List of top Questions asked in UP Board XII

Find the shortest distance between the lines \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \] \text{and} \[ \vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k}. \]
Evaluate: \[ \int \left( \sqrt{\cot x} + \sqrt{\tan x} \right) dx \]
There are 500 students in a school, of which 230 are girls. Also, 10% of 230 girls are studying in class XII. Find the probability that a randomly chosen student is of XII class and is a girl.
Find the equation of the curve which is passing through the point \( (1, 1) \), and whose differential equation is \[ x \, dy = (2x^2 + 1) \, dx, (x \neq 0). \]
By graphical method solve the LPP under the following constraints: \[ x + 2y \geq 10, \] \[ 3x + 4y \leq 24, \] \[ x \geq 0, \, y \geq 0, \] then find the minimum value of \( z = 200x + 500y \).
Find the value of the expression \( i \cdot i + j \cdot j + 2k \cdot k \).
Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]
If \[ \Delta = \begin{vmatrix} 3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3 \end{vmatrix} \] then find the value of $|\Delta|$.
Find the degree of the differential equation \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]
If \( y = A \sin x + B \cos x \), then find the differential equation of it.
Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.
Prove that \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).
Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \] on which the normal is parallel to the x-axis.
If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.
If \( A = \begin{bmatrix} \cos a & -\sin a \\ \sin a & \cos a \end{bmatrix} \) and \( A + A' = I \), then find the value of \( a \).
The value of \[ \int \frac{dx}{x^2 - a^2} \] will be:
Prove that a one-one function \( f : \{2, 3, 4\} \to \{2, 3, 4\} \) is onto.
The value of $\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)$ will be:
The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \] will be:
If $f: X \rightarrow Y$ is an onto function, if and only if the range of $f$ will be:
Prove that \[ \int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\, dx = \frac{\pi^2}{4}. \]
If the area bounded by the curve $x=y^{2}$ and the line $x=4$ is divided into two equal parts by the line $x=a$, find the value of $a$.
Minimize $Z = 6x + 3y$ under the constraints: \[ 4x + y \geq 80, x + 5y \geq 115, 3x + 2y \leq 150, x \geq 0, \; y \geq 0. \]
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}). \]
Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]