List of top Questions asked in UP Board XII

(c) If the unit vectors \( \vec{a}, \vec{b}, \vec{c} \) are such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \):
(b) If \( A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix} \), find \( A^{-1} \).

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \] 

If the matrices \( X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} \) and \( X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} \), then find the matrices \( X \) and \( Y \).
If \( y = (\cot x)^{\sin x} + x^x \), then find \( \frac{dy}{dx} \).
If \[ A = \begin{bmatrix} 3 & 1 & 1 \\ 15 & 6 & 5 \\ 5 & 2 & 2 \end{bmatrix} \] then find \( A^{-1} \).
Solve the following system of equations by using matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]
(d) If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \), show that \( A^2 - 5A + 7I = 0 \) and find \( A^{-1} \):
(d) If \[ \begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}, \] then the values of \( x \) and \( y \) will be:
(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that \[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}. \]
(b) Show that \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \] :
(a) If \( x = a(\theta + \sin \theta) \), \( y = a(1 - \cos \theta) \), find \( \frac{dy}{dx} \):
(d) Find the value of \( \int \log x \, dx \):
(b) Find the value of \( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \):
(e) If \( y = \log_e(\tan x) \), find \( \frac{dy}{dx} \):
(b) Two coins are tossed together. Find the probability of getting both tails:
(a) If \( A = \{1, 2, 3\} \), \( B = \{4, 5\} \), then find the number of relations from \( A \) to \( B \):
(e) If \(A\) and \(B\) are two invertible matrices of order \(n\), then:

If \[ y = 500 e^{7x} + 600 e^{-7x}, \quad \text{then show that} \quad \frac{d^2 y}{dx^2} = 49y. \] 

Find the area of the region enclosed by the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

Prove that \[ \int_0^{\pi} \frac{x \tan x}{\sec x + \tan x} \, dx = \frac{\pi}{2} (\pi - 2). \] 

Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \( R \) is \( \frac{2R}{\sqrt{3}} \). 

If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & -\cos \alpha \end{bmatrix} \] , find \( \text{adj}(A) \) and \( A^{-1} \).

Solve the differential equation \[ (x + y) \, dy + (x - y) \, dx = 0, \quad \text{if} \quad y = 1 \text{ when } x = 1. \] 

Show that the vectors \( 2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}, 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle.