The wavelength of the radiation emitted is \(\lambda_0\) when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be \(\frac{20}{x} \lambda_0\). The value of \(x\) is \(........\)

Match List-I with List-II.

LIST - I	LIST - II
(A) Isobaric	(I) ΔQ = ΔW
(B) Isochoric	(II) ΔQ = ΔU
(C) Adiabatic	(III) ΔQ = zero
(D) Isothermal	(IV) ΔQ = ΔU + PΔV

\(\Delta \mathrm{Q}=\) Heat supplied
\(\Delta \mathrm{W}=\) Work done by the system
\(\Delta \mathrm{U}=\) Change in internal energy
\(\mathrm{P}=\) Pressure of the system
\(\Delta \mathrm{V}=\) Change in volume of the system
Choose the correct answer from the options given below:

Given below are two statements: Statements \(I:\) The temperature of a gas is \(-73^{\circ}\,C\). When the gas is heated to \(527^{\circ}\,C\), the root mean square speed of the molecules is doubled. Statement \(II:\) The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules. In the light of the above statements, choose the correct answer from the options given below :

A thin tube sealed at both ends is \(100\, cm\) long. It lies horizontally, the middle \(20\, cm\) containing mercury and two equal ends containing air at standard atmospheric pressure . If the tube is now turned to a vertical position, by what amount will the mercury be displaced ? (Given : cross-section of the tube can be assumed to be uniform) ........ \(cm\)

Following figure shows two processes \(A\) and \(B\) for a gas. If \(\Delta Q_A\) and \(\Delta Q_B\) are the amount of heat absorbed by the system in two cases, and \(\Delta U_A\) and \(\Delta U_B\) are changes in internal energies, respectively, then

Let \(\eta_{1}\) is the efficiency of an engine at \(T _{1}=447^{\circ}\,C\) and \(T _{2}=147^{\circ}\,C\) while \(\eta_{2}\) is the efficiency at \(T _{1}=947^{\circ}\,C\) and \(T _{2}=47^{\circ}\,C\). The ratio \(\frac{\eta_{1}}{\eta_{2}}\) will be.

A water drop of radius \(1\,\mu m\) falls in a situation where the effect of buoyant force is negligible. Coefficient of viscosity of air is \(1.8 \times 10^{-5}\,Nsm ^{-2}\) and its density is negligible as compared to that of water \(10^{6}\,gm ^{-3}\). Terminal velocity of the water drop is________ \(\times 10^{-6}\,ms ^{-1}\) (Take acceleration due to gravity \(=10\,ms ^{-2}\) )

A water spray gun is attached to a hose of cross sectional area \(30\) cm\(^2\). The gun comprises of \(10\) perforations each of cross sectional area \(15\) mm\(^2\). If the water flows in the hose with the speed of \(50\) cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

The Coefficient of \(x ^{-6}\), in the expansion of \(\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9\), is \(........\).

A wire of resitance \(R\) and length \(L\) is cut into \(5\) equal part. if these parts are joined parts are joined paralley, than result resistance will be:

An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section \(8.0\times10^{-3}\, m^2\) . Initially the gas is at \(300\, K\) and occupies a volume of \(2.4\times10^{-3}\, m^3\) and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by \(0.1\, m\). The force constant of the spring is \(8000\, N/m\) and the atmospheric pressure is \(1.0\times10^5\, N/m^2\) . The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater . The heat capacity of the heater coil is also negligible)

If \([t]\) denotes the greatest integer \(\leq t\), then number of points, at which the function \(f ( x )=4|2 x +3|+\) \(9\left[x+\frac{1}{2}\right]-12[x+20]\) is not differentiable in the open interval \((-20,20)\), is\(.....\)

Let the plane \(ax \,\,+\,\,by \,\,+c z=d\) pass through \((2,3,-5)\) and is perpendicular to the planes \(2 x + y -5 z =10\) and \(3 x+5 y-7 z=12\). If \(a, b, c, d\) are integers \(d>0\) and gcd \((lal, |b|,|c|, d)\) \(=1\), then the value of \(a+7 b+c+20 \,d\) is equal to