A vessel of depth \(d\) is half filled with oil of refractive index \(n_1\) and the other half is filled with water of refractive index \(n_2\). The apparent depth of this vessel when viewed from above will be

A circular coil has moment of inertia \(0.8 \,kg m ^{2}\) around any diameter and is carrying current to roduce a magnetic moment of \(20\, Am ^{2}\). The coil is kept initially in a vertical position and it can rotate freely around a horizontal diameter. When a uniform magnetic field of \(4\, T\) is applied along the vertical, it starts rotating around its horizontal diameter. The angular speed the coil acquires after rotating by \(60^{\circ}\) will be

A prism of refractive index \(\mu\) and angle \(A\) is placed in the minimum deviation position. If the angle of minimum deviation is \(A,\) then the value of \(A\) in terms of \(\mu\) is

10 mole of oxygen is heated at constant volume from \(30^{\circ} C\) to \(40^{\circ} C\). The change in the internal energy of the gas is _________ cal. (The molecular specific heat of oxygen at constant pressure, \(C _{ p }=7 cal . / mol ^{\circ} C\) and \(\left.R =2 cal . / mol ^{\circ} C .\right)\)

A heating element has a resistance of  \(100\,\Omega \) at room temperature. When it is connected to a supply of  \(220\,V,\)  a steady current of \(2\,A\)  passes in it and temperature is \(500\,^oC\) more than room temperature. What is the temperature coefficient resistance of the heating element?

The displacement time graph of a particle executing \(S.H.M.\) is given in figure: (sketch is schematic and not to scale) Which of the following statements is are true for this motion? \((A)\) The force is zero \(t=\frac{3 T}{4}\) \((B)\) The acceleration is maximum at \(t=T\) \((C)\) The speed is maximum at \(t =\frac{ T }{4}\) \((D)\) The \(P.E.\) is equal to \(K.E.\) of the oscillation at \(t=\frac{T}{2}\)

Two waves are simultaneously passing through a string and their equations are : \({y}_{1}={A}_{1} \sin {k}({x}-v {t}), {y}_{2}={A}_{2} \sin {k}\left({x}-{vt}+{x}_{0}\right) .\) Given amplitudes \({A}_{1}=12\, {mm}\) and \({A}_{2}=5\, {mm}\) \({x}_{0}=3.5\, {cm}\) and wave number \({k}=6.28\, {cm}^{-1}\). The amplitude of resulting wave will be \(......\,{mm}\)

Let the refractive index of a denser medium with respect to a rarer medium be \(n_{12}\) and its critical angle be \(\theta_C\).  At an angle of incidence \(A\) when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is \(90^o\).  Angle \(A\) is given by

Match List - I with List - II.

List - I	List - II
(A) Permeability of free space	(I) \(\left[\mathrm{M} \mathrm{~L}^2 \mathrm{~T}^{-2}\right]\)
(B) Magnetic field	(II) \(\left[\mathrm{M} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\)
(C) Magnetic moment	(III) \(\left[\mathrm{M} \mathrm{~L} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]\)
(D) Torsional constant	(IV) \(\left[\mathrm{L}^2 \mathrm{~A}\right]\)

Choose\(\text{ }\)the\(\text{ }\)correct\(\text{ }\)answer\(\text{ }\)from\(\text{ }\)the\(\text{ }\)options\(\text{ }\)given\(\text{ }\)below\(\text{ }\):

Consider two uniform discs of the same thickness and different radii \(R _{1}= R\) and \(R _{2}=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_{1}\) and \(I_{2}\), respectively, about their axes is \(I _{1}: I _{2}=1: 16\) then the value of \(\alpha\) is

A stationary particle breaks into two parts of masses \(m_A\) and \(m_B\) which move with velocities \(v_A\) and \(V_B\) respectively. The ratio of their kinetic energies \(\left(\mathrm{K}_B: \mathrm{K}_{\mathrm{A}}\right)\) is _______.

A radioactive sample decays \(\frac{7}{8}\) times its original quantity in \(15\) minutes. The half-life of the sample is \(......min\)

The time period of a simple pendulum is given by \(T =2 \pi \sqrt{\frac{\ell}{ g }}\). The measured value of the length of pendulum is \(10\, cm\) known to a \(1\, mm\) accuracy. The time for \(200\) oscillations of the pendulum is found to be \(100\) second using a clock of \(1s\) resolution. The percentage accuracy in the determination of \('g'\) using this pendulum is \('x'\). The value of \('x'\) to the nearest integer is ...........\(\%\)