List of practice Questions

The capacitance of a parallel plate capacitor with air as medium is \( 3 \, \mu \text{F} \) . With the introduction of a dielectric medium between the plates, the capacitance becomes \( 15 \, \mu \text{F} \) . The permittivity of the medium in SI units is \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \).

A potentiometer wire is 4 m long and potential difference of 3 V is maintained between the ends. The e.m.f. of the cell which balances against a length of 100 cm of the potentiometer wire is

The mass of the earth is 81 times the mass of the moon and the distance between their centres is \( R \). The distance from the centre of the earth where gravitational force will be zero is

Two wires A and B of equal lengths are connected in left and right gap of a metre bridge, null point is obtained at 40 cm from left end. Diameters of the wires A and B are in the ratio 3:1. The ratio of specific resistance of A to that of B is

The ratio of speed of an electron in the ground state in an atom to the velocity of light (c) is
$(h = \text{Planck's constant}, \epsilon_0 = \text{permittivity of free space})$

If rectangles are inscribed in a circle of radius \( r \) units, then the dimensions of the rectangle which has maximum area are

If a point \( P \) on the line segment joining the points \( (3, 5, -1) \) and \( (6, 3, -2) \) has its \( y \)-coordinate 2, then its \( z \)-coordinate is

The entries in the last column of the truth table for \( \neg (p \wedge q) \) are

If \( A, B, C \) are angles of a triangle ABC, then \( \tan 2A + \tan 2B + \tan 2C = \)

The growth of population is proportional to the number present. If the population of a colony doubles in 50 years, then the population will become triple in

Energy of the incident photon on the metal surface is ' 3W ' and then ' 5W ', where 'W' is the work function for that metal. The ratio of velocities of emitted photoelectrons is

If a fair coin is tossed 8 times, then the probability that it shows heads more than tails is

The distance of the point \( (7, 5, 2) \) from the plane \( 3x + 4y + 2z - 8 = 0 \) measured parallel to the line \( \frac{x - 1}{3} = \frac{y - 2}{6} = \frac{z + 1}{2} \) is

The acute angle between the lines given by \( y - \sqrt{3}x + 1 = 0 \) and \( \sqrt{3}y - x + 7 = 0 \) is

Evaluate the integral \( \int_0^5 \frac{dx}{x^2 + 2x + 10} \)

The volume of a tetrahedron whose vertices are \( A = (-1, 2, 3) \), \( B = (3, -2, 1) \), \( C = (2, 1, 3) \), and \( D = (-1, -2, 4) \) is

The joint equation of the lines through the origin trisecting angles in the first and third quadrant is

Evaluate \( \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \)

If \( \sin x + \sin^2 x = 1 \), then \( \cos^8 x + 2 \cos^6 x + \cos^4 x \) is

The perimeter of the triangle whose vertices have the position vectors \( \hat{i} + \hat{j} + \hat{k} \), \( 5\hat{i} + 3\hat{j} - 3\hat{k} \), and \( 2\hat{i} + 5\hat{j} + 9\hat{k} \) is

Evaluate \( \int_{-4}^{4} \log \left( \frac{8 - x}{8 + x} \right) dx \)

If \( \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \mathbf{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), and \( \mathbf{c} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) are coplanar, then \( \lambda \) is the root of the equation

If the tangent to the curve given by \( x = t^2 - 1 \) and \( y = t^2 - t \) is parallel to the x-axis, then the value of \( t \) is

If \( f \) and \( g \) are differentiable functions satisfying \( g'(a) = 2 \), \( g(a) = b \), and \( f \circ g = 1 \), where 1 is an identity function, then \( f'(b) \) is equal to

If the p.m.f. is given by \( P(X) = k \left( \frac{4}{x} \right) \) for \( x = 0, 1, 2, 3, 4 \) and \( k>0 \), then the value of \( k \) is