A body of mass (\(2 {M}\)) splits into four masses \(\{{m}, {M}-{m}, {m}, {M}-{m}\}\), which are rearranged to form a square as shown in the figure. The ratio of \(\frac{{M}}{{m}}\) for which, the gravitational potential energy of the system becomes maximum is \({x}: 1\). The value of \(x\) is ..... .

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is \(12\,cm\). when viewed from the opposite side, the apparent distance of the bubble is observed as \(4\,cm\). If the side of the ice cube is \(24\,cm\), the refractive index of the ice cube is \(.....\).

If \(t =\sqrt{ x }+4\), then \(\left(\frac{ dx }{ dt }\right)_{ t =4}\) is.

\(ABC\) is a plane lamina of the shape of an equilateral triagnle. \(D,\) \(E\) are mid points of \(AB\), \(AC\) and \(G\) is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through \(G\) and perpendicular to the plane \(ABC\) is \(I _{0} .\) If part \(ADE\) is removed, the moment of inertia of the remaining part about the same axis is \(\frac{ NI _{0}}{16}\) where \(N\) is an integer. Value of \(N\) is

A capacitor of capacitance \(150.0\,\mu F\) is connected to an alternating source of \(emf\) given by \(E =36\) \(\sin (120 \pi t ) V\). The maximum value of current in the circuit is approximately equal to \(......\,A\)

A wire of length \(2\,L\) is made by joining two wires \(A\) and \(B\) of same lengths but different radii \(r\) and \(2r\) and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire \(A\) is \(p\) and that in \(B\) is \(q\) then the ratio \(p : q\) is

All electromagnetic wave is transporting energy in the negative \(z\) direction. At a certain point and certain time the direction of electric field of the wave is along positive y direction. What will be the direction of the magnetic field of the wave at that point and instant?

Four identical solid spheres each of mass \('m'\) and radius \('a'\) are placed with their centres on the four corners of a square of side \('b'\). The moment of inertia of the system about one side of square where the axis of rotation is parallel to the plane of the square is

Work done by a Carnot engine operating between temperatures \(127^{\circ}\,C\) and \(27^{\circ}\,C\) is \(2\,kJ\). The amount of heat transferred to the engine by the reservoir is \(........\,kJ\)

As shown in figures, three identical polaroids \(P_1\), \(P _2\) and \(P _3\) are placed one after another. The pass axis of \(P_2\) and \(P_3\) are inclined at angle of \(60^{\circ}\) and \(90^{\circ}\) with respect to axis of \(P _1\). The source \(S\) has an intensity of \(256 \frac{ W }{ m ^2}\). The intensity of light at point \(O\) is \(...........\frac{ W }{ m ^2}\).