List of top Mathematics Questions asked in UP Board XII

(c) The value of the integral $ \int \sqrt{1 + \sin 2x} \,dx $ is:

(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $

Differentiate $ y = x^x + (\cos x)^{\tan x} $ with respect to $ x $.

The probabilities of solving a question by $ A $ and $ B $ independently are $ \frac{1}{2} $ and $ \frac{1}{3} $ respectively. If both of them try to solve it independently, find the probability that:

Suppose that $ A = \{ 1, 2, 3 \} $, $ B = \{ 4, 5, 6, 7 \} $, and $ f = \{ (1, 4), (2, 5), (3, 6) \} $ be a function from $ A $ to $ B $. Then $ f $ is:

(e) The angle between the vectors $ 3\hat{i} - 2\hat{j} + \hat{k} $ and $ 2\hat{i} + \hat{j} + 3\hat{k} $ will be:

Find the shortest distance between the lines: \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}), \] \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}). \]

If $ A $ and $ B $ are independent events, where $ P(A) = \frac{3}{10} $, $ P(B) = \frac{6}{10} $, then find $ P(A \cup B) $ and $ P(A \cap B) $.

Find the shortest distance between two lines $ l_1 $ and $ l_2 $ whose vector equations are: \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] and \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}). \]

Prove that a relation $ R = \{(a, b) : (a-b) $ is a multiple of $ 5 \} $ is an equivalence relation in the set of integers $ \mathbb{Z} $.

(c) Prove that \[ \begin{vmatrix} 1 \omega & \omega^2 \\ \omega & \omega^2 1 \\ \omega^2 1 \omega \end{vmatrix} = 0 \], where $ \omega $ is a cube root of unity.

Find $ x $ and $ y $ if \[ 2 \left[ \begin{array}{cc} 1 & 3 \\ 0 & x \end{array} \right] + \left[ \begin{array}{cc} y & 0 \\ 1 & 2 \end{array} \right] = \left[ \begin{array}{cc} 5 & 6 \\ 1 & 8 \end{array} \right]. \]

The given events $ A $ and $ B $ are such that $ P(A) = \frac{1}{4} $, $ P(B) = \frac{1}{2} $, and $ P(A \cap B) = \frac{1}{8} $; then find $ P(A' \cap B') $.

Show that the relation:

\[ R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \]

on the set $ \mathbb{Z} $ of integers is an equivalence relation.

Given:

\[ x \sqrt{1 + y} + y \sqrt{1 + x} + x = 0 \]

for $ -1 < x < 1 $, prove that:

\[ \frac{dy}{dx} = -\frac{1}{(1+x)^2}. \]

The differential coefficient of the $ \sin(x^2 + 5) $ with respect to $ x $ will be:

If

\[ A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix} \]

Then find $ AB $ and $ BA $.

Solve:

\[ \int \frac{\sin x}{\sin (x+a)} \, dx. \]

(d) If $ y = x^x $, then find $ \frac{dy}{dx} $.

(d) If $ P(A) = 0.5 $, $ P(B) = 0.4 $, and $ A $ and $ B $ are independent events, then find the values of (A) $ P(A \cap B) $ and (B) $ P(A \cup B) $.

If matrix \[ A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} \] and \[ A + A' = I, \] then the value of $ \alpha $ will be:

The value of the determinant \[ \begin{vmatrix} -2 & 5 \\ -4 & 1 \end{vmatrix} \] will be:

If

\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & -2 \\ -2 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -15 & 5 \\ -1 & 6 & -2 \\ 1 & -5 & 2 \end{bmatrix}, \]

then find the value of $ (AB)^{-1} $.

Differentiate: \[ y = (\tan x)^{\cot x} + (\sin x)^{\cos x}. \]

Find the interval in which the function \[ f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 \] is (A) increasing, (B) decreasing.