List of practice Questions

In which of the following case it was held that there could be no reservation on single post in the cadre?

Which of the following doctrine has been applied to resolve the conflict between Article 25(2)(b) and 26(b) of the Constitution of India?

A $100\, m$ long wire having cross-sectional area $625 \times 10^{-4} \, m ^2$ and Young's modulus is $10^{10} Nm ^{-2}$ is subjected to a load of $250 \, N$, then the elongation in the wire will be:

If \[ A = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \] is a non-singular matrix, then find \( A^{-1} \) by elementary row transformations.

In \( \Delta ABC \), if \( c^2 + a^2 - b^2 = ac \), then \( \angle B = \)

If \( p \land q \) is F, \( p \rightarrow q \) is F, then the truth values of \( p \) and \( q \) are

Which of the following is not included in the word costs as provided under the Code of Civil Procedure, 1908?

A moving coil galvanometer is converted into an ammeter of range 0 to 5mA. The galvanometer resistance is 90Ω and the shunt resistance has a value of 10Ω. If there are 50 divisions in the galvanometer-turnedammeter on either sides of zero, its current sensitivity is

Prove that the volume of a tetrahedron with coterminous edges \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) is \[ \text{Volume} = \frac{1}{6} \left| \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right| \] Hence, find the volume of the tetrahedron whose coterminous edges are \[ \overrightarrow{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \overrightarrow{b} = -\hat{i} + \hat{j} + 2\hat{k}, \quad \overrightarrow{c} = 2\hat{i} + \hat{j} + 4\hat{k}. \]

Find the length of the perpendicular drawn from the point \( P(3, 2, 1) \) to the line \[ \overrightarrow{r} = (7\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda(-2\hat{i} + 2\hat{j} + 3\hat{k}) \]

If \( y = \cos(m^{-1}x)) \), then show that \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2y = 0 \]

Verify Lagrange's mean value theorem for the function \( f(x) = \sqrt{x + 4} \) on the interval \([0, 5]\).

Evaluate: \[ \int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)} \, dx \]

Prove that: \[ \int_0^a f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx \]

If \( y = \sqrt{\tan{x} + \sqrt{\tan{x} + \sqrt{\tan{x} + \dots}}} \), then show that \[ \frac{dy}{dx} = \sec^2{x} \cdot \frac{2y - 1}{y^2 - 1} \] Find \( \frac{dy}{dx} \) at \( x = 0 \).

Find the approximate value of \( \sin{30^\circ 30'} \). Given that \( 1^\circ = 0.0175 \) radians and \( \cos{30^\circ} = 0.866 \).

Evaluate: \[ \int x \tan^{-1}{x} \, dx \]

Find the particular solution of the differential equation: \[ \frac{dy}{dx} = e^{2y} \cos{x}, \quad \text{when} \quad x = \frac{\pi}{6}, \, y = 0 \]

For the following probability density function of a random variable \( X \), find: \[ f(x) = \frac{x + 2}{18}, \quad \text{for} \, -2<x<4 \]

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.