List of top Questions asked in UP Board XII

Find the principal value of \( {cosec}^{-1} \left( -\sqrt{2} \right) \).

Find the differential coefficient of \( \cos(\sin x^2) \) with respect to \( x \).

Solve: \( \frac{dy}{dx} = -4xy^2 \).

Integrate \( \log_e x \) with respect to \( x \).

Evaluate: \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \).

Find the area of the parallelogram whose adjacent sides are represented by the vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).

A relation \( R \) is defined in \( \mathbb{N} \times \mathbb{N} \) as follows: \[ (a, b) \, R \, (c, d) { if and only if } ad = bc. \] Prove that \( R \) is an equivalence relation.

Solve the differential equation \( \frac{dy}{dx} + y = 1 \) (\( y \neq 1 \)).

Prove that: \( 3 \sin^{-1} x = \sin^{-1} (3x - 4x^3) \), \( x \in \left[ -\frac{1}{2}, \frac{1}{2} \right] \).

Show that the points \( A(2, 3, -4), B(1, -2, 3) \) and \( C(3, 8, -11) \) are collinear.

Solve: \( y \, dx - (x + 2y^2) \, dy = 0 \).

If the lines \[ \frac{x - 1}{3} = \frac{y - 2}{2k} = \frac{z - 3}{2} \quad {and} \quad \frac{x - 1}{3k} = \frac{y - 1}{1} = \frac{z - 6}{-5} \] are perpendicular, find the value of \( k \).

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

Show on the map
Country in Europe with the highest Human Development Index (2019).

Show on the map
One important river of North America.

Write the name of the country with the largest population (in 2023).

Write the name of the country in Europe with the highest Human Development Index (2019).

Write the name of the capital city of Japan.