\(2 \mathrm{~kg}\) of ice at \(-20^{\circ} \mathrm{C}\) is mixed with \(5 \mathrm{~kg}\) of water at \(20^{\circ} \mathrm{C}\). Final mass of water formed is

If two soap bubbles each of radius 2 cm combine in vacuum under isothermal conditions, then the radius of the new bubble formed is

When a current of 4 mA passes through an inductor, if the flux linked with it is \(32 \times 10^{-6} \mathrm{~T} \mathrm{~m}^2\), then the energy stored in the inductor is

Currents flowing in each of the following circuits \(A\) and \(B\) respectively are

A neutral ammonia \(\left(\mathrm{NH}_3\right)\) in its vapour state has an electric dipole moment of magnitude \(5 \times 10^{-30} \mathrm{C}\)-m. How far apart are the molecules centres of positive and negative charge.

A launching vehicle carrying an artificial satellite of mass \(m\) is set for launch on the surface of the earth of mass \(M\) and radius \(R\). If the satellite is intended to move in a circular orbit of radius \(7 R\), the minimum energy required to be spent by the launching vehicle on the satellite is
\((\) Gravitational constant \(=G\) )

A uniform chain of mass \(m\) and length \(l\) is on a smooth horizontal table with \(\left(\frac{1}{n}\right)^{\text {th }}\) part of its length is hanging from one end of the table. The velocity of the chain, when it completely slips off the table is

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

\(f(x)=\left|\log _e\right| x \|\) is differentiable at

The de Broglie wavelength associated with a proton under the influence of an electric potential of $100$ volts is

$$\begin{gathered} \mathrm{State} 1 \leftrightarrows \mathrm{State} 2 \leftrightarrows \mathrm{State} 3 \\ \left(\begin{array}{c}\mathrm{T}=300\mathrm{K}\\ \mathrm{P}=15\mathrm{bar}\\ 1\mathrm{mole} \end{array}\right)\left(\begin{array}{c}\mathrm{T}=300\mathrm{K}\\ \mathrm{P}=10\mathrm{bar}\\ 1\mathrm{mole} \end{array}\right) \left(\begin{array}{c}\mathrm{T}=300\mathrm{K}\\ \mathrm{P}=5\mathrm{bar}\\ 1 \mathrm{mole} \end{array}\right) \end{gathered}$$ Above shows a cyclic process. Calculate the total work done during one complete cycle.
(Assume a single step to reach the next state).

The volume of a spherical balloon is increasing at the rate of $30\mathrm{cc}$ per minute. Find the rate of change of surface area of the balloon, when its radius is $6 \mathrm{cm}$.

If two angles of \(\triangle A B C\) are \(45^{\circ}\) and \(60^{\circ}\), then the ratio of the smallest and the greatest sides are