Two stars of masses \(m\) and \(2\, m\) at a distance \(d\) rotate about their common centre of mass in free space. The period of revolution is

A parallel plate capacitor was made with two rectangular plates, each with a length of \(l=3 \mathrm{~cm}\) and breath of \(\mathrm{b}=1 \mathrm{~cm}\). The distance between the plates is \(3 \mu \mathrm{~m}\). Out of the following, which are the ways to increase the capacitance by a factor of 10 ?
A. \(l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
B. \(l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}\)
C. \(l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}\)
D. \(l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}\)
E. \(l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
Choose the correct answer from the options given below:

A solid body of constant heat capacity \(1\ J/^o C\) is being heated by keeping it in contact with reservoirs in two ways : \((i)\) Sequentially keeping in contact with \(2\) reservoirs such that each reservoir supplies same amount of heat. \((ii)\) Sequentially keeping in contact with \(8\) reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature \(100^o C\) to final temperature \(200^o C\). Entropy change of the body in the two cases respectively is :

The diameter of the objective lens of microscope makes an angle \(\beta \) at the focus of the microscope. Further, the medium between the object and the lens is an oil of refractive index \(n\). Then the resolving power of the microscope

A uniformly thick wheel with moment of inertia \(I\) and radius \(R\) is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\left(\mathrm{m}_{1}>\mathrm{m}_{2}\right)\) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when \(\mathrm{m}_{1}\) descents by a distance \(h\) is

A train is moving on a straight track with speed \(20\ ms^{-1}\). It is blowing its whistle at the frequency of \(1000\ Hz\). The percentage change in the frequency heard by a person standing near the track as the train passes him is ( speed of sound \(=320\) \(ms^{-1}\) ) close to .... \(\%\)

A spherical body of mass \(100 \mathrm{~g}\) is dropped from a height of \(10 \mathrm{~m}\) from the ground. After hitting the ground, the body rebounds to a height of \(5 \mathrm{~m}\). The impulse of force imparted by the ground to the body is given by : _______. (given \(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\) )

If \(n\) is the number of irrational terms in the expansion of \(\left(3^{1 / 4}+5^{1 / 8}\right)^{60},\) then \(( n -1)\) is divisible by

The angle of elevation of a cloud \(C\) from a point \(P , 200 m\) above a still lake is \(30^{\circ} .\) If the angle of depression of the image of \(C\) in the lake from the point \(P\) is \(60^{\circ},\) then \(P C\) (in \(m\) ) is equal to

Considering only the principal values of inverse functions, the set \(A = \left\{ {x \geq \,:\,{{\tan }^{ - 1}}\,\left( {2x} \right) + {{\tan }^{ - 1}}\,\left( {3x} \right)\, = \frac{\pi }{4}} \right\}\)

Let \(f(x)=\int_0^x g(t) \log _e\left(\frac{1-t}{1+t}\right) d t\), where \(g\) is a continuous odd function. If \(\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^2 \cos x}{1+e^x}\right) d x=\left(\frac{\pi}{\alpha}\right)^2-\alpha\), then \(\alpha\) is equal to ..............

Let \(P(3,2,3), Q(4,6,2)\) and \(R(7,3,2)\) be the vertices of \(\triangle \mathrm{PQR}\). Then, the angle \(\angle \mathrm{QPR}\) is

The number of turns of the coil of a moving coil galvanometer is increased in order to increase current sensitivity by \(50 \%\). The percentage change in voltage sensitivity of the galvanometer will be \(..........\%\)