List of top Questions asked in CBSE CLASS XII- 2026

Check whether the function \( f : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} \) defined as \( f(x, y) = (2y, 3x) \) is injective (one-to-one) or not.

Solve the following homogeneous differential equation: \( x \frac{dy}{dx} = y - x \sin^2\left(\frac{y}{x}\right) \), given the initial value condition that \( y = \frac{\pi}{6} \) when \( x = 1 \).

Find the general solution of the differential equation: \( y \log_e y \frac{dx}{dy} + x = \frac{y^2}{2} \).

Solve the following linear programming problem graphically: Maximize \( Z = 8000x + 12000y \) subject to the boundary constraints: \( 3x + 4y \le 60 \), \( x + 3y \le 30 \), with non-negativity restrictions \( x \ge 0, y \ge 0 \).

Find the coordinates of the foot of the perpendicular drawn from the origin \((0, 0, 0)\) to the straight line given by the equations \( \frac{x-2}{1} = \frac{y+1}{1} = \frac{z-3}{-1} \).

If \(\vec{AB} = \hat{j} + \hat{k}\) and \(\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}\) represent two vectors along the sides \(AB\) and \(AC\) of \(\Delta ABC\), prove that the median vector satisfies \(\vec{AB} + \vec{AC} = 2\vec{AD}\), where \(D\) is the midpoint of \(BC\). Hence, find the length of the median \(AD\).

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

Find the absolute maximum value of \( f(x) = \cos x + \sin^2 x \) in the closed interval \( x \in [0, \pi] \).

If $e^{x + y} = 3x$, then $\frac{dy}{dx}$ is equal to:

Assertion (A): $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.
Reason (R): $|\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b})\tan\theta$, where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ ($\theta \neq \frac{\pi}{2}$).

Assertion (A): A line can have direction cosines $\langle 1, 1, 1 \rangle$.
Reason (R): $\cos\theta = 1$ is possible for $\theta = 0$.

For two events $A$ and $B$ such that $P(A) \neq 0$ and $P(B) \neq 1$, the conditional probability $P(A'/B')$ is equal to:

If $A$ and $B$ are square matrices of the same order, then which of the following statements is/are always true?
(i) $(A + B)(A - B) = A^2 - B^2$
(ii) $AB = BA$
(iii) $(A + B)^2 = A^2 + AB + BA + B^2$
(iv) $AB = 0 \implies A = 0$ or $B = 0$

A relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 2), (2, 1), (2, 2)\}$ is:

For a square matrix $A$ of order $n$, the inverse matrix $(3A)^{-1}$ is equal to:

The direction cosines of the line given by $x = y = 1 - z$ are:

If $|\vec{a}| = 5$ and $-2 \le \lambda \le 1$, then the sum of the greatest and the smallest value of $|\lambda \vec{a}|$ is:

A vector of magnitude 3 making equal angles with the $x$ and $y$ axes and perpendicular to the $z$ axis is:

The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is:

In a linear programming problem, the linear function which has to be maximized or minimized is called:

For a feasible region bounded by the corner points $(0,3)$, $(3,2)$, and $(5,0)$ in the first quadrant, the non-trivial constraints of the linear programming problem are:

The indefinite integral $\int \frac{1}{1 + \cos x} \, dx$ is equal to:

For the function $f(x) = x + \frac{1}{x}$ ($x \neq 0$), which of the following statements is true?

Which of the following expressions will give the area of the region bounded by the curve $y = x^2$ and the line $y = 16$?

The general solution of the differential equation $x \, dy - y \, dx = 0$ is: