List of top Mathematics Questions asked in CBSE CLASS XII

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 \( P' = -P \).

The anti-derivative of \( \sqrt{1 + \sin 2x}, \, x \in \left[ 0, \frac{\pi}{4} \right] \) 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 \( M - m \).

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 \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right]. \)

Find the domain of the function \( f(x) = \sin^{-1}(x^2 - 4) \). Also, find its range.

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) If \[ A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] find \( (AB)^{-1} \). Also, find \( |AB|^{-1} \).

(b) Given \[ A = \begin{bmatrix} 2 & 3 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{bmatrix}, \text{ find } A^{-1}. \text{ Use it to solve the following system of equations:} \] \[ x + y + z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ x + y + 2z = 4. \]

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.