A circular coil of radius \(8.0 \,{cm}\) and \(20\) turns is rotated about its vertical diameter with an angular speed of \(50\, {rad} {s}^{-1}\) in a uniform horizontal magnetic field of \(3.0 \times 10^{-2}\, {T}\). The maximum \(emf\) induced the coil will be \(\ldots \ldots \ldots \times 10^{-2}\) \(volt\) (rounded off to the nearest integer)

The centre of a wheel rolling on a plane surface moves with a speed \(v_{0} .\) A particle on the rim of the wheel at the same level as the centre will be moving at a speed \(\sqrt{x}\, v_{0}\). Then the value of \(x\) is ...... .

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

The time period of a satellite, revolving above earth's surface at a height equal to \(R\) will be (Given \(g =\pi^2 m / s ^2, R =\) radius of earth)

In a carnot engine, the temperature of reservoir is \(527^{\circ} C\) and that of \(\operatorname{sink}\) is \(200 \; K\). If the workdone by the engine when it transfers heat from reservoir to sink is \(12000 \; kJ\), the quantity of heat absorbed by the engine from reservoir is \(\times 10^{6} \; J\)

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is \(M,\) radius of its top, \(R\) and height, \(H\), then its moment of inertia about its axis is

If a radioactive element having half-life of \(30\,min\) is undergoing beta decay, the fraction of radioactive element remains undecayed after \(90\,min\). will be :

In an experiment with a convex lens. The plot of the image distance (\(v'\)) against the object distance \(\left(\mu^{\prime}\right)\) measured from the focus gives a curve \(v^{\prime} \mu^{\prime}=225\). If all the distances are measured in \(cm\). The magnitude of the focal length of the lens is \(cm\).

Match List \(I\) with List \(II\) :

List \(I\) (Physical Quantity)	List \(II\) (Dimensional Formula)
\((A)\) Pressure gradient	\((I)\) \(\left[ M ^0 L ^2 T ^{-2}\right]\)
\((B)\) Energy density	\((II)\) \(\left[ M ^1 L ^{-1} T ^{-2}\right]\)
\((C)\) Electric Field	\((III)\) \(\left[ M ^1 L ^{-2} T ^{-2}\right]\)
\((D)\) Latent heat	\((IV)\) \(\left[ M ^1 L ^1 T ^{-3} A ^{-1}\right]\)

Choose the correct answer from the options given below:

There is a uniform electrostatic field in a region. The potential at various points on a small sphere centred at \(P\), in the region, is found to vary between in the limits \(589.0\,V\) to \(589.8\, V\). What is the potential at a point on the sphere whose radius vector makes an angle of \(60^o\) with the direction of the field ?........\(V\)

A uniform solid cylinder with radius \(R\) and length L has moment of inertia \(I_1\), about the axis of cylinder. A concentric solid cylinder of radius \(R^{\prime}=\frac{R}{2}\) and length \(L^{\prime}=\frac{L}{2}\) is caned out of the original cylinder. If \(I_2\) is the moment of inertia of the carved out portion ot the cylinder then \(\frac{I_1}{I_2}=..........\) (Both \(I_1\) and \(I_2\) are about the axis of the cylinder)