List of top Mathematics Questions asked in UP Board XII

(d) If \( (\cos x)^y = (\cos y)^x \), find \( \frac{dy}{dx} \):
(b) Minimize \( Z = x + 2y \) under the following constraints: \[ 2x + y \geq 3, \quad x + 2y \geq 6, \quad x \geq 0, \quad y \geq 0. \]
(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):
(c) Solve: \[ \int \frac{3x + 5}{x^3 - x^2 - x + 1} \, dx. \]
(a) If a die is thrown three times, find the probability of getting one appearing number in them will be odd:
(a) Prove that for the two vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}| \):
(c) Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \):
(d) Prove that the function \( f(x) = |x| \) is not differentiable at \( x = 0 \):
(b) Find the differential coefficient of the function \( x^x \) with respect to \( x \):
(c) If the vectors \( \vec{v_1} = 3\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{v_2} = \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular, find the value of \( \lambda \):
(a) Prove that the function \( f(x) = |x - 1| \) is continuous at \( x = 1 \):
(d) The value of the expression: \[ \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \]
(b) Order of the differential equation: \[ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 \]
(c) The value of the integral: \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{dx}{1+x^2} \]
Solve by matrix method the system of equations \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \] \
The value of \( \int \sin^2 x \, dx \) is:
Prove that: \[ \int_0^{\pi/2} \sin 2x \tan^{-1} (\sin x) \,dx = \left(\frac{\pi}{2} - 1\right) \]
Evaluate: \[ I = \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \]
Solve the differential equation: \[ \frac{dy}{dx} + \frac{y}{x} = x^2, \quad y = 1 \text{ when } x = 1 \]
Solve the differential equation: \[ (\tan^{-1} y - x) \, dy = (1 + y^2) \, dx \]
Solve the system of equations by matrix method: \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \]

If matrix \[ A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 3 & -3 \\ -2 & -4 & -4 \end{bmatrix}, \] then find \( A^{-1} \).

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}, \] where \( n \in \mathbb{N} \).

Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad \text{and} \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \] and find the sine of the angle between them.
Find the area of the circle \( x^2 + y^2 = a^2 \).