List of practice Questions

A circular and a square coil is prepared from two identical metal wires and a current is passed through them. Ratio of magnetic dipole moment associated with circular coil to that with square coil is

Light of incident frequency 2 times the threshold frequency is incident on a photosensitive material. If the incident frequency is made \( \frac{1}{3} \)rd and intensity is doubled then the photoelectric current will

Evaluate the integral \( \int_0^1 \frac{x^2}{1 + x^2} dx \)

The general solution of the differential equation \( (1 - x^2) \frac{dy{dx} + 2xy = x(1 - x^2)^{\frac{1}{2}} \) is}

Simplify the expression \( \sin \left( \frac{\pi}{3} + x \right) - \cos \left( \frac{\pi}{6} + x \right) \)

If \( y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \), then \[ \frac{dy}{dx} = \]

Evaluate the integral \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \log |9e^{2x} - 4| + c \), then (where \( c \) is the constant of integration)

Two galvanometers 'G1' and 'G2' require 2 mA and 3 mA respectively to produce the same deflection. Then

The shortest distance between the lines \( 1 + x = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) is

The particular solution of the differential equation \( \cos \left( \frac{dy}{dx} \right) = a \), under the conditions \( a \in \mathbb{R} \) and \( y(0) = 2 \) is

If \( \sqrt{x + y} + \sqrt{y - x} = 5 \), then \( \frac{d^2y}{dx^2} = \)

The domain of the function \( f(y) = \frac{\cos^{-1(y - 5)}{\sqrt{25 - y^2}} \) is}

In a box containing 100 bulbs, 10 are defective. The probability that out of 20 bulbs selected at random, none is defective is

If \( \vec{a} = \hat{i} + 5\hat{k}, \, \vec{b} = 2\hat{i} + 3\hat{k}, \, \vec{c} = 4\hat{i} - \hat{j} + 2\hat{k}, \, \vec{d} = \hat{i} - \hat{j} \), then \( (\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d}) = \)

If \( f(x) = x^2 - 3x + 4 \) and \( f(x) = f(2x + 1) \), then \( x = \)

The area bounded by the parabola \( y^2 = 16x \) and its latus-rectum in the first quadrant is

20 meters of wire is available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, then the radius of the circle is

If the radius of a circle \( x^2 + y^2 - 4x + 6y - k = 0 \) is 5, then \( k = \)

The equation of the normal to the curve \( 2x^2 + 3y^2 - 5 = 0 \) at \( P(1, 1) \) is

The angle between the lines \( \frac{x - 1}{4} = \frac{y - 3}{1} = \frac{z}{8} \) and \( \frac{x - 2}{2} = \frac{y + 1}{2} = \frac{z - 4}{1} \) is

If the direction cosines of a line are \( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \), then

If \( \frac{1}{4}, a, b, \frac{1}{19} \) form a harmonic progression (H.P.), then the values of \( a \) and \( b \) are respectively

Evaluate \( \int_0^{\frac{\pi}{2}} \left( e^{\sin x} - e^{\cos x} \right) \, dx \)

The minimum value for the LPP \( Z = 6x + 2y \), subject to \( 2x + y \geq 16, x \geq 6, y \geq 1 \) is

A fair coin is tossed 2 times. A person receives \( X^3 \) if he gets \( X \) number of heads. His expected gain is