List of top Questions asked in CBSE CLASS XII

Find the probability that the airplane will not crash.
(b) Find \( \frac{dy}{dx} \), \text{ if } \[ 5^x + 5^y = 5^{x + y}. \]
Evaluate: \[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
If vectors \( \vec{a}, \vec{b} \) and \( 2\vec{a} + 3\vec{b} \) are unit vectors, then find the angle between \( \vec{a} \) and \( \vec{b} \).
A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
Find: \[ \int x^2 \sin^{-1}(x^{3/2}) \, dx. \]
(a) Check the differentiability of \( f(x) = | \cos x | \) at \( x = \frac{\pi}{2} \).
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]
Evaluate: \[ \int_0^a \frac{x^2}{x^6 + a^6} \, dx \]
Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
If \[ \int_0^2 2e^{2x} \, dx = \int_0^a e^x \, dx, \text{ the value of 'a' is:} \]
If a line makes an angle of \( 30^\circ \) with the positive direction of \( x \)-axis, \( 120^\circ \) with the positive direction of \( y \)-axis, then the angle which it makes with the positive direction of \( z \)-axis is:
The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) is:
Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
The derivative \( \frac{d}{dx} [\cos(\log x + e^x)] \) at \( x = 1 \) is:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
The common region determined by all the constraints of a linear programming problem is called:
Find: \[ \int x^2 \sin^{-1}(x^{3/2}) \, dx. \]
Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.}
A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as: \[ (a, b) \, R \, (c, d) \iff a - c = b - d. \] Show that \( R \) is an equivalence relation.