CBSE Class 12 Mathematics Compartment Set 2 - (65/7/2) Question Paper 2026 PDF

For \( f(x) = -|x+1| + 5 \), which of the following statements is true ?

• (A) Max \( f(x) = 0 \) at \( x = 4 \).
• (B) Max \( f(x) = 5 \) at \( x = -1 \).
• (C) Max \( f(x) = 4 \) at \( x = 0 \).
• (D) \( f(x) \) has no maximum value.

If \(f(x) = \begin{cases} \dfrac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}\), then which of the following statements is not correct ?

• (A) \(f\) is continuous for every \(x < 0\).
• (B) \(f\) is continuous for every \(x \in R\).
• (C) \(f\) is continuous for every \(x > 0\).
• (D) \(f\) is continuous for each point except \(x = 0\).

The value of \(\cos^{-1}\left[\cos\frac{7\pi}{6}\right]\) is :

• (A) \(\frac{7\pi}{6}\)
• (B) \(\frac{5\pi}{6}\)
• (C) \(\frac{\pi}{3}\)
• (D) \(\frac{\pi}{6}\)

If \(\begin{vmatrix} 3x & x+2 \\ 3(x-1) & x-2 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix}\), then the value of \(x\) is :

• (A) \(2\)
• (B) \(\frac{1}{3}\)
• (C) \(5\)
• (D) \(3\)

If \(A\) is a square matrix, then which of the following is not true ?

• (A) \(A + A^{\prime}\) is a symmetric matrix
• (B) \(AA^{\prime}\) and \(A^{\prime}A\) are symmetric matrices
• (C) \(A + A^{\prime}\) is a skew symmetric matrix
• (D) \(A - A^{\prime}\) is a skew symmetric matrix

If \(A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\) satisfies \(A^{\prime} + A = I\), then the value of \(\theta\) is :

• (A) \(\frac{\pi}{6}\)
• (B) \(\frac{\pi}{3}\)
• (C) \(\frac{2\pi}{3}\)
• (D) \(\frac{5\pi}{6}\)

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

• (A) \( \pi \)
• (B) \( -\dfrac{\pi}{3} \)
• (C) \( \dfrac{\pi}{3} \)
• (D) \( \dfrac{2\pi}{3} \)

Differential of \( \cos^{-1}(e^{x}) \) with respect to \( x \) is :

• (A) \( \dfrac{e^{x}}{\sqrt{1 - e^{2x}}} \)
• (B) \( \dfrac{-e^{x}}{\sqrt{1 + e^{x}}} \)
• (C) \( \dfrac{-e^{x}}{\sqrt{1 - e^{2x}}} \)
• (D) \( \dfrac{e^{x}}{\sqrt{1 + e^{2x}}} \)

If \( \displaystyle\int_{0}^{1} (9x^{2} - 4x - 2k)\,dx = 0 \), then the value of \( k \) is :

• (A) 1
• (B) 0
• (C) \( \dfrac{1}{2} \)
• (D) \( -\dfrac{1}{2} \)

If \( A \) and \( B \) are square matrices of order 3 such that \( |A| = 4 \) and \( |B| = 7 \), then \( |2AB| \) is :

• (A) 112
• (B) 56
• (C) 28
• (D) 224

If the rate of change of the area of a circle is equal to the rate of change of its diameter, then its radius is :

• (A) \(2\pi\) units
• (B) \(\pi\) units
• (C) \(\frac{\pi}{2}\) units
• (D) \(\frac{1}{\pi}\) units

If vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}\) and \(\vec{b} = \lambda\hat{i} + \hat{j}\) are perpendicular to each other, then the value of \(\lambda\) is :

• (A) \(3\)
• (B) \(-3\)
• (C) \(\frac{2}{3}\)
• (D) \(-\frac{2}{3}\)

\(\int \left[\sin(\log x)+\cos(\log x)\right] dx\) is equal to :

• (A) \(-\cos(\log x)+C\)
• (B) \(\frac{1}{\cos x}+\frac{1}{\sin x}+C\)
• (C) \(\sin(\log x)+C\)
• (D) \(x\sin(\log x)+C\)

If \(y=\sqrt{y+\cos x}\), then \(\frac{dy}{dx}\) is equal to :

• (A) \(\frac{1}{2\sqrt{y+\cos x}}\)
• (B) \(\frac{\cos x}{2y-1}\)
• (C) \(\frac{-\sin x}{2\sqrt{y+\cos x}}\)
• (D) \(\frac{\sin x}{1-2y}\)

If \( A = \begin{bmatrix} 1 & -2 & 3 \\ 3 & 2 & -1 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 3 \\ -1 & 2 \\ 4 & -5 \end{bmatrix} \) and \( C = [C_{ij}] = AB \), then \( C_{12} \) is :

• (A) 22
• (B) -16
• (C) 4
• (D) 0

If two events \(A\) and \(B\) are independent, then which of the following statements is not correct ?

• (A) \(\overline{A}\) and \(B\) are independent.
• (B) Being independent does not indicate that they are mutually exclusive.
• (C) \(P(\overline{A}\cap\overline{B}) = [1 - P(\overline{A})]\,[1 - P(\overline{B})]\).
• (D) \(P(\overline{A}\cap\overline{B}) = [1 - P(A)]\,[1 - P(B)]\).

If \(\displaystyle\int_{0}^{4}\frac{1}{x+\sqrt{x}}\,dx = k\log 3\), then the value of \(k\) is :

• (A) \(4\)
• (B) \(-2\)
• (C) \(-4\)
• (D) \(2\)

For events A and B, if \(P(B)=\frac{3}{5}\), \(P(A/B)=\frac{1}{2}\) and \(P(A\cup B)=\frac{4}{5}\), then \(P(A)\) is :

• (A) \(\frac{1}{2}\)
• (B) \(\frac{3}{5}\)
• (C) \(\frac{1}{5}\)
• (D) \(\frac{3}{10}\)

Assertion (A) : Let \(R\) be a relation on set \(A\) of all books in a school library given by \(R = \{(x, y) : x, y \in A,\ x \text{ and } y \text{ have the same number of pages}\}\) and \(S\) is a relation on set \(A\) from the same library such that \(S = \{(p, q) : p \text{ and } q \text{ are Mathematics books}\}\). Also, \(R\cap S \neq \phi\). Then, \(R\cap S\) is an equivalence relation.

Reason (R) : The intersection of two equivalence relations on a set is an equivalence relation on the set.

• (A) Both Assertion (A) and Reason (R) are true and 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) : Points \(A(6, -7, 0)\), \(B(16, -19, -4)\) and \(C(1, -1, 2)\) are collinear.

Reason (R) : Three points \(A\), \(B\) and \(C\) are collinear, when point \(C\) is the mid-point of line segment joining points \(A\) and \(B\).

• (A) Both Assertion (A) and Reason (R) are true and 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.

Solve the given matrix equation for \(x\), given \(x > 0\):

\[ x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 & 0 \end{bmatrix} = 2\begin{bmatrix} x^2+8 & 24 \\ 10 & 8 \end{bmatrix} \]

Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation \(R\) is defined on the set \(A\) of thumb pins as

\[ R = \{(P, Q) : P, Q \in A \text{ and distance between } P \text{ and } Q \text{ is less than } 3 \text{ cm}\} \]

Determine if \(R\) is an equivalence relation.

A box contains cards numbered 1 to 22. Two cards are drawn simultaneously without replacement and their numbers are noted. What is the probability that both numbers are even ?

If \(P(A) = \frac{3}{8}\), \(P(B) = \frac{1}{2}\) and \(P(A \cap B) = \frac{1}{4}\), find \(P(\overline{A} / \overline{B})\).

For the given Linear Programming Problem, shade the feasible region on the graph and write the coordinates of the corner points.

Minimize \(Z = 5x + 7y\)

subject to constraints

\[ 2x + y \geq 8 \]

\[ x + 2y \geq 10 \]

\[ x,\ y \geq 0. \]

Determine \(a\) and \(b\) if \(f(x)\) is a continuous function, where

\[ f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases} \]

If \(2y^3 + y = 2x\), then show \(\dfrac{d^2y}{dx^2} = \dfrac{-48y}{(6y^2 + 1)^3}\).

Express the matrix \(A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}\) as the sum of a symmetric matrix and a skew symmetric matrix.

Two lines are represented by equations \(5x - 7y - 2 = 0\) and \(7x - 5y - 3 = 0\) in the xy-plane. Using matrix method, find their point of intersection.

If \(y = \left\{x + \sqrt{x^2 - a^2}\right\}^n\), then prove that \(\dfrac{dy}{dx} = \dfrac{ny}{\sqrt{x^2 - a^2}}\).

Find the sum of maximum and minimum values of \(f(x) = \sin x\,(1 + \cos x)\) in the interval \([0, \pi]\).

Solve the following Linear Programming Problem graphically :

Minimize \(Z = 4x + y\)

subject to

\(x + y \geq 3\)

\(x + 2y \geq 4\)

\(x, y \geq 0\).

Find the equation of a line passing through the point \((1, -1, 2)\) and parallel to x-axis. Also, find the shortest distance between this line and the line \(\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k})\).

Find the equations of the lines AC and BD, if coordinates of points A, B, C and D respectively, are \((3, -4, 2)\), \((-1, -2, 0)\), \((4, 1, -1)\) and \((4, 0, 5)\). Show that the lines AC and BD are perpendicular to each other.

If R is a relation on set of natural numbers N such that

\( R = \{(x, y) : x, y \in N,\ x + 2y = 25\} \),

then find the Domain and Range of the relation R. Also, determine if it is an equivalence relation.

Determine whether the function \(f : \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = 2x^3 - 7\), \(\forall\, x \in \mathbb{R}\) (Real numbers) is bijective or not.

Evaluate :

\[ \int_{0}^{\pi/2} \frac{\sin^{2} x}{1+\sin x \cos x}\, dx \]

Let \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) be three non-zero vectors, such that \( \vec{\alpha} \times \vec{\beta} = \vec{\gamma} \) and \( \vec{\beta} \times \vec{\gamma} = \vec{\alpha} \). Show that \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) are mutually perpendicular vectors such that \( |\vec{\beta}| = 1 \) and \( |\vec{\alpha}| = |\vec{\gamma}| \).

Find projection of vector \( \vec{\alpha} \) on \( \vec{\beta} \). Give reasons to support your answer.

If points \( A(3, -1, 2) \), \( B(6, -3, 6) \), \( C(4, -3, 1) \) and \( D(1, -1, -3) \) are vertices of a quadrilateral ABCD, write the position vectors of A, B, C and D and show that ABCD represents a parallelogram. Also, find the area of the parallelogram ABCD.

Solve the differential equation \( (x+y)\,dy + (x-y)\,dx = 0 \), given that \( y(1) = 1 \).

Using integration, find the area bounded by the curve \( y = x^2 \) and the line \( y = 2 \).

Using integration, find the area bounded by the curve \( 9x^2 + 16y^2 = 144 \) and the lines \( x = 2 \), \( x = -2 \).

(i) If the perimeter of the window is 18 m, write the relation between x and y.

(ii) From (i), write an expression for the area of the window as a function of x only.

(iii) (a) Find the dimensions of the rectangle that will allow the maximum light through the window.

(iii) (b) If it is given that the area of the window is \(100\ \text{m}^2\), then find the expression for its perimeter in terms of x.

An organisation needs to select a building construction company to make houses in cities. Three companies X, Y and Z are short-listed with the probability of their selection being \(\frac{1}{3}\), \(\frac{2}{9}\) and \(\frac{4}{9}\) respectively. It was found that there were 20%, 40% and 30% chances respectively, of delivering poor construction quality.

Based on the above information, answer the following questions :

An organisation needs to select a building construction company to make houses in cities. Three companies X, Y and Z are short-listed with the probability of their selection being \(\frac{1}{3}\), \(\frac{2}{9}\) and \(\frac{4}{9}\) respectively. It was found that there were 20%, 40% and 30% chances respectively, of delivering poor construction quality.

Based on the above information, answer the following questions :

(i) If the path of student A is represented by line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-3p}\), find the value of p.

(ii) Write the vector equation of the path of student B swimming parallel to the path of student A, if at a certain instant he passes through the point with coordinates \((2, 4, -1)\).

(iii) (a) Find the foot of the perpendicular drawn from \((2, 4, -1)\) to the path represented by student A.

(iii) (b) If a bird flies across the top of the swimming pool with its path represented by \(\frac{x+4}{2}=\frac{y+3}{8}=\frac{z-5}{-18}\), then find the shortest distance between its path and the path of student A.