A ball of radius r and density \(\rho\) dropped through a viscous liquid of density \(\sigma\) and viscosity \(\eta\) attains its terminal velocity at time t , given by \(t = A \rho^{ a } r ^{ b } \eta^{ c } \sigma^{ d }\), where A is a constant and a , b c and d are integers. The value of \(\frac{b+c}{a+d}\) is ___________.

A parallel plate capacitor has plate of length \('l',\) width \('w'\) and separation of plates is \('d'.\) It is connected to a battery of emf \(V\). A dielectric slab of the same thickness '\(d\)' and of dielectric constant \(k =4\) is being inserted between the plates of the capacitor. At what length of the slab inside plates, will be energy stored in the capacitor be two times the initial energy stored\(?\)

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at \(30^{\circ} C\). The coefficient of linear expansion of aluminium and steel are \(24 \times 10^{-6} /{ }^{\circ} C\) and \(1.2 \times 10^{-5} /{ }^{\circ} C\), respectively. The length of this composite rod when its temperature is raised to \(100^{\circ} C\), is _________ cm .

What is the current through the battery in the circuit shown below?

If an optical medium possesses a relative permeability of \(\frac{10}{\pi}\) and relative permittivity of \(\frac{1}{0.0885}\), then the velocity of light is greater in vacuum than that in this medium by ________ times.
\(\left(\mu_0=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}, \epsilon_0=8.85 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right.\)
\(\left.\mathrm{c}=3 \times 10^8 \mathrm{~m} / \mathrm{s}\right)\)

In the given circuit below inductance values of \(L_1\), \(L_2\) and \(L_3\) are same. The magnetic energy stored in the entire circuit is \((U_t)\) and that stored in the \(L_2\) inductor is \((U_l)\). \(U_t / U_l\) is __________.
(Ignore the mutual inductance if any)

When an object is placed 40 cm away from a spherical mirror an image of magnification \(\frac{1}{2}\) is produced. To obtain an image with magnification of \(\frac{1}{3}\), the object is to be moved :

According to kinetic theory of gases, \(A\). The motion of the gas molecules freezes at \(0^{\circ} C\) \(B\). The mean free path of gas molecules decreases if the density of molecules is increased. \(C\). The mean free path of gas molecules increases if temperature is increased keeping pressure constant. \(D\). Average kinetic energy per molecule per degree of freedom is \(\frac{3}{2} k_{B} T\) (for monoatomic gases) Choose the most appropriate answer from the options given below

The velocity time graph of a body moving in a straight line is shown in figure.The ratio of displacement to distance travelled by the body in time \(0\) to \(10\,s\) is

A mirror is used to produce an image with magnification of \(\frac{1}{4}\). If the distance between object and its image is 40 cm , then the focal length of the mirror is ________

Two rods \(A\) and \(B\) of identical dimensions are at temperature \(30\,^oC\). If a heated upto \(180\,^oC\) and \(B\) upto \(T\,^oC\), then the new lengths are the same. If the ratio of the coefficients of linear expansion of \(A\) and \(B\) is \(4:3\), then the value of \(T\) is........\(^oC\)

A motor operating on 100 V draws a current of 1 A. If the efficiency of the motor is \(91.6 \%\), then the loss of power in units of \(\mathrm{cal} / \mathrm{s}\) is