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A cube is placed inside an electric field, \( \vec{E}=150y^{2}\,\hat{j} \). The side of the cube is \( 0.5 \, \text{m} \) and is placed in the field as shown in the given figure. The charge inside the cube is:

Two resistor \( R_{1}=(4 \pm 0.8)\, \Omega \) and \( R_{2}=(4 \pm 0.4)\, \Omega \) are connected in parallel. The equivalent resistance of their parallel combination will be:

The ranges and heights for two projectiles projected with the same initial velocity at angles \( 42^\circ \) and \( 48^\circ \) with the horizontal are \( R_{1}, R_{2} \) and \( H_{1}, H_{2} \) respectively. Choose the correct option:

A student determined Young’s Modules of elasticity using the formula \( Y=\dfrac{MgL^{3}}{4bd^{3}\,\delta} \). The value of \( g \) is taken to be \( 9.8 \, \text{m} / \text{s}^{2} \), without any significant error, his observation are as following.

Then the fractional error in the measurement of \( Y \) is:

A glass tumbler having inner depth of \( 17.5 \, \text{cm} \) is kept on a table. A student starts pouring water \( (\mu = \tfrac{4}{3}) \) into it while looking at the surface of water from the above. When he feels that the tumbler is half filled, he stops pouring water. Up to what height, the tumbler is actually filled?

A body of mass \( m \) dropped from a height \( h \) reaches the ground with a speed of \( 0.8\sqrt{g h} \). The value of work done by the air-friction is:

A mass of \( 5 \, \text{kg} \) is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length \( 4 \, \text{m} \) has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?

Following plots show magnetization \( (M) \) vs magnetizing field \( (H) \) and magnetic susceptibility \( (\chi) \) vs temperature \( (T) \) graph: Which of the following combination will be represented by a diamagnetic material?

Electric field of a plane electromagnetic wave propagating through a non-magnetic medium is given by \( E = 20 \, \sec(2 \times 10^{10}(t - 200x)) \, \text{V}/\text{m} \). The dielectric constant of the medium is equal to: (Take \( \mu_r = 1 \))

An object of mass \( m \) is being moved with a constant velocity under the action of an applied force of \( 2 \, \text{N} \) along a frictionless surface with following surface profile.

The correct applied force vs distance graph will be:

Due to cold weather a \( 1 \, \text{m} \) water pipe of cross-sectional area \( 1 \, \text{cm}^2 \) is filled with ice at \( -10^\circ \text{C} \). Resistive heating is used to melt the ice. Current of \( 0.5 \, \text{A} \) is passed through \( 4 \, \text{k}\Omega \) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required?
(Given latent heat of fusion for water/ice \( 3.33 \times 10^{5} \, \text{J} \, \text{kg}^{-1} \), specific heat of ice \( = 2 \times 10^{3} \, \text{J} \, \text{kg}^{-1} \) and density of ice \( = 10^{3} \, \text{kg} / \text{m}^{3} \))

A square loop of side \( 20 \, \text{cm} \) and resistance \( 1 \, \Omega \) is moved towards right with a constant speed \( v_0 \). The right arm of the loop is in a uniform magnetic field of \( 5 \, \text{T} \). The field is perpendicular to the plane of the loop and is going into it. The loop is connected to a network of resistors each of value \( 4 \, \Omega \). What should be the value of \( v_0 \) so that a steady current of \( 2 \, \text{mA} \) flows in the loop?

For the given circuit the current \( i \) through the battery when the key in closed and the steady state has been reached is:

Four particles each of mass \( M \), move along a circle of radius \( R \) under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is:

A block of mass \( m \) slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is:
Given \( m = 8 \, \text{kg} \), \( M = 16 \, \text{kg} \)
Assume all the surfaces shown in the figure to be frictionless

The temperature of an ideal gas in \( 3 \)-dimensional is \( 300 \, \text{K} \). The corresponding de-Broglie wavelength of the electron approximately at \( 300 \, \text{K} \), is:
[ \( m_{e} = \) mass of electron \( = 9 \times 10^{-31} \, \text{kg} \)
\( h = \) Planck constant \( = 6.6 \times 10^{-34} \, \text{J s} \)
\( k_{B} = \) Boltzmann constant \( = 1.38 \times 10^{-23} \, \text{J K}^{-1} \) ]

In the given figure, each diode has a forward bias resistance of \( 30 \, \Omega \) and infinite resistance in reverse bias. The current \( I_{1} \) will be:

There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductors is \( 1:1 \). The magnetic field at point \( P \) is:

The half life period of a radioactive element \( x \) is same as the mean life time of another radioactive element \( y \). Initially they have the same number of atoms. Then:

A uniform heating wire of resistance \( 36\,\Omega \) is connected across a potential difference of \( 240\,\text{V} \). The wire is then cut into half and a potential difference of \( 240\,\text{V} \) is applied across each half separately. The ratio of power dissipation in first case to the total power dissipation in the second case would be \( 1:x \), where \( x \) is:

An engine is attached to a wagon through a shock absorbed of length \( 1.5\,\text{m} \). The system with a total mass of \( 40{,}000\,\text{kg} \) is moving with a speed of \( 72\,\text{km}\,\text{h}^{-1} \) when the brakes are applied to bring it to rest. In the process of the system being through to rest, the spring of the shock absorber gets compressed by \( 1.0\,\text{m} \). If \( 90\% \) of energy of the wagon is lost due to friction, the spring constant is ______ \( \times 10^5\,\text{N/m} \).

When a body slides down from rest along a smooth inclined plane making an angle of \( 30^\circ \) with the horizontal, it take time \( T \). When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time \( \alpha T \), where \( \alpha \) is a constant greater than \( 1 \). The co-efficient of friction between the body and the rough plane is \(\dfrac{1}{\sqrt{x}}(\dfrac{a^2-1}{a^2}) \) where \(x=\)___________.

A steel rod with \( y=2.0\times10^{11}\ \text{Nm}^{-2} \) and \( \alpha=10^{-5}\,^\circ\text{C}^{-1} \) of length \( 4\,\text{m} \) and area of cross-sectional \( 10\,\text{cm}^2 \) is heated from \( ^\circ\text{C} \) and \( 400^\circ\text{C} \) without being allowed to extend. The tension produced in the rod is \( x\times10^5\,\text{N} \) where the value of \( x \) is:

The temperature of \( 3.00 \) mol of an ideal diatomic gas is increased by \( 40.0^\circ \text{C} \) without changing the pressure of the gas. The molecular in the gas rotate but do not oscillate. If the ratio of change in internal energy of the gas to the amount of work done by the gas is \( \frac{x}{10} \) . Then the value of \( x \) is:

Two satellites revolved around a plane in coplanar circular orbits in anticlockwise direction. Their period of revolutions are \( 1 \) hour and \( 8 \) hours respectively. The radius of the orbit of nearer satellite is \( 2 \times 10^3 \, \text{km} \). The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closet if \( \frac{\pi}{x} \, \text{rad} \, \text{h}^{-1} \) where \( x \) is:

The width of one of the two slits in a Young’s double slit experiment is three times the other slit. If the amplitude of the light coming from a slit is proportional to the slit-width, the ratio of minimum to maximum intensity in the interference pattern is \( x : 4 \) where \( x \) is:

The average translational kinetic energy of \( \mathrm{N_2} \) gas molecules at _____ \( ^\circ \text{C} \) becomes equal to the K.E. of an electron accelerated from rest through a potential difference of \( 0.1 \) volt. (Given \( k_B = 1.38 \times 10^{-23} \, \text{J} / \text{K} \) ).

A carrier wave with amplitude of \( 250 \, \text{V} \) is amplitude modulated by a sinusoidal base band signal of amplitude \( 150 \, \text{V} \). The ratio of minimum amplitude to maximum amplitude for the amplitude modulated wave is \( 50 : x \), then value of \( x \) is:

A \( 2 \, \text{kg} \) steel rod of length \( 0.6 \, \text{m} \) is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is _____ \( \text{ms}^{-1} \). (Take \( g = 10 \, \text{ms}^{-2} \))