List of top Questions asked in UP Board XII

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

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

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

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

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One important river of North America.

Write the name of the capital city of Japan.

Write the name of an important river of North America.

Write the name of an important sea port of South-East Asia.

Write the name of the State with the second largest population in India.

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Country with the largest population (in 2023).

Write the name of the capital city of Arunachal Pradesh.

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Capital city of Japan.

Write the name of an important sea port of Gujarat.

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An important sea port of South-East Asia.

Write the name of an oil refinery of this State (Gujarat).