If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately ........ minutes (Take : \(g =10 \,ms ^{-2},\) the radius of earth, \(R =6400 \times 10^{3}\, m ,\) Take \(\left.\pi=3.14\right)\)

A metallic ring is uniformly charged as shown in figure. AC and BD are two mutually perpendicular diameters. Electric field due to \(\operatorname{arc} \mathrm{AB}\) to ' O ' is ' E ' is magnitude. What would be the magnitude of electric field at ' O ' due to arc ABC ?

A particle executes \(S.H.M.\) of amplitude A along \(x\)-axis. At \(t =0\), the position of the particle is \(x=\frac{A}{2}\) and it moves along positive \(x\)-axis the displacement of particle in time \(t\) is \(x=A \sin (\omega t+\delta)\), then the value \(\delta\) will be

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of \(24\,W\) The radius of curvature of hemisphere is \(10\,cm\) and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is \(..........\times 10^{-8}\,N\).

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of \(20 \mathrm{~cm}\) from its center is \(1.5 \times 10^{-5} \  \mathrm{Tm}\). The magnetic moment of the dipole is _______ \(\mathrm{Am}^2\). (Given : \(\frac{\mu_0}{4 \pi}=10^{-7} \  \mathrm{TmA}^{-1}\) )

The driver of a bus approaching a big wall notices that the frequency of his bus's horn changes from \(420\, Hz\) to \(490\, Hz ,\) when he hears it after it gets reflected from the wall. Find the speed of the bus (in \(kmh^{-1}\)) if speed of the sound is \(330\, ms ^{-1}\).

The magnetic field vector of an electromagnetic wave is given by \({B}={B}_{o} \frac{\hat{{i}}+\hat{{j}}}{\sqrt{2}} \cos ({kz}-\omega {t})\); where \(\hat{i}, \hat{j}\) represents unit vector along \({x}\) and \({y}\)-axis respectively. At \(t=0\, {s}\), two electric charges \(q_{1}\) of \(4\, \pi\) coulomb and \({q}_{2}\) of \(2 \,\pi\) coulomb located at \(\left(0,0, \frac{\pi}{{k}}\right)\) and \(\left(0,0, \frac{3 \pi}{{k}}\right)\), respectively, have the same velocity of \(0.5 \,{c} \hat{{i}}\), (where \({c}\) is the velocity of light). The ratio of the force acting on charge \({q}_{1}\) to \({q}_{2}\) is :-

A force \(\mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}}\) acts on a particle in a plane \(\mathrm{x}+\mathrm{y}=10\). The work done by this force during a displacement from \((0,0)\) to \((4 \mathrm{~m}, 2 \mathrm{~m})\) is Joule (round off to the nearest integer)

If \(1, \log _{10}\left(4^{x}-2\right)\) and \(\log _{10}\left(4^{x}+\frac{18}{5}\right)\) are in
arithmetic progression for a real number \(x\) then the value of the determinant \(\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|\) is equal to ...... .

Let \(A\) be a \(2 \times 2\) real matrix and \(I\) be the identity matrix of order \(2\) . If the roots of the equation \(|A-x I|=0\) be \(-1\) and \(3\) , then the sum of the diagonal elements of the matrix \(A^2\) is ...........

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of \(P_1\) and \(P_2\) are orthogonal to each other. The polarizer \(P_3\) covers both the slits with its transmission axis at \(45^{\circ}\) to those of \(P_1\) and \(P_2\). An unpolarized light of wavelength \(\lambda\) and intensity \(I_0\) is incident on \(P_1\) and \(P_2\). The intensity at a point after \(P_3\) where the path difference between the light waves from \(s_1\) and \(s_2\) is \(\frac{\lambda}{3}\), is _______.

As shown in the figure. a configuration of two equal point charges \(\left( q _0=+2 \mu C \right)\) is placed on an inclined plane. Mass of each point charge is \(20\,g\). Assume that there is no friction between charge and plane. For the system of two point charges to be in equilibrium (at rest) the height \(h = x \times 10^{-3}\,m\) The value of \(x\) is \(..........\).(Take \(\left.\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9\,Nm ^2 C ^{-2}, g=10\,ms ^{-1}\right)\)

Three masses \(m,\,2\,m\) and \(3\,m\) are moving in \(x-y\) plane with speed \(3\,u,\,2\,u\,\) and \(u\) respectively as shown in figure. The three masses collide at the same point at \(P\) and stick together. The velocity of resulting mass will be