List of top Questions asked in UP Board XII

Find the area of a parallelogram whose adjacent sides are given by vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).

The volume of a cube is increasing at the rate of 9 cm\(^3\)/s. If the length of its edge is 10 cm, then its surface area is increasing with which rate?

If \( y = x^{x^{x^{.... \text{ad inf}}}} \), then prove that \( x\frac{dy}{dx} = \frac{y^2}{1 - y \log x} \).

The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :

The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :

Find the principal value of \( \sec^{-1}(-\sqrt{2}) \).

Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).

If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.

If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).

Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)

The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :

If three vectors \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) satisfying the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \). If \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \) and \( |\vec{c}| = 2 \), then find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

The radius of an air bubble is increasing at the rate of \( \frac{1}{2} \) cm/s. At what rate is the volume of the bubble increasing while the radius is 1 cm?

The probability of solving a question by the three students A, B, C are respectively \( \frac{3}{10}, \frac{1}{5} \) and \( \frac{1}{10} \). Find the probability of solving the question.

Find the unit vector perpendicular to each of the vectors \( (\vec{a} + \vec{b}) \) and \( (\vec{a} - \vec{b}) \) where \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \).

If the Cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then find its equation in vector form.

There are 4 white and 2 black balls in a bag and in another bag 3 white and 5 black balls. Find the probability of getting both black balls if a ball is drawn from each bag.

Differentiate \( \tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right) \) with respect to x.

Prove that the function f(x) = |x| is continuous at x = 0.

Find the degree of the differential equation \( xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \)

If \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) be functions defined by \( f(x) = \cos x \) and \( g(x) = 3x^2 \) respectively, then prove that \( gof \neq fog \).

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

The value of expression \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} \) is

Explain with reason:
(i) Cr²⁺ is a reducing agent while Mn³⁺ is an oxidising agent, while both have d⁴ configuration.