A \(2 \,{kg}\) steel rod of length \(0.6\, {m}\) is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity, Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is \(\ldots \ldots \ldots \ldots \,{ms}^{-1}\). (Take \(g = 10\, {ms}^{-2}\) )

One end of a metal wire is fixed to a ceiling and a load of \(2 \mathrm{~kg}\) hangs from the other end. A similar wire is attached to the bottom of the load and another load of \(1 \mathrm{~kg}\) hangs from this lower wire. Then the ratio of longitudinal strain of upper wire to that of the lower wire will be _______. [Area of cross section of wire \(=0.005 \mathrm{~cm}^2\), \(\mathrm{Y}=2 \times 10^{11}\  \mathrm{Nm}^{-2}\) and \(\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right]\)

Three containers \(\mathrm{C}_{1}, \mathrm{C}_{2}\) and \(\mathrm{C}_{3}\) have water at different temperatures. The table below shows the final temperature \(T\) when different amounts of water (given in litres) are taken from each containers and mixed (assume no loss of heat during the process) \(\begin{array}{|c|c|c|c|}\hline \mathrm{C_{1 }} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}} & {\mathrm{T}} \\ \hline {1 l} & {2 l} & {-} & {60^{\circ} \mathrm{C}} \\ \hline {-} & {1 l} & {2 l} & {30^{\circ} \mathrm{C}} \\ \hline  {2 l} & {-} & {1 l} & {60^{\circ} \mathrm{C}} \\ \hline  {1 l} & {1 l} & {1 l} & {\theta} \\ \hline\end{array}\) The value of \(\theta\) (in \(^{\circ} \mathrm{C}\) to the nearest integer) is

Asseretion \(A:\) If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is \(5\, {mm}\) and there are \(50\) total divisions on circular scale, then least count is \(0.001\, {cm}\). Reason \(R:\) Least Count \(=\frac{\text { Pitch }}{\text { Total divisions on circular scale }}\) In the light of the above statements, choose the most appropriate answer from the options given below:

A spherical surface of radius of curvature \(R\), separates air from glass (refractive index \(=1.5\) ). The centre of curvature is in the glass medium. A point object ' \(O\) ' placed in air on the optic axis of the surface, so that its real image is formed at ' \(I\) ' inside glass. The line OI intersects the spherical surface at P and \(\mathrm{PO}=\mathrm{PI}\). The distance PO equals to ________.

A Carnot's engine working between \(400\, K\) and \(800\, K\) has a work output of \(1200\, J\) per cycle. The amount of heat energy supplied to the engine from the source in each cycle is ........... \(J\)

When the temperature of a metal wire is increased from \(0^{\circ} \,C\) to \(10^{\circ}\, C\), its length increases by \(0.02 \% .\) The percentage change in its mass density will be closest to:

A \(10\,\mu C\) charge is divided into two parts and placed at \(1\,cm\) distance so that the repulsive force between them is maximum. The charges of the two parts are :

The self produced \(emf\) of a coil is \(25\,volts\). When the current in it is changed at uniform rate from \(10\,A\) to \(25\, A\) in \(1\,s\), the change in the energy of the inductance is......\(J\)

A passenger sitting in a train A moving at \(90\,km / h\) observes another train \(B\) moving in the opposite direction for \(8\,s\). If the velocity of the train \(B\) is \(54\,km / h\), then length of train \(B\) is \(..........\,m\)

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is