\(27\) similar drops of mercury are maintained at \(10 \,V\) each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is ....... times that of a smaller drop.

Two Carnot engines \(A\) and \(B\) are operated in series. Engine \(A\) receives heat from a reservoir at \(600\,K\) and rejects heat to a reservoir at temperature \(T\). Engine \(B\) receives; heat rejected by engine \(A\) and in turn rejects it to a reservoir at \(100\,K\). If the efficiencies of the two engines \(A\) and \(B\) are represented by \({\eta _A}\) and \({\eta _B}\) respectively, then what is the value of \(\frac{{{\eta _A}}}{{{\eta _B}}}\)

The half-life of \({ }^{198} {Au}\) is \(3 \,days.\) If atomic weight of \({ }^{198} {Au}\) is \(198\, {g} / {mol}\) then the activity of \(2 \,{mg}\) of \({ }^{198} {Au}\) is ..... \(\times 10^{12}\,disintegration/second\)

Two charges \(7 \mu C\) and \(-2 \mu C\) are placed at \((-9,0,0) cm\) and \((9,0,0) cm\) respectively in an external field \(E=\frac{ A }{ r ^2} \hat{ r }\), where \(A=9 \times 10^5 N / C . m ^2\).
Considering the potential at infinity is 0 , the electrostatic energy of the configuration is ___________ J .

A capacitor has air as dielectric medium and two conducting plates of area \(12 \mathrm{~cm}^2\) and they are \(0.6 \mathrm{~cm}\) apart. When a slab of dielectric having area \(12 \mathrm{~cm}^2\) and \(0.6 \mathrm{~cm}\) thickness is inserted between the plates, one of the conducting plates has to be moved by \(0.2 \mathrm{~cm}\) to keep the capacitance same as in previous case. The dielectric constant of the slab is _______. (Given \(\left.\epsilon_0=8.834 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right)\)

The equivalent resistance between \(A\) and \(B\) as shown in figure is \(........\,k\Omega\)

In the circuit shown in the figure, the ratio of the quality factor and the band width is \(.............\) s.

An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density \(1000 \mathrm{~kg} / \mathrm{m}^3\). If the pressure inside the bubble is \(2100 \mathrm{~N} / \mathrm{m}^2\) greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

A particle of mass \(1\, {kg}\) is hanging from a spring of force constant \(100\, {Nm}^{-1 .}\) The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period \({T}\). The time when the kinetic energy and potential energy of the system will become equal, is \(\frac{{T}}{{x}}\). The value of \({x}\) is ..... .

\(\mathrm{A}, \mathrm{B}\) and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about \(P Q\) is \(\frac{X}{15} I\), where \(I\) is the moment of inertia of the disc about its diameter. The value of x is _______.

The force required to stretch a wire of crosssection \(1 cm ^{2}\) to double its length will be ........ \( \times 10^{7}\,N\) (Given Yong's modulus of the wire \(=2 \times 10^{11}\,N / m ^{2}\) )

A capacitor P with capacitance \( 10\times10^{-6} F \) is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor P is connected across another capacitor Q with capacitance \( 20\times10^{-6} F \). The charge on capacitor Q when equilibrium is established will be \( \alpha\times10^{-5} C \).(assume capacitor \(Q\) does not have any charge initially),The value of \( \alpha \) is __________ .

A circular table is rotating with an angular velocity of \(\omega\) \(\mathrm{rad} / \mathrm{s}\) about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of \(1 \mathrm{~m}\) on the groove. All the surface are smooth. If the radius of the table is \(3 \mathrm{~m}\), the radial velocity of the ball w.r.t. the table at the time ball leaves the table is \(x \sqrt{2} \omega\) \(\mathrm{m} / \mathrm{s}\), where the value of \(x\) is _______.