List of top Mathematics Questions asked in UP Board XII

Prove that the semi-vertical angle of a cone with given slant height and maximum volume is \( \tan^{-1}(\sqrt{2}) \).
If \( f(x) = \begin{cases} -2 & \text{if } x \le -1 \\ 2x & \text{if } -1 < x \le 1 \\ 2 & \text{if } x > 1 \end{cases} \), then test the continuity of the function at \( x = -1 \) and at \( x = 1 \).
If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \), then find \( P(A \cup B) \).
If P(A) = 0.12, P(B) = 0.15 and P(B/A) = 0.18, then find the value of P(A \( \cap \) B).
For two invertible matrices A and B of order n, prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).
Prove that \( \int_{0}^{\pi/2} \sqrt{\frac{1+\cos 4x}{2}} \, dx = 1 \).
Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
The modulus function f: R \( → \) R\(^+\) given by f(x) = |x| is
Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).
Write \( \cot^{-1}\left\{\frac{1}{\sqrt{x^2-1}}\right\}; x>1 \) in the simplest form.
The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left(\frac{dy}{dx}\right)^2 \right\}^{\tfrac{3}{2}} \] is
Solve: \( \int \frac{(3x + 5)dx}{x^3 - x^2 - x + 1} \)
Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).
Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).
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 \(\int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2\).
If \(A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}\), find \(A^{-1}\). Using \(A^{-1}\) solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.
If \( f: \mathbb{R} \to \mathbb{R} \), is given by \( f(x) = (3 - x^3)^{1/3} \), then \( f \circ f(x) \) is equal to :
If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), then show that \(A^2 - 5A + 7I = O\). Using this, obtain \(A^{-1}\).
If \( y = \sin^{-1} x \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0 \).
If \(A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}\), then find \(A^{-1}\).
Solve: \( (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \).
Prove that \(\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2\).
At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is