List of top Mathematics Questions asked in UP Board XII

If the coordinates of the points \( A \), \( B \), \( C \), and \( D \) are \( (1, 2, 3) \), \( (4, 5, 7) \), \( (-4, 3, -6) \), and \( (2, 9, 2) \) respectively, then find the angle between the lines AB and CD.

Find the value of \[ \int e^x \left( \tan^{-1} x + \frac{1}{1 + x^2} \right) dx. \] 

Prove that the number of equivalence relations in the set \( \{1, 2, 3\} \) including \( \{(1, 2)\} \) and \( \{(2, 1)\} \) is 2.

If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \] 

Integrate \[ \int \frac{\sin(\tan^{-1} x)}{1 + x^2} \, dx. \] 

Solve: \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \] 

If \( e^y(x + 1) = 1 \), show that \[ \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2. \] 

\[\cot^{-1} \left( - \frac{1}{\sqrt{3}} \right)\]

A relation \( R = \{(a, b) : a = b - 2, b \geq 6 \} \) is defined on the set \( \mathbb{N} \). Then the correct answer will be:

The order of the differential equation \[ \left( \frac{d^3 y}{dx^3} \right)^2 + x \left( \frac{dy}{dx} \right)^3 + 8y = \log_e x \] is:
If \( |\vec{a}| = \sqrt{3} \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{6} \), then the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is:
A relation \( R \) is defined on the set \( \mathbb{Q}^* \) of non-zero rational numbers as follows: \[ a \, R \, b \quad \text{if} \quad a = \frac{1}{b}. \] Then on \( \mathbb{Q}^* \), this relation \( R \) is:
Find the value of the integral: \[ I = \int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx. \]
Find the value of the integral: \[ I = \int \frac{x^2 + x + 1}{(x+2)(x^2 + 1)} dx. \]
Show that the points \( (2, -1, 1) \), \( (1, -3, -5) \) and \( (3, -4, -4) \) are the vertices of a right-angled triangle.
Find the value of \( p \) such that the lines \[ \frac{1 - x}{3} = \frac{7y - 14}{2p} = \frac{z - 3}{2} \] and \[ \frac{7 - 7x}{3p} = \frac{y - 5}{1} = \frac{6 - z}{5} \] are mutually perpendicular.
Find the maximum value of \( Z = 4x + y \) under the given constraints by graphical method: \[ x + y \leq 50, \quad 3x + y \leq 90, \quad x \geq 0, \quad y \geq 0. \]
Find the general solution of the differential equation \( x \frac{dy}{dx} + 2y = x^2 \); \( x \neq 0 \).
Find such two positive numbers whose sum is 15 and sum of their squares is minimum.
Find the value of \( \int \frac{x+2}{2x^2+6x+5} \,dx \).
Find the area of the circle \( x^2 + y^2 = a^2 \) surrounded by it.
If the position vectors of the points \( A, B, C, D \) are successively \( \hat{i} + \hat{j} + \hat{k} \), \( 2\hat{i} + 5\hat{j} \), \( 3\hat{i} + 2\hat{j} - 3\hat{k} \) and \( \hat{i} - 6\hat{j} + \hat{k} \), then find the angle between the lines \( AB \) and \( CD \).
Find the area of a parallelogram whose adjacent sides are \( \vec{a} = 3\hat{i} + \hat{j} + 4\hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \).
Find the value of \( \int x^2 e^{x^3} dx \).
Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).