List of top Mathematics Questions asked in UP Board XII

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: \[ 4x + y \geq 80, x + 5y \geq 115, 3x + 2y \leq 150, x \geq 0, \; y \geq 0. \]
Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]
Prove that cone of the maximum volume inscribed in a sphere of radius $R$ is $\tfrac{8}{27}$ of the volume of the sphere.
From a well-shuffled pack of 52 cards, three cards were drawn one after another without any replacement. What is the probability that the first two cards be King and the third be Ace?
The probabilities of solving a particular problem by $A$ and $B$ independently are respectively $\tfrac{1}{2}$ and $\tfrac{1}{3}$. If both try to solve the problem independently, find the probability that (i) the problem is solved, (ii) only one of them solves the problem.
Find the intervals in which the function $f(x)=\sin x + \cos x,\; 0\leq x \leq 2\pi$ is increasing or decreasing.
Define symmetric and skew-symmetric matrices. If $A$ and $B$ are symmetric matrices, prove that $(AB-BA)$ is a skew-symmetric matrix.
Solve by matrix method the system of equations: \[\begin{aligned} 2x - 3y + 5z &= 11 \\ 3x + 2y - 4z &= -5 \\ x + y - 2z &= -3 \end{aligned}\]
Find the equation of the curve passing through $(-2,3)$ on which the gradient of the tangent at any point $(x,y)$ is \[ \frac{dy}{dx} = \frac{2x}{y^2}. \]
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 \[ \vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}). \]
Eliminating arbitrary constants $a$ and $b$, find the differential equation represented by the family of curves: \[ y^2 = a(b^2 - x^2) \]
Evaluate: \[ \int \sqrt{x^2 - 8x + 7}\, dx \]
Prove that \[ \begin{vmatrix} x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]
Find the point on the curve \[ y = x^3 - 11x + 5 \] at which the tangent is \[ y = x - 11. \]
If \[ A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, \] verify that $A'A = I$.
Prove that in the set $\mathbb{Z}$ of integers the relation $R$ given by $R = \{(a,b): 5 \text{ divides } (a-b)\}$ is an equivalence relation.
Find the vector and Cartesian equation of the plane which passes through the point $(1,0,-2)$ and on which the vector $\hat{i} + \hat{j} - \hat{k}$ is perpendicular.
If \[ x = a\left(\cos t + \log \tan \frac{t}{2}\right), y = a \sin t, \] find the value of $\dfrac{dy}{dx}$.
If $y = e^{a \cos^{-1}x}, \; -1 \leq x \leq 1$, then prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2 y = 0. \]
Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]
Evaluate: \[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \]
Test whether the function $f : \mathbb{R} \to \mathbb{R}$ defined by \[f(x) = \begin{cases} x^3 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}\] is continuous at $x=0$.
If $A$ and $B$ are two events such that $P(A) = 0.6$, $P(B) = 0.3$ and $P(A \cap B) = 0.15$, find $P(A|B)$.