List of practice Questions

Judiciary has become the centre of controversy, in the recent past, on account of the sudden 'Me' in the level of judicial intervention. The area of judicial intervention has been steadily expanding through the device of public interest litigation. The judiciary has shed its pro-status-quo approach and taken upon itself the duty to enforce the basic rights of the poor and vulnerable sections of society, by progressive interpretation and positive action. The Supreme Court has developed new methods of dispensing justice to the masses through the public interest litigation. Former Chief Justice PN. Bhagwat, under whose leadership public interest litigation attained a new dimension comments that "the Supreme Court has developed several new commitments. It has carried forward participative justice".

\( f(x) = \left\{ \begin{array}{ll} \sin \frac{1}{x} , & \text{for} \, x \neq 0 \\ 0, & \text{for} \, x = 0 \end{array} \right. \) is:

In the questions below, a sentence is given, a part of which is printed in bold and underlined. This part may contain a grammatical error. Each sentence is followed by phrases a, b, c and d. Find out which phrase should replace the phrase given in bold/underline to correct the error, if there is any, to make the sentence grammatically meaningful and correct.
Recent incidents of tigers straying have brought to focus the lack of proper regulatory mechanism and powers with the forest department to take action against the resorts mushroom in forest fringes.

Choose the best pronunciation of the word 'Mischievous' from the following options:

The value of \[ \int_{-1}^{1} (x - \lfloor x \rfloor) dx \quad (\text{where} \, \lfloor \cdot \rfloor \, \text{denotes the greatest integer function}) \] is:

The correct evaluation of \[ \int_0^{\frac{\pi}{2}} \sin^4 x \, dx \] is:

The order and degree of the differential equation whose solution is \( y = cx + c^2 - 3c^3y^2 + 2 \), where \( c \) is a parameter, is:

The solution of \[ \frac{dv}{dt} + \frac{k}{m} v = -g \] is:

A unit vector perpendicular to the plane formed by the points \( (1, 0, 1) \), \( (0, 2, 2) \), and \( (3, 3, 0) \) is:

If \( \mathbf{a} = ( \hat{i} + \hat{j} + \hat{k} ) \), \( \mathbf{a} \times \mathbf{b} = \hat{i} - \hat{j} \), then \( \mathbf{b} \) is:

The mean and variance of a random variable \( X \) having binomial distribution are 4 and 2 respectively, then \( P(X = 1) \) is:

The radius of the sphere \[ x^2 + y^2 + z^2 = 49, \quad 2x + 3y - z - 5\sqrt{14} = 0 \] is:

One of the values of \[ \left( \frac{1 + i}{\sqrt{2}} \right)^{2/3} \] is:

The value of \( \lambda \) does the line \( y = x + \lambda \) touches the ellipse \( 9x^2 + 16y^2 = 144 \) is:

The combined equation of the asymptotes of the hyperbola \[ 2x^2 + 5xy + 2y^2 + 4x + 5y = 0 \] is:

The two curves \( x^3 - 3x^2 + 2 = 0 \) and \( 3x y - y^3 - 2 = 0 \) intersect at an angle of:

If at \( x = 1 \), the function \( x^4 - 62x^2 + ax + 9 \) attains its maximum value on the interval \( [0, 2] \), then the value of \( a \) is:

The value of \( c \) in Rolle's Theorem for the function \[ f(x) = e^x \sin x, \, x \in [0, \pi] \] is:

The area of the region bounded by the curve \( x = 2y + 3 \) and lines \( y = 1 \) and \( y = -1 \) is:

A signal which can be green or red with probability \( \frac{4}{5} \) and \( \frac{1}{5} \) respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is \( \frac{3}{4} \). If the signal received at station B is given, then the probability that the original signal is green is:

The value of the determinant \( \Delta = \begin{vmatrix} 1 & 4 & 3 \\ 0 & 12 & 9 \\ 1 & 2 & 2 \end{vmatrix} \) is:

If the equations \( x + ay = 0 \), \( 2x + y + az = 0 \), \( ax + y + 2z = 0 \) have non-trivial solutions, then \( a = \):