A binary star consists of two stars \(A\) (mass \(2.2 M_S\) ) and \(B\) (mass \(11 M_s\) ), where \(M_S\) is the mass of the sun. They are separated by distance \(d\) and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of \(\operatorname{star} B\) about the centre of mass is

A massless rod \(B D\) is suspended by two identical massless strings \(A B\) and \(C D\) of equal lengths. A block of mass ' \(m\) ' is suspended point \(P\) such that \(B P\) is equal to ' \(x\) ', if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of \(x\) is

Two identical glass rods ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light $\mathrm{P}$ is placed inside rod ${\mathrm{S}}_1$ on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside ${\mathrm{S}}_2$ . The distance $\mathrm{d}$ is

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
The work done in bringing the dipole from infinity to a distance$r$from the centre of the loop by the given process is proportional to:

Statement 1 It is easier to pull a heavy object than to push it on a level ground.
and Statement 2 The magnitude of frictional force depends on the nature of the two surfaces in contact.

A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air column is the second resonance. Then,

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

Four combinations of two thin lenses are given in List I. The radius of curvature of all curved surfaces is r and the refractive index of all the lenses is 1.5. Match lens combinations in List I with their focal length in List II and select the correct answer using the code given below the lists.

	List I		List II
A.		$\mathrm{P}.$	$2\mathrm{r}$
B.		$\mathrm{Q}.$	$\frac{\mathrm{r}}{2}$
C.		$\mathrm{R}.$	$-\mathrm{r}$
D.		$\mathrm{S}.$	$\mathrm{r}$

One mole of an ideal gas is taken from \(a\) to \(b\) along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is \(W_s\) and that along the dotted line path is \(W_d\), then the integer closest to the ratio \(\frac{W_d}{W_s}\) is

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $\left(\mathrm{V}-\mathrm{T}\right)$ diagram. The correct statement(s) is/are:
[ $\mathrm{R}$ is the gas constant]

A thin flexible wire of length \(L\) is connected to two adjacent fixed points and carries a current \(I\) in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength \(B\) going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

The reaction of ${\mathrm{Xe}}_{}$ and ${\mathrm{O}}_2{\mathrm{F}}_2$ gives a $\mathrm{Xe}$ compound $\mathrm{P}$. The number of moles of $\mathrm{HF}$ produced by the complete hydrolysis of $1 \mathrm{mol}$ of $\mathrm{P}$ is _____ .

Let \(I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x\).
Then, for an arbitrary constant \(C\), the value of \(J-I\) equals