In the isothermal expansion of \(10\,g\) of gas from volume \(V\) to \(2V\) the work done by the gas is \(575\,J\). What is the root mean square speed of the molecules of the gas at that temperature ..... \(m/s\)?

In Young's double slit experiment, two slits \(S_1\) and \(S_2\) are ' \(d\) ' distance apart and the separation from slits to screen is \(D\) (as shown in figure). Now if two transparent slabs of equal thickness \(0.1\,mm\) but refractive index \(1.51\) and \(1.55\) are introduced in the path of beam \((\lambda=4000 \mathring A)\) from \(S_1\) and \(S_2\) respectively. The central bright fringe spot will shift by  \(..........\) number of fringes.

Three parallel plate capacitors \(\mathrm{C}_1, \mathrm{C}_2\) and \(\mathrm{C}_3\) each of capacitance \(5 \mu \mathrm{~F}\) are connected as shown in figure. The effective capacitance between points A and B, when the space between the parallel plates of \(C_1\) capacitor is filled with a dielectric medium having dielectric constant of 4, is :

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is:

\(5\) moles of unknown gas is heated at constant volume from \(10 \, °C\) to \(20 \, °C\). The molar specific heat of this gas at constant pressure \(c_p = 8\) cal/mol.°C and \(R = 8.36\) J/mol.°C. The change in the internal energy of the gas is _______ calorie.

Two blocks of masses \(m\) and \(M,(M \gt m)\), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
(\(\mu=\) coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is \(\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}\)
(B) The acceleration of the blocks is \(\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}\) (\(\mathrm{x}=\) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \(\frac{m \mu|x|}{M+m}\)
(D) The maximum amplitude of the upper block, if it does not slip, is \(\frac{\mu(M+m) g}{k}\)
(E) Maximum frictional force can be \(\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}\).
Choose the correct answer from the options given below:

A hypothetical gas expands adiabatically such that its volume changes from \(8\) litres to \(27\) litres. If the ratio of final pressure of the gas to initial pressure of the gas is \(\frac{16}{81}\). Then the ratio of \(\frac{C_P}{C_V}\) will be

The length of the latus rectum of a parabola, whose vertex and focus are on the positive \(x\)-axis at a distance \(\mathrm{R}\) and \(\mathrm{S}(\,>\,\mathrm{R})\) respectively from the origin, is:

If \(\alpha\) satisfies the equation \(x^2+x+1=0\) and \((1+\alpha)^7=\mathrm{A}+\mathrm{B} \alpha+\mathrm{C}^2, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \geq 0\), then \(5(3 A-2 B-C)\) is equal to ...........

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

Let \(f(\mathrm{x})=\mathrm{x} \cos ^{-1}(-\sin |\mathrm{x}|), \quad \mathrm{x} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) then which of the following is true?

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\) , then the probability that he is unable to solve less than two problems is

A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M\). At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2}\) , and the other mass, in a circular orbit of radius \(\frac{3R}{2}\). The difference between the final and initial total energies is