In the circuit below, $$A$$ and $$B$$ represent two inputs and $$C$$ represents the output.

The circuit represents

Then which of the following is true?

This question contains Statement- 1 and Statement- 2. Of the four choices given after the statements, choose the one that best describes the two statements:
Statement- 1:
Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion and

Statement- 2:
For heavy nuclei, binding energy per nucleon increases with increasing $$Z$$ while for light nuclei it decreases with increasing $$Z.$$

A student measures the focal length of a convex lens by putting an object pin at a distance $$'u'$$ from the lens and measuring the distance $$'v'$$ of the image pin. The graph between $$'u'$$ and $$'v'$$ plotted by the student should look like

In an experiment, electrons are made to pass through a narrow slit of width $$'d'$$ comparable to their de Broglie wavelength. They are detected on a screen at a distance $$'D'$$ from the slit (see figure).

Which of the following graphs can be expected to represent the number of electrons $$'N'$$ detected as a function of the detector position $$'y'\left( {y = 0} \right.$$ corresponds to the middle of the slit$$\left. \, \right)$$

An experiment is performed to find the refractive index of glass using a travelling microscope. In this experiment distances are measured by

Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross-sectional area $$A=$$ $$10\,\,c{m^2}$$ and length $$=20$$ $$cm$$ . If one of the solenoid has $$300$$ turns and the other $$400$$ turns, their mutual inductance is
$$\left( {{\mu _0} = 4\pi \times {{10}^{ - 7}}\,Tm\,{A^{ - 1}}} \right)$$

A horizontal overhead powerline is at height of $$4m$$ from the ground and carries a current of $$100A$$ from east to west. The magnetic field directly below it on the ground is
$$\left( {{\mu _0} = 4\pi \times {{10}^{ - 7}}\,\,Tm\,\,{A^{ - 1}}} \right)$$

Relative permittivity and permeability of a material $${\varepsilon _r}$$ and $${\mu _r},$$ respectively. Which of the following values of these quantities are allowed for a diamagnetic material?

A $$5V$$ battery with internal resistance $$2\Omega $$ and a $$2V$$ battery with internal resistance $$1\Omega $$ are connected to a $$10\Omega $$ resistor as shown in the figure.

The current in the $$10\Omega $$ resistor is

Consider a block of conducting material of resistivity $$'\rho '$$ shown in the figure. Current $$'I'$$ enters at $$'A'$$ and leaves from $$'D'$$. We apply superposition principle to find voltage $$'\Delta V'$$ developed between $$'B'$$ and $$'C'$$. The calculation is done in the following steps:
(i) Take current $$'I'$$ entering from $$'A'$$ and assume it to spread over a hemispherical surface in the block.
(ii) Calculate field $$E(r)$$ at distance $$'r'$$ from A by using Ohm's law $$E = \rho j,$$ where $$j$$ is the current per unit area at $$'r'$$.
(iii) From the $$'r'$$ dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$.
(iv) Repeat (i), (ii) and (iii) for current $$'I'$$ leaving $$'D'$$ and superpose results for $$'A'$$ and $$'D'.$$

$$\Delta V$$ measured between $$B$$ and $$C$$ is

A body is at rest at $$x=0.$$ At $$t=0,$$ it starts moving in the positive $$x$$-direction with a constant acceleration. At the same instant another body passes through $$x=0$$ moving in the positive $$x$$ direction with a constant speed. The position of the first body is given by $${x_1}\left( t \right)$$ after time $$'t';$$ and that of the second body by $${x_2}\left( t \right)$$ after the same time interval. Which of the following graphs correctly describes $$\left( {{x_1} - {x_2}} \right)$$ as a function of time $$'t'$$ ?

Consider a block of conducting material of resistivity $$'\rho '$$ shown in the figure. Current $$'I'$$ enters at $$'A'$$ and leaves from $$'D'$$. We apply superposition principle to find voltage $$'\Delta V'$$ developed between $$'B'$$ and $$'C'$$. The calculation is done in the following steps:
(i) Take current $$'I'$$ entering from $$'A'$$ and assume it to spread over a hemispherical surface in the block.
(ii) Calculate field $$E(r)$$ at distance $$'r'$$ from A by using Ohm's law $$E = \rho j,$$ where $$j$$ is the current per unit area at $$'r'$$.
(iii) From the $$'r'$$ dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$.
(iv) Repeat (i), (ii) and (iii) for current $$'I'$$ leaving $$'D'$$ and superpose results for $$'A'$$ and $$'D'.$$

For current entering at $$A,$$ the electric field at a distance $$'r'$$ from $$A$$ is

This question contains Statement - $$1$$ and Statement - $$2$$. of the four choices given after the statements, choose the one that best describes the two statements.

Statement - $$1$$:

For a mass $$M$$ kept at the center of a cube of side $$'a'$$, the flux of gravitational field passing through its sides $$4\,\pi \,GM.$$

Statement - 2:

If the direction of a field due to a point source is radial and its dependence on the distance $$'r'$$ from the source is given as $${1 \over {{r^2}}},$$ its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.

Shown in the figure below is a meter-bridge set up with null deflection in the galvanometer.

The value of the unknown resister $$R$$ is

A thin spherical shell of radius $$R$$ has charge $$Q$$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $$E(r)$$ produced by the shell in the range $$0 \le r < \infty ,$$ where $$r$$ is the distance from the center of the shell?

A parallel plate capacitor with air between the plates has capacitance of $$9$$ $$pF.$$ The separation between its plates is $$'d'.$$ The space between the plates has dielectric constant $${k_1}$$ $$=3$$ and thickness $${d \over 3}$$ while the other one has dielectric constant $${k_2} = 6$$ and thickness $${{2d} \over 3}$$. Capacitance of the capacitor is now

A wave travelling along the $$x$$-axis is described by the equation $$y(x, t)=0.005$$ $$\cos \,\left( {\alpha \,x - \beta t} \right).$$ If the wavelength and the time period of the wave are $$0.08$$ $$m$$ and $$2.0s$$, respectively, then $$\alpha $$ and $$\beta $$ in appropriate units are

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of $$18$$ $$cm$$ during winter. Repeating the same experiment during summer, she measures the column length to be $$x$$ $$cm$$ for the second resonance. Then

A jar is filled with two non-mixing liquids $$1$$ and $$2$$ having densities $${\rho _1}$$ and $${\rho _2}$$ respectively. A solid ball, made of a material of density $${\rho _3}$$, is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $${\rho _1}$$ , $${\rho _1}$$ and $${\rho _3}$$ ?

A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in a soap-water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?

A thin rod of length $$'L'$$ is lying along the $$x$$-axis with its ends at $$x=0$$ and $$x=L$$. Its linear density (mass/length) varies with $$x$$ as $$k{\left( {{x \over L}} \right)^n},$$ where $$n$$ can be zero or any positive number. If the position $${X_{CM}}$$ of the center of mass of the rod is plotted against $$'n',$$ which of the following graphs best approximates the dependence of $${X_{CM}}$$ on $$n$$?

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $${V_1}$$ and contains ideal gas at pressure $${P_1}$$ and temperature $${T_1}$$. The other chamber has volume $${V_2}$$ and contains ideal gas at pressure $${P_2}$$ and temperature $${T_2}$$. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

The speed of sound in oxygen $$\left( {{O_2}} \right)$$ at a certain temperature is $$460\,\,m{s^{ - 1}}.$$ The speed of sound in helium $$(He)$$ at the same temperature will be (assume both gases to be ideal)

A spherical solid ball of volume $$V$$ is made of a material of density $${\rho _1}$$. It is falling through a liquid of density $${\rho _2}\left( {{\rho _2} < {\rho _1}} \right)$$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $$v,$$ i.e., $${F_{viscous}} = - k{v^2}\left( {k > 0} \right).$$ The terminal speed of the ball is

A planet in a distant solar system is $$10$$ times more massive than the earth and its radius is $$10$$ times smaller. Given that the escape velocity from the earth is $$11\,\,km\,{s^{ - 1}},$$ the escape velocity from the surface of the planet would be

Consider a uniform square plate of side $$' a '$$ and mass $$'m'$$. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

A block of mass $$0.50$$ $$kg$$ is moving with a speed of $$2.00$$ $$m{s^{ - 1}}$$ on a smooth surface. It strike another mass of $$1.0$$ $$kg$$ and then they move together as a single body. The energy loss during the collision is :

A block of mass $$0.50$$ $$kg$$ is moving with a speed of $$2.00$$ $$m{s^{ - 1}}$$ on a smooth surface. It strike another mass of $$1.0$$ $$kg$$ and then they move together as a single body. The energy loss during the collision is :

An athlete in the olympic games covers a distance of $$100$$ $$m$$ in $$10$$ $$s.$$ His kinetic energy can be estimated to be in the range

Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of − 0.03 mm while measuring the diameter of a thin wire, a student notes the main scale reading of 3 mm and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is

A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms⁻¹. The magnitude of its momentum is recorded as