List of top Mathematics Questions asked in CBSE CLASS XII

Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
(a) If \( y = (\log x)^2 \), prove that \( x^2 y'' + x y' = 2 \).
If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
If \( f(x) = |\tan 2x| \), then find the value of \( f'(x) \) at \( x = \frac{\pi}{3} \).
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
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 \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
Assertion (A): For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R): A square matrix \( P \) is skew-symmetric if \( P' = -P \).
Determine whether the function \( f(x) = x^2 - 6x + 3 \) is increasing or decreasing in \( [4, 6] \).
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The anti-derivative of \( \sqrt{1 + \sin 2x}, \, x \in \left[ 0, \frac{\pi}{4} \right] \) is:
The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is:
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
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:
Let \( f : \mathbb{R}_+ \to [-5, \infty) \) be defined as \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. Then, \( f \) 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:
If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:
If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and \( [F(x)]^2 = F(kx) \), then the value of \( k \) is:
Find the value of \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right]. \)
If \( A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix} \), show that \( A^T A^{-1} = \begin{bmatrix} -\cos 2x & -\sin 2x \\ \sin 2x & -\cos 2x \end{bmatrix}. \)
If \( M \) and \( m \) denote the local maximum and local minimum values of the function \( f(x) = x + \frac{1}{x} \) (\( x \neq 0 \)) respectively, find the value of \( M - m \).

Airplanes are by far the safest mode of transportation when the number of transported passengers is measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is \( 0.00001\% \). Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let \( E_1 \) be the event that there is a plane crash and \( E_2 \) be the event that there is no crash. Let \( A \) be the event that passengers survive after the journey. On the basis of the above information, answer the following questions:
(i) Find the probability that the airplane will not crash.
(ii) Find \( P(A \,|\, E_1) + P(A \,|\, E_2) \).
(iii)
(a) Find \( P(A) \).
OR
(b) Find \( P(E_2 \,|\, A) \).