CBSE Class 12 Mathematics 2026 Question Paper

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

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

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)

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

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

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

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

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

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

The area bounded by the curve $y = x|x|$, x-axis and the ordinates $x = -1 \text{ and } x = 1$ is given by

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

The order and degree of the differential equation $\frac{d}{dx}(e^y) = 0$ respectively are

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

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

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

The length of the perpendicular drawn from the point $(2, 5, 7)$ on the line $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$ is

The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below. The maximum value of Z - minimum value of Z is

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

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, $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

Assertion (A): Lines given by $x = py + q$, $z = ry + s$ and $x = p'y + q'$, $z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$. 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) Check whether the function f(x) defined as $f(x) = \begin{cases} \dfrac{|x - 3|}{2(x - 3)}, & x < 3 \\ \dfrac{x-6}{6}, & x \ge 3 \end{cases}$ is continuous at $x = 3$ or not. OR (b) If $\sqrt{3} (x^2 + y^2) = 4xy$, then find $\dfrac{dy}{dx}$ at $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

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 $\pi/6$.

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

Vectors $a = 3i - 2j + 2k$ and $b = i + 2k$ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

(a) Simplify: $\tan^{-1}\left[\dfrac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x}\right]$, $0 < x < \pi/4$. OR (b) Evaluate: $\tan\left[\sin^{-1} 1 - \cos^{-1}(-1/2)\right]$.

Evaluate: integral from 0 to 1 of $x \tan^{-1} x \, dx$.

(a) Find: $\int \sqrt{\dfrac{x+2}{x-2}} \, dx$. OR (b) Find: $\int \dfrac{x^2}{(x^2+9)(x^2+16)} \, dx$.

If I_1 = integral from -pi/4 to pi/4 of dx/(1 + cos 2x) and I_2 = integral from -1/2 to 1/2 of |x| dx, then show that $I_1 - 4 I_2 = 0$.

(a) Find the general solution of the following differential equation: $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ OR (b) Find the particular solution of the differential equation $xy \frac{dy}{dx} = (x + 2)(y + 2)$, given that $y(1) = -1$.

Solve the following linear programming problem graphically: Minimize $Z = 13x - 15y$ subject to constraints $x + y \le 7$, $2x - 3y + 6 \ge 0$, $x \ge 0$, $y \ge 0$.

(a) 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. OR (b) The probability of simultaneous occurrence of at least 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) A relation R is defined on Z, the set of integers, as R = {(x, y) : |x - y| is divisible by a prime number 'p', x, y in Z}. Check whether R is an equivalence relation or not. OR (b) A function $f : \mathbb{R} - \{3/5\} \to \mathbb{R} - \{3/5\}$ is defined as $f(x) = \frac{3x + 2}{5x - 3}$. Show that f is one-one and onto.

(a) 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: $-2y + z = 7$, $2x - y - z = 8$, $x - 2y = 10$. OR (b) 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 $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0$.

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

How many subscribers will discontinue after an increase of Rs x in annual fee?

If $R(x)$ denotes the total revenue collected after the increase of Rs x in subscription fee, express $R(x)$ as a function of x.

(a) Find the value of x for which $R(x)$ is maximum. OR (b) Find the sub-intervals of $(0, 5000)$ in which $R(x)$ is increasing and decreasing.

What are the possible amounts, the person can win?

(a) What is the probability that the person wins atleast Rs 2,00,000? OR (b) What is the probability that the person does not win any amount?

In another jackpot, Rohan also bought a ticket having a prize money of Rs 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.

Represent the given equations C1 and C2 with the help of a diagram.

Express y as a function of x, ($y = f(x)$), for both C1 and C2.

(a) Using integration find the area of region covered by the roundabout. OR (b) Using integration, find the area of region covered by circular pond.