CBSE Class 12 Mathematics Set 2- (65/1/2) Question Paper 2026 with Solution PDF

If \( \int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K \), then the value of \( A \) is

• (A) \( 3a \)
  
• (B) \( \frac{3a}{2b^2} \)
  
• (C) \( \frac{3a}{b^2c^2} \)
  
• (D) \( \frac{3a}{2c^2} \)

The value of \( \int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1} dx \) is

• (A) \( 0 \)
  
• (B) \( \log 2 \)
  
• (C) \( 2 \log 2 \)
  
• (D) \( \frac{1}{2} \log 2 \)

The area bounded by the curve \( y = x|x| \), x-axis and the ordinates \( x = -1 \) and \( x = 1 \) is given by

• (A) \( 0 \)
  
• (B) \( \frac{1}{3} \)
  
• (C) \( \frac{2}{3} \)
  
• (D) \( \frac{4}{3} \)

The integrating factor of differential equation \( R \frac{dx}{dy} + Px = Q \) where \( P, Q, R \) are functions of \( y \) is

• (A) \( e^{\int \frac{P}{Q} dy} \)
  
• (B) \( e^{\int P dy} \)
  
• (C) \( e^{\int \frac{P}{R} dy} \)
  
• (D) \( e^{\int \frac{P}{R} dx} \)

The order and degree of the differential equation : \(\dfrac{d}{dx}(\sin y) = y^2\) respectively are

• (A) 1, 1
• (B) 2, 1
• (C) 2, 2
• (D) 1, 2

The value of \( p \) for which vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( 2\hat{i} - p\hat{j} + \hat{k} \) are perpendicular to each other is

• (A) \( 0 \)
  
• (B) \( 1 \)
  
• (C) \( \frac{5}{2} \)
  
• (D) \( -\frac{5}{2} \)

The value of \( m \) for which the points with position vectors \( -\hat{i} - \hat{j} + 2\hat{k} \), \( 2\hat{i} + m\hat{j} + 5\hat{k} \) and \( 3\hat{i} + 11\hat{j} + 6\hat{k} \) are collinear, is

• (A) \( 8 \)
  
• (B) \( -8 \)
  
• (C) \( 2 \)
  
• (D) \( \frac{5}{2} \)

If \( |\vec{a}| = 8 \), \( |\vec{b}| = 3 \) and \( |\vec{a} \times \vec{b}| = 12 \), then the value of \( |\vec{a} \cdot \vec{b}| \) is

• (A) \( 6\sqrt{3} \)
  
• (B) \( 8\sqrt{3} \)
  
• (C) \( 12\sqrt{3} \)
  
• (D) \( 3\sqrt{12} \)

The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is

• (A) \(2\)
• (B) \(6\)
• (C) \(\sqrt{10}\)
• (D) \(\sqrt{14}\)

The feasible region of a linear programming problem with objective function \( Z = 5x + 7y \) is shown below :

• (A) \( 8 \)
  
• (B) \( 29 \)
  
• (C) \( 35 \)
  
• (D) \( 43 \)

The degree of an objective function of a linear programming problem is

• (A) \( 0 \)
  
• (B) \( 1 \)
  
• (C) \( 2 \)
  
• (D) Any natural number

If \(\tan^{-1} x = 3y\), then

• (A) \(-\dfrac{\pi}{2} < y < \dfrac{\pi}{2}\)
• (B) \(-\dfrac{3\pi}{2} < y < \dfrac{3\pi}{2}\)
• (C) \(-\dfrac{\pi}{6} < y < \dfrac{\pi}{6}\)
• (D) \(-\dfrac{\pi}{6} \leq y \leq \dfrac{\pi}{6}\)

Which of the following cannot be an order of a column-matrix ?

• (A) \(1 \times 2\)
• (B) \(2 \times 1\)
• (C) \(1 \times 1\)
• (D) \(m \times 1\), where \(m \in N\)

Which of the following properties is/are true for two matrices of suitable orders ? (i) \( (A + B)' = A' + B' \)
(ii) \( (A - B)' = B' - A' \)
(iii) \( (AB)' = A'B' \)
(iv) \( (kAB)' = kB'A' \) (k is a scalar)

• (A) (i) only
  
• (B) (i), (ii) and (iii)
  
• (C) (i) and (ii)
  
• (D) (i) and (iv)

If \( \Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} \) and \( \Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix} \), then

• (A) \( \Delta_1 = 2\Delta_2 \)
  
• (B) \( \Delta_2 = -2\Delta_1 \)
  
• (C) \( \Delta_1 = \Delta_2 \)
  
• (D) \( \Delta_2 = -\Delta_1 \)

One of the values of \( x \) for which \( \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 \) is

• (A) \( 0 \)
  
• (B) \( \frac{\pi}{4} \)
  
• (C) \( \frac{\pi}{3} \)
  
• (D) \( \frac{\pi}{2} \)

If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?

• (A) \(A^2 + B^2\)
• (B) \(A^2 - B^2\)
• (C) \(AB + BA\)
• (D) \(AB - BA\)

The absolute maximum value of \(f(x) = x^2 + 1\) in \([-5, 2]\) is

• (A) 26
• (B) 1
• (C) 5
• (D) 2

Assertion (A) : Lines given by \(x = py + q\), \(z = ry + s\) and \(x = p^{\prime}y + q^{\prime}\), \(z = r^{\prime}y + s^{\prime}\) are perpendicular to each other when \(pp^{\prime} + rr^{\prime} = 1\).

Reason (R) : Two lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\) are perpendicular to each other if \(\vec{b}_1 \cdot \vec{b}_2 = 0\).

• (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 and Reason (R) is false.
• (D) Assertion (A) is false and Reason (R) is true.

Assertion (A) : In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is \( \frac{2}{3} \).
Reason (R) : For any two events \( A \) and \( B \), \( P(A|B) = \frac{P(A \cup B)}{P(B)} \).

• (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 and Reason (R) is false.
  
• (D) Assertion (A) is false and Reason (R) is true.

A vector \(\vec{a}\) of magnitude 14 has direction ratios \(\langle 2, 3, -6 \rangle\). Find the projection of the vector \(\vec{a}\) on \(\hat{i}\).

Vectors \( \vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{k} \) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Simplify : \( \tan^{-1} \left( \frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x} \right) \), \( 0 < x < \frac{\pi}{4} \).

Evaluate : \( \tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right) \).

Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not. 

If \( \sqrt{3}(x^2 + y^2) = 4xy \), then find \( \frac{dy}{dx} \) at \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

Simplify : \(\cot^{-1}\sqrt{\dfrac{1 + \cos 2x}{1 - \cos 2x}}\), \(x \in \left(0, \dfrac{\pi}{2}\right)\).

Evaluate : \(\sin\left(\tan^{-1}(-\sqrt{3}) - \sec^{-1} 2\right)\)

If \( I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x} \) and \( I_2 = \int_{-1/2}^{1/2} |x| dx \), then show that \( I_1 - 4I_2 = 0 \).

Find the general solution of the differential equation :

\(y^2\,dx + (x^2 - xy + y^2)\,dy = 0\)

Find the particular solution of the differential equation \(\dfrac{dy}{dx} = y\tan x\), given that \(y = 2\) if \(x = 0\).

Solve the following linear programming problem graphically :
Minimize \( Z = 13x - 15y \)
Subject to constraints
\( x + y \leq 7, \)
\( 2x - 3y + 6 \geq 0, \)
\( x \geq 0, y \geq 0 \)

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

The probability of simultaneous occurrence of atleast one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that \( P(X') + P(Y') = 2 - 2a + b \).

Evaluate : \( \displaystyle\int_{0}^{1} x \sin^{-1}x \, dx \)

Find \( \int \sqrt{\frac{x + 2}{x - 2}} dx \)

Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

If \( x = 3\sin t - \sin 3t,\ y = 3\cos t - \cos 3t \), find \( \dfrac{dy}{dx} \) and prove that \( \dfrac{d^2y}{dx^2} = \dfrac{-\operatorname{cosec}^3 2t \cdot \operatorname{cosec} t}{3} \).

Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.

A relation \(R\) is defined on \(Z\), the set of integers, as \[ R = \{(x,y) : |x-y| \text{ is divisible by a prime number } 'p', x,y \in Z\}. \] Check whether \(R\) is an equivalence relation or not.

A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

If \( A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix} \), find \( A^{-1} \) and use it to solve the following system of equations :
\( -2y + z = 7, 2x - y - z = 8, x - 2y = 10 \)

If \( \begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix} \) is a singular matrix, then find all values of \( x \) where \( x \in [0, \pi/2] \).

In an online jackpot, there is one first prize of Rs 3,00,000, two second prizes of Rs 2,00,000 each and three third prizes of Rs 50,000 each.

Roundabouts are often made on busy roads to ease the traffic and avoid red lights.

An online delivery company in a city has 5000 subscribers and collects annual subscription fees of Rs 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of Rs 1, ten subscribers will discontinue. Assume that the company increased the annual fee by Rs x.