Surface tension of two liquids (having same densities), \(T_1\) and \(T_2\), are measured using capillary rise method utilizing two tubes with inner radii of \(r_1\) and \(r_2\) where \(r_1>r_2\). The measured liquid heights in these tubes are \(h_1\) and \(h_2\) respectively. [Ignore the weight of the liquid about the lowest point of miniscus]. The heights \(h _1\) and \(h _2\) and surface tensions \(T _1\) and \(T _2\) satisfy the relation :

The radius of a nucleus of mass number \(64\) is \(4.8\) fermi. Then the mass number of another nucleus having radius of \(4\) fermi is \(\frac{1000}{x}\), where \(x\) is _______.

A potentiometer wire of length \(10 \,m\) and resistance\(20 \,\Omega\) is connected in series with a \(25 \,V\) battery and an external resistance \(30\, \Omega\). A cell of emf \(E\) in secondary circuit is balanced by \(250\, cm\) long potentiometer wire. The value of \(E\) (in volt) is \(\frac{x}{10}\). The value of \(x\) is.......

Match List-\(I\) with List-\(II\) :-

List - \(I\) (EM-Wave)	List - \(II\) (Wavelength) (Range)
\((A)\) infra-red	\((i)\) \(<10^{-3} \mathrm{~nm}\)
\((B)\) Ultraviolet	\((ii)\) \(400 \mathrm{~nm}\) to \(1 \mathrm{~nm}\)
\((C)\) X-rays	\((iii)\) \(1 \mathrm{~mm}\) to \(700 \mathrm{~nm}\)
\((D)\) Gamma rays	\((iv)\) \(1 \mathrm{~nm}\) to \(10^{-3} \mathrm{~nm}\)

Choose the correct answer from the options given below:

Three containers \(\mathrm{C}_{1}, \mathrm{C}_{2}\) and \(\mathrm{C}_{3}\) have water at different temperatures. The table below shows the final temperature \(T\) when different amounts of water (given in litres) are taken from each containers and mixed (assume no loss of heat during the process) \(\begin{array}{|c|c|c|c|}\hline \mathrm{C_{1 }} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}} & {\mathrm{T}} \\ \hline {1 l} & {2 l} & {-} & {60^{\circ} \mathrm{C}} \\ \hline {-} & {1 l} & {2 l} & {30^{\circ} \mathrm{C}} \\ \hline  {2 l} & {-} & {1 l} & {60^{\circ} \mathrm{C}} \\ \hline  {1 l} & {1 l} & {1 l} & {\theta} \\ \hline\end{array}\) The value of \(\theta\) (in \(^{\circ} \mathrm{C}\) to the nearest integer) is

A charge of \(4\,\mu C\) is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be.

Two iron solid discs of negligible thickness have radius \(R_1\) and \(R_2\) and moment of intertia \(I_1\) and \(I_2\), respectively. For \(R_2=2 R_1\), the ratio of \(I_1\) and \(I_2\) would be \(1 / x\), where \(\mathrm{x}=\) ________ -

A cricket ball of mass \(0.15\, kg\) is thrown vertically up by a bowling machine so that it rises to a maximum height of \(20 \;m\) after leaving the machine. If the part pushing the ball applies a constant force \(F\) on the ball and moves horizontally a distance of \(0.2\, m\) while launching the ball, the value of \(F(\) in \(N)\) is  \(\left(g=10\, m s^{-2}\right)\)

A bottle has an opening of radius \(a\) and length \(b\). A cork of length band radius \(\left( {a + \Delta a} \right)\) where \(\left( {\Delta a <  < a} \right)\) is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is \(B\) and frictional coefficient between the bottle and cork is \(\mu \) then the force needed to push the cork into the bottle is

If \(\epsilon_0\) denotes the permittivity of free space and \(\Phi_{\mathrm{E}}\) is the flux of the electric field through the area bounded by the closed surface, then dimension of \(\left(\epsilon_0 \frac{\mathrm{~d} \phi_{\mathrm{E}}}{\mathrm{dt}}\right)\) are that of :

In a nuclear fission process, a high mass nuclide \((A \approx 236)\) with binding energy \(7.6\  \mathrm{MeV} /\) Nucleon dissociated into middle mass nuclides \((A \approx 118)\), having binding energy of \(8.6 \ \mathrm{MeV} / \mathrm{Nucleon}\). The energy released in the process would be \(\mathrm{MeV}\).

The relation between time \(t\) and distance \(x\) for a moving body is given as \(t=m x^{2}+n x\), where \({m}\) and \({n}\) are constants. The retardation of the motion is - (Where \(v\) stands for velocity)