interest rates namely, fixed rate, floating rate, and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate, or variable rate with probabilities 10%, 20%, and 70% respectively. A customer after availing a loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate, and variable rate is 5%, 3%, and 1% respectively.

Based on the above information, answer the following:

• (i) What is the probability that a customer after availing the loan will default on the loan repayment?
• (i) **What is the probability that a customer after availing the loan will default on the loan repayment?**
  To find the total probability of defaulting on loan repayment, we use the law of total probability: \[ P(D) = P(D|F) \cdot P(F) + P(D|Fl) \cdot P(Fl) + P(D|V) \cdot P(V). \] Substitute the given values: \[ P(D) = (0.05 \times 0.1) + (0.03 \times 0.2) + (0.01 \times 0.7) \] \[ P(D) = 0.005 + 0.006 + 0.007 = 0.018. \] Thus, the probability that a customer after availing the loan will default on the loan repayment is \( 0.018 \) or 1.8%.

A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections.

Let the length of the side perpendicular to the partition be \( x \) metres and the side parallel to the partition be \( y \) metres.


Based on this information, answer the following questions:

(i) Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of \( x \) and \( y \).

(ii) Write the area of the solar panel as a function of \( x \).

(iii) (a) Find the critical points of the area function. Use the second derivative test to determine critical points at the maximum area. Also, find the maximum area.

OR

(iii) (b) Using the first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.

• (i) The total boundary material used includes the perimeter of the rectangular solar panel with an additional partition running parallel to one of the sides. The perimeter of the rectangle is \( 2x + 2y \), and the length of the partition is \( y \). Therefore, the total boundary material used is: \[ Total Boundary Material = 2x + 2y + y = 2x + 3y. \] We are given that the total boundary material used is 300 metres, so: \[ 2x + 3y = 300. \]

A classroom teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set \( A = \{1, 2, 3\} \): \[ R_1 = \{(2, 3), (3, 2)\}, \quad R_2 = \{(1, 2), (1, 3), (3, 2)\}, \quad R_3 = \{(1, 2), (2, 1), (1, 1)\}, \] \[ R_4 = \{(1, 1), (1, 2), (3, 3), (2, 2)\}, \quad R_5 = \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}. \]
The students are asked to answer the following questions about the above relations:

• (i) Identify the relation which is reflexive, transitive but not symmetric.
• (i) Identify the relation which is reflexive, transitive but not symmetric:
  - Reflexive: A relation \( R \) is reflexive if for all \( a \in A \), \( (a, a) \in R \).
  - Transitive: A relation \( R \) is transitive if for all \( (a, b) \in R \) and \( (b, c) \in R \), \( (a, c) \in R \).
  - Symmetric: A relation \( R \) is symmetric if for all \( (a, b) \in R \), \( (b, a) \in R \).
  Now, let's check each relation for reflexivity, transitivity, and symmetry.
  - \( R_1 = \{(2, 3), (3, 2)\} \):
  - Not reflexive (missing \( (1, 1) \), \( (2, 2) \), \( (3, 3) \)).
  - Not transitive because we don't have \( (2, 2) \) or \( (3, 3) \) for transitivity.
  - Not symmetric since \( (2, 3) \) is in \( R_1 \), but \( (3, 2) \) is not.
  - \( R_2 = \{(1, 2), (1, 3), (3, 2)\} \):
  - Not reflexive (missing \( (2, 2) \), \( (3, 3) \)).
  - Transitive: It is transitive since if we have \( (1, 2) \) and \( (2, 3) \), we also have \( (1, 3) \), and similarly for other combinations.
  - Not symmetric, because \( (1, 2) \) is in \( R_2 \), but \( (2, 1) \) is not.
  -\( R_3 = \{(1, 2), (2, 1), (1, 1)\} \):
  - Reflexive because it includes \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).
  - Symmetric because for every pair \( (a, b) \), \( (b, a) \) is also present.
  - Not transitive because there is no \( (1, 3) \), which makes it not transitive.
  Thus, **\( R_3 \)** is reflexive and symmetric but not transitive.