List of top Mathematics Questions asked in UP Board XII

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: \[ I = \int \frac{x^2 + x + 1}{(x+2)(x^2 + 1)} dx. \]
Find the value of the integral: \[ I = \int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx. \]
A die is thrown two times. It is found that the sum of the appeared numbers is 6. Find the conditional 64abecb1454319c1ac04f9e6 that the number 4 appeared at least one time.
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 \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}). \]
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.
Show that the points \( (2, -1, 1) \), \( (1, -3, -5) \) and \( (3, -4, -4) \) are the vertices of a right-angled triangle.
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 \( x \frac{dy}{dx} + 2y = x^2 \); \( x \neq 0 \).
Prove that a relation \( R = \{(a, b) : (a-b) \) is a multiple of \( 5 \} \) is an equivalence relation in the set of integers \( \mathbb{Z} \).
Find the value of \( \int \frac{x+2}{2x^2+6x+5} \,dx \).
Find such two positive numbers whose sum is 15 and sum of their squares is minimum.
There are two children in a family. If it is known that at least one child is a boy, find the 64abecb1454319c1ac04f9e6 that both children are boys.
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 \( CD \).
If \( A = \begin{bmatrix} 1 & -2 & 3
-4 & 2 & 5 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 3
4 & 5
2 & 1 \end{bmatrix} \), then find \( AB \) and \( BA \).
Prove that the 6266899d2bbfcb1799af2d57 \( f(x) = x^2 \) is continuous at \( x \neq 0 \).
Solve: \( \int \frac{\sin x}{\sin (x+a) dx \).}
Find the value of \( \int x^2 e^{x^3} dx \).
Find the area of a parallelogram whose adjacent sides are \( \vec{a} = 3\hat{i} + \hat{j} + 4\hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \).
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 \( P(A \cap B) \).}
Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).
If \( y = x^3 + \tan x \), then find \( \frac{d^2 y}{dx^2} \).