If the time period \(t\) of the oscillation of a drop of liquid of density \(d\), radius \(r\), vibrating under surface tension \(s\) is given by the formula \(t = \sqrt {{r^{2b}}\,{s^c}\,{d^{a/2}}} \) . It is observed that the time period is directly proportional to \(\sqrt {\frac{d}{s}} \) . The value of \(b\) should therefore be

The average kinetic energy of a monatomic molecule is \(0.414 \mathrm{eV}\) at temperature : (Use \(\mathrm{K}_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) )

Two guns \(A\) and \(B\) can fire bullets at speed \(1\, km/s\) and \(2\, km/s\) respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively \(1.5\%\) and \(1\%\), the maximum error in determining the density is   ........ \(\%\)

A conducting circular loop made of a thin wire, has area \(3.5 \times 10^{-3}\, m^2\) and resistance \(10\,\Omega \). It is placed perpendicular to a time dependent magnetic field \(B(t) = (0.4\,T)\, sin\, (50\, \pi t)\). The field is uniform in space. Then the net charge flowing through the loop during \(t = 0\, s\) and \(t = 10\, ms\) is close to.......\(mC\)

An object is taken to a height above the surface of earth at a distance \(\frac{5}{4} R\) from the centre of the earth. Where radius of earth, \(R=6400\,km\). The percentage decrease in the weight of the object will be \(....\%\)

For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is _______.

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In Young's double slit experiment, the fringes produced by red light are closer as compared to those produced by blue light.
Reason (R): The fringe width is directly proportional to the wavelength of light.
In the light of the above statements, choose the correct answer from the options given below.

The mid points of two small magnetic dipoles of length \(d\) in end-on positions, are separated by a distance \(x, (x > > d)\). The force between them is proportional to \(x^{-n}\) where \(n\) is

In the given circuit, the charge on \(4\, \mu F\) capacitor will be.....\(\mu C\)

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

The velocity of a particle is \(v = v _{0}+ gt + Ft ^{2}\). Its position is \(x=0\) at \(t=0\); then its displacement after time \(( t =1)\) is :

An electromagnetic wave travels in free space along the \(x\)-direction. At a particular point in space and time, \(\vec{B} = 2 \times 10^{-7} \hat{j}\) T is associated with this wave. The value of corresponding electric field \(\vec{E}\) at this point is _______ V/m.