List of practice Questions

The work done in bringing a unit positive charge from infinity to a given point against the direction of electric field is known as:
Phase difference between a node and an adjacent antinode in a stationary wave is:
The dimensional formula of surface tension is:
The moment of inertia (MI) of a disc of radius \( R \) and mass \( M \) about its central axis is:
Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x \quad (x \neq 0), \quad \text{when } y = 0 \text{ if } x = \frac{\pi}{2}. \]
Find the value of \[ \int \frac{x}{(x-a)(x-b)(x-c)} \, dx. \]
Find the area bounded by the circle \[ x^2 + y^2 - 2x = 8. \]
There are 4 red and 4 black balls in a bag and another bag contains 2 red and 6 black balls. One bag is randomly selected and a red ball is drawn. Find the probability that the red ball is drawn from the first bag.
If the position vectors of the vertices \( A, B, C \) of a triangle \( \Delta ABC \) are \( \vec{a}, \vec{b}, \vec{c} \), respectively, then prove that the area of \( \Delta ABC \) is \[ \frac{1}{2} \left| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \right|. \]
Prove that \[ \int_0^{\pi/2} \log \sin x \, dx = -\frac{\pi}{2} \log 2. \]
Find the value of \[ \int_0^{\pi/2} \frac{x}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx. \]
Solve the system of linear equations by matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]
(b) Show that the points \( A(2,3,4) \), \( B(-1,-2,1) \), and \( C(5,8,7) \) are collinear.
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).
Minimize \( Z = 3x + 9y \) under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]
(d) Solve: \( \frac{dy}{dx} = \frac{1 + x}{1 - y} \).
(e) Find the slope of the curve \( y = 2x^2 - 3\cos x \) at \( x = 0 \).
(b) Solve the differential equation \( \sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0 \).
(a) Find the slope of the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \).
(b) The value of \( \int x \sin x \, dx \) will be:
(c) The modulus function \( f: \mathbb{R} \to \mathbb{R}^+ \) is given by \( f(x) = |x| \); then it will be:
(a) Find the direction cosines of the vector \( \hat{i} + \hat{j} - 2\hat{k} \).
(a) Differential coefficient of \( \sin^{-1}(e^{-x}) \) will be:
(b) Find the value of: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]
(e) Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height is \( \tan^{-1} (\sqrt{2}) \):