The resistance of the meter bridge \(AB\) in given figure is \(4\,\Omega \). With a cell of emf  \(\varepsilon \, = 0.5\,\,V\) and rheostat resistance \(R_h = 2\,\Omega \) the null point is obtained at some point \(J.\) When the cell is replaced by another one of emf \(\varepsilon \, = {\varepsilon _2}\) the same null point \(J\) is found for \(R_h = 6\,\Omega .\) The \(emf\)  \({\varepsilon _2}\) is ................. \(V\)

Two particles are projected from the same point with the same speed \(u\) such that they have the same range \(R\), but different maximum heights, \(h_1\) and \(h_2\). Which of the following is correct ?

A ball of mass \(10\, kg\) moving with a velocity \(10 \sqrt{3} m / s\) along the \(x-\)axis, hits another ball of mass \(20\, kg\) which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along \(y-\)axis with a speed of \(10\) \(m / s\). The second piece starts moving at an angle of \(30^{\circ}\) with respect to the \(x-\)axis. The velocity of the ball moving at \(30^{\circ}\) with \(x-\) axis is \(x\, m / s\). The configuration of pieces after collision is shown in the figure below. The value of \(x\) to the nearest integer is

In an \(LC\) oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes \(x\) times its initial resonant frequency \(\omega_0\). The value of \(x\) is:

In a building there are \(15\) bulbs of \(45\; \mathrm{W}, 15\) bulbs of \(100\; \mathrm{W}, 15\) small fans of \(10 \;\mathrm{W}\) and \(2\) heaters of \(1 \;\mathrm{kW}\). The voltage of electric main is \(220\; \mathrm{V}\). The minimum fuse capacity (rated value) of the building will be: .......... \(A\)

Two capacitors of capacitances \(C\) and \(2\, C\) are charged to potential differences \(V\) and \(2\, V\), respectively. These are then connected in parallel in such a manner that the positive terminal of one is connected to the negative terminal of the other. The final energy of this configuration is\(.....CV^2\)

Consider two ideal diatomic gases \(\mathrm{A}\) and \(\mathrm{B}\) at some temperature \(T\). Molecules of the gas \(A\) are rigid, and have a mass \(m\). Molecules of the gas \(\mathrm{B}\) have an additional vibrational mode, and have a mass \(\frac{\mathrm{m}}{4} .\) The ratio of the specific heats \((\mathrm{C}_{\mathrm{v}}^{\mathrm{A}}\) and \(\mathrm{C}_{\mathrm{v}}^{\mathrm{B}})\) of gas \(\mathrm{A}\) and \(\mathrm{B}\), respectively is

A uniform rectangular thin sheet \(ABCD\) of mass \(M\) has length \(a\) and breadth \(b\), as shown in the figure. If the shaded portion \(HBGO\) is cut off, the coordinates of the centre of mass of the remaining portion will be

A parallel plate capacitor with width \(4\,cm\), length \(8\,cm\) and separation between the plates of \(4\,mm\) is connected to a battery of \(20\,V\). A dielectric slab of dielectric constant \(5\) having length \(1\,cm\), width \(4\,cm\) and thickness \(4\,mm\) is inserted between the plates of parallel plate capacitor. The electrostatic energy of this system will be......... \(\in_{0}\,J\). (Where \(\epsilon_{0}\) is the permittivity of free space)

A beam of monochromatic light is used to excite the electron in \(Li ^{++}\)from the first orbit to the third orbit. The wavelength of monochromatic light is found to be \(x \times 10^{-10} \; m\). The value of \(x\) is \(\dots.\) [Given hc \(=1242 \; eV nm ]\)

In a double slit experiment, when one of the slits is covered by a transparent mica sheet of refractive index \(1.56\), the central fringe shifts to the position of \(7^{th}\) bright fringe, obtained with both slits uncovered. If the light source wavelength is \(450\) nm, the thickness of mica sheet is \(\alpha \times 10^{-9}\) m. The value of \(\alpha\) is ______.

The energy released in the fusion of \(2 \mathrm{~kg}\) of hydrogen deep in the sun is \(\mathrm{E}_{\mathrm{H}}\) and the energy released in the fission of \(2 \mathrm{~kg}\) of \({ }^{235} \mathrm{U}\) is \(E_U\). The ratio \(\frac{E_H}{E_U}\) is approximately _______. (Consider the fusion reaction as \(4{ }_1^1 \mathrm{H}+2 \mathrm{e}^{-} \rightarrow{ }_2^4 \mathrm{He}+2 \mathrm{v}+6 \gamma+26.7 \mathrm{MeV}\), energy released in the fission reaction of \({ }^{235} \mathrm{U}\) is \(200 \mathrm{MeV}\) per fission nucleus and \(\mathrm{N}_{\mathrm{A}}=6.023 \times 10^{23}\) )

A bus moving along a straight highway with speed of \(72 \mathrm{~km} / \mathrm{h}\) is brought to halt within \(4 \mathrm{~s}\) after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is __________\(\mathrm{m}\).