A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius \(R _{ e }\). By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that is become \(\sqrt{\frac{3}{2}}\) times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is \(R\), value of \(R\) is\(....R_e\)

An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii \(r_e,r_p\) and \({r_\alpha }\) respectively in a uniform magnetic field \(B\). The relation between \(r_e,r_p\) and \(\;{r_\alpha }\) is

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

Two p-n junction diodes \(D _1\) and \(D _2\) are connected as shown in figure. A and B are input signals and C is the output. The given circuit will function as a ___________.

Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of \(3\,s\) and \(4\,s\) respectively. If their moments of inertia are in the ratio of \(3: 2\) then the ratio of their magnetic moments will e.

To mop-clean a floor, a cleaning machine presses a circular mop of radius \(R\) vertically down with a total force \(F\) and rotates it with a constant angular speed about its axis. If the force \(F\) is distributed uniformly over the mop and the floor is \(\mu \), the torque, applied by the machine on the mop is

A cone of base radius \(R\) and height \(h\) is located in a uniform electric field \(\vec E\) parallel to its base. The electric flux entering the cone is

\(0.08 \mathrm{~kg}\) air is heated at constant volume through \(5^{\circ} \mathrm{C}\). The specific heat of air at constant volume is \(0.17 \mathrm{kcal} / \mathrm{kg}^{\circ} \mathrm{C}\) and \(\mathrm{J}=4.18\) joule \(/ \mathrm{cal}\). The change in its internal energy is approximately.

Let \(\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in R\). If \(x(\alpha, 1,2)+y(1, \beta, 2)\) \(+z(2,3, \gamma)=(0,0,0)\) for some \(x, y, z \in R, x y z \neq\) 0 , then \(6 \alpha+4 \beta+\gamma\) is equal to ..............

The \(V-I\) characteristic of a diode is shown in the figure. The ratio of forward to reverse bias resistance is

Let \(\mathrm{A}=\left[\mathrm{a}_{i j}\right]=\left[\begin{array}{cc}\log _5 128 & \log _4 5 \\ \log _5 8 & \log _4 25\end{array}\right]\).
If \(\mathrm{A}_{i j}\) is the cofactor of \(\mathrm{a}_{i j}, \mathrm{C}_{i j}=\sum_{\mathrm{k}=1}^2 \mathrm{a}_{i \mathrm{k}} \mathrm{A}_{j \mathrm{k}}, 1 \leq i, j \leq 2\), and \(\mathrm{C}=\left[\mathrm{C}_{i j}\right]\), then \(8|\mathrm{C}|\) is equal to :

If the matrix \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{array}\right]\) satisfies the equation \(A ^{20}+\alpha A ^{19}+\beta A =\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{array}\right]\) for some real numbers \(\alpha\) and \(\beta\), then \(\beta-\alpha\) is equal to ........ .

In an adiabatic process, which of the following statements is true?