Consider a solid sphere of radius \(\mathrm{R}\) and mass density \(\rho(\mathrm{r})=\rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right),  0<\mathrm{r} \leq \mathrm{R} .\) The minimum density of a liquid in which it will float is

The \(Q\)-value of a nuclear reaction and kinetic energy of the projectile particle, \(K _{ p }\) are related as

If \(‘M’\) is the mass of water that rises in a capillary tube of radius \(‘r’,\) then mass of water which will rise in a capillary tube of radius \(‘2r’\) is

Which of the following pair of nuclei are isobars of the element?

Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current \(I=10 \sin (\omega t ) A\), where \(\omega=1000 rad . / s\). A circular conducting loop (B) of radius 1 cm coaxially slided through the solenoid at a speed \(v =1 cm / s\). The r.m.s. current through the loop when the coil B is inserted 10 cm inside the solenoid is \(\alpha / \sqrt{2} \quad \mu A\). The value of \(\alpha\) is ___________.
[Resistance of the loop \(=10 \Omega\) ]

The electric field in a region of space is given by, \(\overrightarrow E  = {E_0}\hat i + 2{E_0}\hat j\) where \(E_0\, = 100\, N/C\). The flux of the field through a circular surface of radius \(0.02\, m\) parallel to the \(Y-Z\) plane is nearly

The height \({ }^{\prime} h ^{\prime}\) at which the weight of a body will be the same as that at the same depth \('h'\) from the surface of the earth is (Radius of the earth is \(R\) and effect of the rotation of the earth is neglected):

A charge \(Q\) is placed at a distance \(a/2\) above the centre of the square surface of edge \(a\) as shown in the figure. The electric flux through the square surface is

Solid sphere \(A\) is rotating about an axis \(PQ\). If the radius of the sphere is \(5\,cm\) then its radius of gyration about \(P Q\) will be \(\sqrt{x} \;cm\). The value of \(x\) is \(................\).

An electron, a doubly ionized helium ion \(\left( He ^{++}\right)\) and a proton are having the same kinetic energy. The relation between their respective de\(-\)Broglie wavelengths \(\lambda_{ e }, \lambda_{ He ^{++}}\) and \(\lambda_{ P }\) is

A 20 m long uniform copper wire held horizontally is allowed to fall under the gravity ( \(g =10 m / s ^2\) ) through a uniform horizontal magnetic field of 0.5 Gauss perpendicular to the length of the wire. The induced EMF across the wire it travells a vertical distance of 200 m is ___________ mV .

Match the LIST-I with LIST-II

List-I	List-II
A. Planck's constant	I. \(ML^2T^{-2}\)
B. Stopping potential	II. \(T^{-1}\)
C. Work function	III. \(ML^2T^{-1}\)
D. Threshold frequency	IV. \(ML^2T^{-3}A^{-1}\)

Choose the correct answer from the options given below:

The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\)