CBSE Class 12 Mathematics Compartment Set 3 - (65/7/3) Question Paper 2026 PDF

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

• (A) \(\frac{3\pi}{5}\)
• (B) \(\frac{\pi}{5}\)
• (C) \(\frac{7\pi}{5}\)
• (D) \(\frac{2\pi}{5}\)

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 \(A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\) 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) \(\frac{5\pi}{6}\)

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

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

The value of \( \left[ \tan^{-1}(-1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right) \right] \) is :

• (A) \( \frac{7\pi}{12} \)
• (B) \( \frac{5\pi}{12} \)
• (C) \( -\frac{5\pi}{12} \)
• (D) \( \frac{\pi}{3} \)

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

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

The motion of a particle is described as \( S = 2t^{3} - 6t^{2} + 8 \). Its velocity \( \frac{dS}{dt} \), when the acceleration \( \frac{d^{2}S}{dt^{2}} \) vanishes is :

• (A) \( -6 \) unit/sec
• (B) \( 0 \) unit/sec
• (C) \( 18 \) unit/sec
• (D) \( 6 \) unit/sec

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) \(2\)

\( \int e^{x}\left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx \) is equal to :

• (A) \( \log \cos 2x + C \)
• (B) \( \sec^{2} 2x + C \)
• (C) \( e^{x}\tan x + C \)
• (D) \( e^{x}\cot x + C \)

Differential of \( \log \sin x \) with respect to \( \sqrt{\cos x} \) is :

• (A) \( -2\sqrt{\cos x}\cdot\cot x\cdot\operatorname{cosec} x \)
• (B) \( \sqrt{\cos x}\cdot\cot x\cdot\operatorname{cosec} x \)
• (C) \( \frac{2\sqrt{\cos x}}{\sin x} \)
• (D) \( \frac{1}{\sin x\sqrt{\cos x}} \)

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) \(P(\overline{A}\cap\overline{B}) = [1 - P(A)]\,[1 - P(B)]\).

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 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) \(-\frac{2}{3}\)

If \(P(A) = 0.3\), \(P(B) = 0.7\) and \(P(B/A) = 0.5\), then \(P(A \cup B)\) is :

• (A) \(0.35\)
• (B) \(0.85\)
• (C) \(0.5\)
• (D) \(0.21\)

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.

Find the value of \(x\) if \(A = B\), where

\[ A = \begin{bmatrix} 6 & 3x \\ 8 & -6 \end{bmatrix} \text{ and } B = \begin{bmatrix} 6 & x^2+2 \\ 8 & x^2-5x \end{bmatrix}. \]

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 \(P(\overline{A} / \overline{B})\).

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 \(\dfrac{d^2y}{dx^2} = \dfrac{-48y}{(6y^2 + 1)^3}\).

For the following Linear Programming Problem, shade the feasible region on the graph and find the coordinates of its corner points :

Minimize \(Z = 18x + 10y\)

subject to constraints

\[ \begin{aligned} 4x + y &\ge 20 \\ 2x + 3y &\ge 30 \\ x, y &\ge 0. \end{aligned} \]

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.

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 \(y = \sqrt{\log \left\{ \sin \left( \dfrac{x^2}{3} + 1 \right) \right\}}\), then show that

\[ \frac{dy}{dx} = \frac{x \cot \left( \dfrac{x^2}{3} + 1 \right)}{3y}. \]

Find the product of the maximum and minimum values of the function \( f(x) = \cos x - \frac{1}{2}\cos 2x,\ x \in \left[0, \frac{\pi}{2}\right] \).

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.

Solve the following Linear Programming Problem graphically :

Minimize \(Z = 4x + y\)

subject to

\(x + y \geq 3\)

\(x + 2y \geq 4\)

\(x, y \geq 0\).

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.

Evaluate :

\[ I = \int_{-\pi}^{\pi} \frac{2x\,(1+\sin x)}{1+\cos^2 x}\,dx \]

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 \( x = 2 \), \( x = -2 \).

Solve the differential equation \( y + \frac{d}{dx}(xy) = x(\sin x + \log x) \).

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.

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.

(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.