List of practice Questions

A car moves at a speed of \( 20 { m/s} \) on a banked track and describes an arc of a circle of radius \( 40\sqrt{3} \) m. The angle of banking is: (Take \( g = 10 { m/s}^2 \))

What will be the maximum speed of a car on a road turn of radius 30m if the coefficient of friction between the tyres and the road is 0.4? (Take \( g = 9.8 { m/s}^2 \))

A person aiming to reach the exactly opposite point on the bank of a stream is swimming with speed of \( 0.5 \) m/s at an angle of \( 120^\circ \) with the direction of flow of water. The speed of water in the stream is:

Let the probability distribution of random variable X be

X	-2	-1	1	2	3
P(X=x)	k	2k	2k	k	3k

Then, the value of E(X²) is

The argument of the complex number \[ \left( \frac{i}{2} - \frac{2}{i} \right) \] is equal to:

The lines $$ p(p^2 + 1)x - y + q = 0 \quad {and} \quad (p^2 + 1)^2 x + (p^2 + 1)y + 2q = 0 $$ are perpendicular to a common line for:

If 

and $(A + B)^2 = A^2 + B^2$ then $x + y$ is:

The area of the parallelogram whose diagonals are \[ \mathbf{d_1} = \frac{3}{2} \hat{i} + \frac{1}{2} \hat{j} - \hat{k}, \quad \mathbf{d_2} = 2 \hat{i} - 6 \hat{j} + 8 \hat{k} \] is:

Find the missing number in the sequence: \[ 285, 253, 221, 189, ? \]

Write the principle of Wheatstone's Bridge. Find the current measured by the ammeter in the given circuit diagram.

Explain Biot-Savart law and find the unit of \( \mu_0 \) with the help of the Biot-Savart equation.

Let \(\vec{a}\) = 2\(\widehat{i}\)+3\(\widehat{j}\)+4\(\widehat{k}\), \(\vec{b}\) = \(\widehat{i}\)-2\(\widehat{j}\)-2\(\widehat{k}\), \(\vec{c}\) = -\(\widehat{i}\)+4\(\widehat{j}\)+3\(\widehat{k}\) and \(\vec{d}\) is a vector perpendicular to \(\vec{b}\) and \(\vec{c}\),  \(\vec{a}\).\(\vec{d}\) = 18, then find |\(\vec{a}\)x\(\vec{d}\)|²

Use any one of the following pairs of words in your own sentences to make the difference in their meanings clear: Birth — Berth
Dose — Doze

If the vertex of a parabola is \( (2, -1) \) and the equation of its directrix is \[ 4x - 3y = 21, \] then the length of its latus rectum is:

Eccentricity of ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if it passes through point $(9, 5)$ and $(12, 4)$ is:

The particular solution of \[ \log \frac{dy}{dx} = 3x + 4y, \quad y(0) = 0 \] is:

The angle between the two lines: \[ \frac{x + 1}{2} = \frac{y + 3}{2} = \frac{z - 4}{-1} \] \[ \frac{x - 4}{1} = \frac{y + 4}{2} = \frac{z + 1}{2} \] is:

If \[ \left| \frac{\sec(x - y)}{\sec(x + y)} \right| = a \] then \( \frac{dy}{dx} \) is:

Evaluate the definite integral: \[ I = \int_{0}^{\frac{\pi}{2}} (\sqrt{\tan x} + \sqrt{\cot x})dx \]

The derivative of \[ \sin^{-1} \left(\frac{2x}{1 + x^2} \right) \] with respect to \[ \cos^{-1} \left(\frac{1 - x^2}{1 + x^2} \right) \] is equal to:

Evaluate the integral: \[ I = \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx \]