List of top Mathematics Questions asked in CBSE CLASS XII

Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors. Prove that \( |\vec{a} \times \vec{b}| \leq |\vec{a}| |\vec{b}| \). State the condition under which equality holds, i.e., \( |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \).
Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( | \overrightarrow{AB} | : | \overrightarrow{BC} | \).
Given \[ \frac{d}{dx} F(x) = \frac{1}{\sqrt{2x - x^2}} \] and \( F(1) = 0 \), find \( F(x) \).
Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx \]
Check the differentiability of \( f(x) \) at \( x = 1 \), where: \[ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1, \\ 3 - x, & 1 \leq x \leq 2. \end{cases} \]
If \( x = e^{y^2} \), prove that \[ \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2}. \]
Evaluate: \[ \sec^2 \left( \tan^{-1} \frac{1}{2} \right) + \csc^2 \left( \cot^{-1} \frac{1}{3} \right) \]
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?
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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 \( |a| = 2 \) and \( -3 \leq k \leq 2 \), then \( |a| |k| \in: \)
The derivative of \( 2^x \) w.r.t. \( 3^x \) is:
If \[ \begin{bmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \] has a determinant of \( \pm 6 \), then the value of \( k \) is:
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 \( \alpha, \beta \), and \( \gamma \) are the angles which a line makes with the 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 \( \vec{b} = \hat{i} - 2\hat{j} + \hat{k} \), 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:
If the sides of a square are decreasing at the rate of \( 1.5 \, \mathrm{cm/s} \), the rate of decrease of its perimeter is:
A function \( f(x) = |1 - x + |x|| \) is:
Let \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] be a square matrix such that \[ \text{adj } A = A. \] Then, \( (a + b + c + d) \) is equal to:
Let \( \mathbb{R}_+ \) denote the set of all non-negative real numbers. Then the function \( f : \mathbb{R}_+ \to \mathbb{R}_+ \) defined as \( f(x) = x^2 + 1 \) is:
If \( A = [a_{ij}] \) is an identity matrix, then which of the following is true?