List of top Mathematics Questions asked in UP Board XII

Find the shortest distance between the lines: \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}), \] \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}). \]

(c) If \( 2P(A) = P(B) = \frac{5{13} \) and \( P(A / B) = \frac{2}{5} \), then find \( P(A \cap B) \).}

(a) If \( x^y = e^{x-y} \), then prove that \( \frac{dy}{dx} = \frac{\log_e x}{(1 + \log_e x)^2 \).}

(b) The value of \( \int x \sin x \, dx \) will be:

(e) The angle between the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} \) and \( 2\hat{i} + \hat{j} + 3\hat{k} \) will be:

(c) The modulus function \( f: \mathbb{R} \to \mathbb{R}^+ \) is given by \( f(x) = |x| \); then it will be:

(b) If \( \sin^{-1} \left( \frac{1}{2} \right) = \tan^{-1 x \), then find the value of \( x \).}

(a) Differential coefficient of \( \sin^{-1}(e^{-x}) \) will be:

(b) Find the value of: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]

(b)(i) Find the angle between the pair of lines: \[ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}, \quad \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}. \]

(a) Solve the system of linear equations by the matrix method: \[ 2x - 3y + 5z = 11, \quad 3x + 2y - 4z = -5, \quad x + y - 2z = -3. \]

(b)(ii) If the coordinates of midpoints of the sides of a triangle are: \[ (1, 5, -1), \quad (0, 4, -2), \quad (2, 3, 4), \] find the coordinates of its vertices.

(a) Find the shortest distance between the lines: \[ \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}, \quad \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}. \]

(a) Prove that: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \sin x}{1 + \cos^2 x} \, dx = \frac{\pi^2}{4}. \]

(b) Minimize \( Z = x + 2y \) under the following constraints: \[ 2x + y \geq 3, \quad x + 2y \geq 6, \quad x \geq 0, \quad y \geq 0. \]