For the thin convex lens, the radii of curvature are at \(15 \mathrm{~cm}\) and \(30 \mathrm{~cm}\) respectively. The focal length the lens is \(20 \mathrm{~cm}\). The refractive index of the material is _______.

A uniform string oflength \(20\ m\) is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the supports is (take \(g= 10 \) \(ms^{-2}\) )

Two forces \(\vec{F}_1\) and \(\vec{F}_2\) are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between \(\vec{F}_1\) and \(\overrightarrow{\mathrm{F}}_2\) is \(\cos ^{-1}\left(\frac{1}{\mathrm{n}}\right)\). The value of \(|\mathrm{n}|\) is __________.

A particle starts with an initial velocity of \(10.0\,ms ^{-1}\) along \(x\)-direction and accelerates uniformly at the rate of \(2.0\,ms ^{-2}\). The time taken by the particle to reach the velocity of \(60.0\,ms ^{-1}\) is \(.......\,s\)

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

List \(I\)	List \(II\)
\((A)\) Torque	\((I)\) \(Nms^{-1}\)
\((B)\) Stress	\((II)\) \(J\,kg^{-1}\)
\((C)\) Latent Heat	\((III)\) \(Nm\)
\((D)\) Power	\((IV)\) \(Nm^{-2}\)

Choose the correct answer from the options below.

The magnetic field existing in a region is given by \(\overrightarrow{\mathrm{B}}=0.2(1+2 \mathrm{x}) \hat{\mathrm{k} T}\). A square loop of edge \(50 \mathrm{~cm}\) carrying \(0.5 \mathrm{~A}\) current is placed in \(x-y\) plane with its edges parallel to the \(x-y\) axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is___________ \(\mathrm{mN}\).

As shown in the figure, a plane mirror is fixed at a height of \(50\,cm\) from the bottom of tank containing water \(\left(\mu=\frac{4}{3}\right)\). The height of water in the tank is \(8\,cm\). A small bulb is placed at the bottom of the water tank. The distance of image of the bulb formed by mirror from the bottom of the tank is \(......cm\)

A Carnot's engine works as a refrigerator between \(250\, K\) and \(300\, K\). It receives \(500\, cal\) heat from the reservoir at the lower temperature. The amount of work done in each cycle to operate the refrigerator is ..... \(J\)

A block of \(\sqrt{3}\,kg\) is attached to a string whose other end is attached to the wall. An unknown force \(F\) is applied so that the string makes an angle of \(30^{\circ}\) with the wall. The tension \(T\) is \(...........\,N\) :(Given \(g =10\,ms ^{-2}\) )

A block of mass \(100 \mathrm{~kg}\) slides over a distance of \(10 \mathrm{~m}\) on a horizontal surface. If the co-efficient of friction between the surfaces is \(0.4\) , then the work done against friction (in \(\mathrm{J}\) ) is :

At very high frequencies, the effective impendance of the given circuit will be \(....\Omega\)

A heat engine operates with the cold reservoir at temperature \(324 K\). The minimum temperature of the hot reservoir, if the heat engine takes \(300 \; J\) heat from the hot reservoir and delivers \(180 \; J\) heat to the cold reservoir per cycle, is \(\dots \; K .\)