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 ................. \(V\)

For a solid rod, the Young's modulus of elasticity is \(3.2 \times 10^{11}\,Nm ^{-2}\) and density is \(8 \times 10^3\,kg\,m ^{-3}\). The velocity of longitudinal wave in the rod will be \(......... \times 10^{3}\,ms^{-1}\)

A balloon carries a total load of \(185\; {kg}\) at normal pressure and temperature of \(27^{\circ} {C}\). What load will the balloon carry on rising to a height at which the barometric pressure is \(45\; {cm}\) of \({Hg}\) and the temperature is \(-7^{\circ} {C}\). Assuming the volume constant? (in \({kg}\))

An ideal battery of \(4\, V\) and resistance \(R\) are connected in series in the primary circuit of a potentiometer of length \(1\, m\) and resistance \(5\,\Omega \) . The value of \(R\), to give a difference of \(5\, mV\) across \(10\, cm\) of potentiometer wire, is: ................ \(\Omega\)

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)

A potential barrier of \(0.4 \,V\) exists across a p-n junction. An electron enters the junction from the \(n\)-side with a speed of \(6.0 \times 10^{5} \,ms ^{-1}\). The speed with which electron enters the \(p\) side will be \(\frac{x}{3} \times 10^{5} \,ms ^{-1}\) the value of \(x\) is .............. (Given mass of electron \(=9 \times 10^{-31} \,kg\), charge on electron \(=1.6 \times 10^{-19} \,C\).)

According to Bohr atom model, in which of the following transitions will the frequency be maximum ?

The parabola \(y^2=4 x\) divides the area of the circle \(x^2+y^2=5\) in two parts. The area of the smaller part is equal to :

As shown in the figure, a network of resistors is connected to a battery of \(2\,V\) with an internal resistance of \(3\,\Omega\). The currents through the resistors \(R_4\) and \(R_5\) are \(I_4\) and \(I_5\) respectively. The values of \(I_4\) and \(I_5\) are :

If the mean and variance of the frequency distribution

\(x_i\)	\(2\)	\(4\)	\(6\)	\(8\)	\(10\)	\(12\)	\(14\)	\(16\)
\(f_i\)	\(4\)	\(4\)	\(\alpha\)	\(15\)	\(8\)	\(\beta\)	\(4\)	\(5\)

are \(9\) and \(15.08\) respectively, then the value of \(\alpha^2+\beta^2-\alpha \beta\) is \(............\).

Two marbles are drawn in succession from a box containing \(10\) red, \(30\) white, \(20\) blue and \(15\) orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

A bullet of \(10\, {g}\), moving with velocity \(v\), collides head-on with the stationary bob of a pendulum and recoils with velocity \(100 \, {m} / {s}\). The length of the pendulum is \(0.5\, {m}\) and mass of the bob is \(1\, {kg}\). The minimum value of \(v=\) \(....{m} / {s}\) so that the pendulum describes a circle. (Assume the string to be inextensible and \({g}=10\, {m} / {s}^{2}\) )

\(\int \limits_{\pi / 6}^{\pi / 3} \tan ^{3} x \cdot \sin ^{2} 3 x\left(2 \sec ^{2} x \cdot \sin ^{2} 3 x+3 \tan x \cdot \sin 6 x\right) d x\) is equal to