List of top Questions asked in CBSE CLASS XII- 2026

Find the general solution of the differential equation \( (x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy \).

Solve the following Linear Programming Problem graphically :
Minimize $Z = 20x + 10y$
subject to the constraints:
$x + 2y \le 40$
$3x + y \ge 30$
$4x + 3y \ge 60$
$x, y \ge 0$

In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

Find whether the function $f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \ge 2 \end{cases}$ at $x = 2$ is differentiable or not.

Show that the function $f(x) = \begin{cases} \frac{\cos x}{\frac{\pi}{2} - x}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.

Let $n$ be a fixed positive integer. A relation $R$ is defined in the set of integers $\mathbb{Z}$ such that $R = \{(x, y) : (x - y) \text{ is divisible by } n, x, y \in \mathbb{Z}\}$. Determine if $R$ is an equivalence relation.

Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. A function $f : A \to B$ is defined by $f(x) = \frac{x - 2}{x - 3}$. Find whether $f$ is one-one and onto.

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then compute $A^2 - 4A - 5I$.

Find the interval(s) in which the function $f(x) = \frac{x}{\log x}$, where $x \in (0, 1) \cup (1, \infty)$, is increasing.

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

Evaluate : $\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\left(\frac{2\pi}{3}\right)\right)$}

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with the $x$-axis, $\frac{\pi}{3}$ with the $y$-axis and an acute angle $\theta$ with the $z$-axis. Find $\theta$ and the components of $\vec{a}$.

Let two rods placed on the ground be represented by vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $-2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

If $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(2\hat{i} + \hat{j} - \hat{k})$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : A function f : N $\to$ N given by f(x) = $x^3$ + 2, $\forall$ x $\in$ N is one-one but not onto.
Reason (R) : Since $\forall$ y $\in$ N (Codomain), there does not exist x = $(y - 2)^{1/3}$ in N (Domain) such that f(x) = $x^3$ + 2 = y.

The probability that it will rain tomorrow in cities A, B and C is \(60\%\), \(70\%\) and \(80\%\) respectively. Assuming these events are independent, the probability that it will rain tomorrow in at least one of the cities is:

The region represented by the system of linear inequalities \( 3x + y \ge 3 \), \( 2x - y \ge -5 \), and \( x, y \ge 0 \) is:

The greatest integer function, \(f(x) = [x]\), evaluated across the open domain boundary \(0 < x < 3\), is not differentiable at how many points?

The area of the region bounded by the curve \(y = x\) and the x-axis, between the boundaries \(x = 0\) and \(x = 2\) is:

The sum of the order and the degree of the differential equation \(\left[\frac{d^2y}{dx^2}\right]^2 + \left[\frac{dy}{dx}\right]^4 + 3\left(\frac{dy}{dx}\right) + 1 - y^2 = 0\) is:

Three points \( A(0, 1, 1) \), \( B(2, 0, -1) \) and \( C(1, 0, 3) \) form \(\Delta ABC\). The area of \(\Delta ABC\) is:

In a linear programming problem, the feasible region representing the maximization of the objective function \( Z = px + qy \), where \( p, q > 0 \), is a shaded polygonal region. If all points on the line segment joining the corner points \( A(0, 3) \) and \( B(3, 2) \) give the maximum value of \( Z \), then which of the following is true?

The differential of \(e^{e^x}\) with respect to \(x\) is:

If the distance travelled by a particle in \(t\) seconds is given by \(S = 72t + 3t^2 - t^3\), then the time taken by the particle to come to rest is:

If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1}e + k\), then the constant value of \(k\) is: