List of top Mathematics Questions asked in UP Board XII

Find the differential coefficient of \( \tan^{-1}\left( \frac{2x}{1-x^2} \right) \) \text{with respect to} \( x \).

Find the equation of normal at the point \( (1, 1) \) of the curve \( x^{2/3} + y^{2/3} = 2 \).

Evaluate: \[ \int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx. \]

Find the equation of the plane passing through the points \((-2i + 6j - 6k), (-3i + 10j - 9k), (-5i - 6j - 6k)\).

If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.

Find the shortest distance between the lines \[ \vec{r} = (i + 2j + k) + \lambda(i - j + k) \] and \[ \vec{r} = (2i - j - k) + \mu(2i + j + 2k) \]

If two dice are thrown together, then find the probability of getting at least one 6.

Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).

Two integers among 1 to 11 are selected at random. If their sum is even, then find the probability that both integers are odd.

If \( \vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \), and \( \vec{c} = 3\hat{i} + \hat{j} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then find the value of \( \lambda \).

Find the area of the triangle whose two sides are represented by the vectors \[ \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - \hat{k}. \]

If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).

If \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \), prove that the events \( A \) and \( B \) are independent.

Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.

Prove: \[ \left| \begin{matrix} 1+\alpha & 1 & 1 \\ 1+\beta & 1 & 1 \\ 1 & 1 & 1+\gamma \\ \end{matrix} \right| = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}, \] prove that \[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix}. \]

If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), then the function from A to B will be

Solve the inequality \( 8x + 4<7x + 8 \).

Prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1 \), when \( x>2 \), and \( f(1) = f(2) = 1 \), is onto but not one-to-one.

The angle between the vectors \[ \mathbf{A} = 2\hat{i} + \hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \] will be

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

The value of \[ \int \cos^2 x \, dx \] will be

The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] will be