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

If \( 2\cos^{-1}x = y \), then

• (A) \( 0 \leq y \leq \pi \)
• (B) \( -\pi \leq y \leq \pi \)
• (C) \( 0 \leq y \leq 2\pi \)
• (D) \( -\pi \leq y \leq 0 \)

Which of the following cannot be the order of a row-matrix ?

• (A) \( 2 \times 1 \)
• (B) \( 1 \times 2 \)
• (C) \( 1 \times 1 \)
• (D) \( 1 \times n \)

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 skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

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

The least value of \( f(x) = x^3 - 12x,\ x \in [0, 3] \) is

• (A) \( -16 \)
• (B) \( -9 \)
• (C) \( 0 \)
• (D) \( 16 \)

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 \( \dfrac{d}{dx}\left(e^{y}\right) = 0 \) respectively are

• (A) \( 0,\ 1 \)
• (B) \( 1,\ 1 \)
• (C) \( 2,\ 1 \)
• (D) \( 1,\ \text{not defined} \)

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 point \( (2, 5, 7) \) on line \( \dfrac{x}{1} = \dfrac{y}{0} = \dfrac{z}{0} \) is

• (A) \( 2 \)
• (B) \( 5 \)
• (C) \( \sqrt{74} \)
• (D) \( \sqrt{78} \)

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

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.

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 Reason (R) is the correct explanation of Assertion (A).
• (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
• (C) Assertion (A) is true, but Reason (R) is false.
• (D) Assertion (A) is false, but Reason (R) is true.

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 \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of \(1\ \text{mm}^3/\text{min}\). Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is \(10\ \text{mm}\), if the semi-vertical angle of conical bottle is \(\dfrac{\pi}{6}\).

Find the vector of magnitude \(14\) in the direction of \(\overrightarrow{QP}\), where P and Q are the points \((1,\ 3,\ 2)\) and \((-1,\ 0,\ 8)\) respectively.

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

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

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

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

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 following differential equation:

\(x^{2}\dfrac{dy}{dx} = x^{2} + xy + y^{2}\)

Find the particular solution of the differential equation

\(xy\dfrac{dy}{dx} = (x+2)(y+2)\), given that \(y(1) = -1\).

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 \( P(X') + P(Y') = 2 - 2a + b \).

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 \( x \in [0, \pi/2] \).

If \(x = \cos t\), \(y = \cos mt\), prove that \((1-x^{2})\dfrac{d^{2}y}{dx^{2}} - x\dfrac{dy}{dx} + m^{2}y = 0\).

Check whether the lines given by \(\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}\) and \(\dfrac{x-4}{5} = \dfrac{y-1}{2} = z\) are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

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.

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.