List of top Mathematics Questions asked in UP Board XII

A die was thrown twice and it was found that the sum of the numbers that appeared was 6. Find the conditional probability that the number 4 appeared at least once.

Maximize \( z = 8000x + 12000y \) subject to constraints: \[ 3x + 4y \leq 60, \quad x + y \leq 30, \quad x \geq 0, \quad y \geq 0. \]

Let \[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}, \quad \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}, \quad \vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}, \] show that \[ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}. \]

Solve: \( \frac{dy}{dx} = \frac{x^2 + y^2}{2xy} \).

Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.

Prove: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \frac{\pi}{8} \log_e 2. \]

Evaluate: \[ \int \frac{x^2 + x + 1}{(x + 2)(x^2 + 1)} \, dx. \]