List of top Mathematics Questions asked in CBSE CLASS XII

Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( |\vec{AB}| : |\vec{BC}| \).
Given \( \frac{d}{dx}F(x) = \frac{1}{\sqrt{2x - x^2}} \) and \( F(1) = 0 \), find \( F(x) \).
Check the differentiability of \( f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1 \\ 3 - x, & 1 \leq x \leq 2 \end{cases} \) at \( x = 1 \).
If \( x = e^{x/y} \), prove that \( \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2} \).
Evaluate: \[ \cot^2 \left\{ \csc^{-1}(3) \right\} + \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right\}. \]
\( \vec{a}, \vec{b}, \vec{c} \) are three mutually perpendicular unit vectors. If \( \theta \) is the angle between \( \vec{a} \) and \( (2\vec{a} + 3\vec{b} + 6\vec{c}) \), find the value of \( \cos \theta \).

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.

If \( A \) and \( B \) are two skew-symmetric matrices, then \( AB + BA \) is:
Let \( E \) and \( F \) be two events such that \( P(E) = 0.1 \), \( P(F) = 0.3 \), \( P(E \cup F) = 0.4 \). Then \( P(F|E) \) is:
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:

If \( \alpha, \beta, \gamma \) are the angles which a line makes with positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
 

If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} - \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:
\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:
\( \int_{-a}^a f(x) \, dx = 0 \), if:
The rate of change of surface area of a sphere with respect to its radius \( r \), when \( r = 4 \, {cm} \), is:
Let \( \Delta = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a square matrix such that \( \text{adj} A = A \). Then, \( (a + b + c + d) \) is equal to:
Let \( \mathbb{Z} \) denote the set of integers. Then, the function \( f : \mathbb{Z} \to \mathbb{Z} \) defined as \( f(x) = x^3 - 1 \) is:
If \( A = [a_{ij}] \) is an identity matrix, then which of the following is true?
If \( A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \), then \( A^{-1} \) is:

Of the following, which group of constraints represents the feasible region given below? 

If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:
If \( |\vec{a}| = 2 \) and \( -3 \leq k \leq 2 \), then \( |k\vec{a} | \in: \)
If \( y = \sin^{-1} x \), then \( \frac{d^2 y}{dx^2} \) is:
The value of \[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix} \textbf{is:} \]
Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx. \]