List of top Mathematics Questions asked in UP Board XII

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

Show that the given function \( f, f(x) = x^3 - 3x^2 + 4x, x \in \mathbb{R} \) is an increasing function in \( \mathbb{R} \).

Find the value of the determinant \( \begin{vmatrix} x+y+2z & x & y \\ z & y+z+2x & y \\ z & x & z+x+2y \end{vmatrix} \).

Evaluate: \( \int \frac{dx}{(x+1)(x+2)} \).

If \(A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}\), find \((A+2B)'\).

If \(P(B) = \frac{9}{13}\) and \(P(A \cap B) = \frac{4}{13}\), find \(P(A|B)\).

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \), verify that \( A'A=I \).

Find the value of the determinant \( \begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{vmatrix} \).

If \(y=(\tan^{-1} x)^2\), show that \((x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2\).

Prove that \(\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x\), where \(-\frac{1}{\sqrt{2}} \leq x \leq 1\).

Test whether the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \begin{cases} x+5, & \text{if } x \le 1 \\ x-5, & \text{if } x > 1 \end{cases} \] is continuous at \(x=1\).

If \(R^*\) be the set of all non-zero real numbers, then the mapping \(f: R^* \to R^*\) defined by \(f(x) = \frac{1}{x}\) is

The degree of the differential equation \(7\left(\frac{d^3y}{dx^3}\right)^2 + 5\left(\frac{d^2y}{dx^2}\right)^3 + x\frac{dy}{dx} + y = 0\) will be

The relation \(R = \{ (a,b): b = a + 2 \}\) defined in the set \(A = \{1, 2, 3, 4, 5\}\) is

Solve: \((1+y^2) dx = (\tan^{-1} y - x) dy\).

If \( I = \int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \), find the value of I.

Solve: \(\frac{dy}{dx} = \frac{x+y+1}{2x+2y+3}\).

Prove that \( \int_{0}^{\pi/2} \sqrt{\frac{1+\cos 4x}{2}} \, dx = 1 \).

Find the shortest distance between the lines \(\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})\) and \(\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k})\).

A car is started to move from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \(x = t^2\left(3-\frac{t}{2}\right)\). Find the time required by the car to reach at the point Q and also find the distance between P and Q.

Solve the following system of equations by matrix method :
2x + 3y + 3z = 5
x - 2y + z = -4
3x - y - 2z = 3

If x, y, z are three independent events, then prove that \(P(X \cap Y \cap Z) = P(X) \cdot P\left(\frac{Y}{X}\right) \cdot P\left(\frac{Z}{X \cap Y}\right)\).

If \( I = \int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} \), then prove that \( I = \frac{\pi}{12} \).

For the two vectors \( \vec{a} \) and \( \vec{b} \), prove that \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) when \( \vec{a} \neq \vec{0} \) and \( \vec{b} \neq \vec{0} \).

Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).