A container has a base of $50\mathrm{cm}\times 5\mathrm{cm}$ and height $50\mathrm{cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50\mathrm{cm}\times 50\mathrm{cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant $3$at a uniform rate of $250{\mathrm{cm}}^3{\mathrm{s}}^{-1}$. What is the value of the capacitance of the container after $10$seconds?
[Given: Permittivity of free space ${\epsilon }_0=9\times {10}^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]

Three objects \(A, B\) and \(C\) are kept in a straight line on a frictionless horizontal surface. These have masses \(m, 2 m\) and \(m\), respectively. The object \(A\) moves towards \(B\) with a speed \(9 \mathrm{~ms}^{-1}\) and makes an elastic collision with it. Thereafter, \(B\) makes completely inelastic collision with \(C\). All motions occur on the same straight line. Find the final speed (in \(\mathrm{ms}^{-1}\) ) of the object \(C\).

A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

A small object is placed at the centre of a large evacuated hollow spherical container. Assume the container is maintained at $0 \mathrm{K}$. At time $t=0$, the temperature of the object is $200 \mathrm{K}$. The temperature of the object becomes $100 \mathrm{K}$ at $t=t_1$ and $50 \mathrm{K}$ at $t=t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(\frac{t_2}{t_1}\right)$ _________ .

A thin ring of mass \(2 \mathrm{~kg}\) and radius \(0.5 \mathrm{~m}\) is rolling without slipping on a horizontal plane with velocity 1 \(\mathrm{m} / \mathrm{s}\). A small ball of mass \(0.1 \mathrm{~kg}\), moving with velocity \(20 \mathrm{~m} / \mathrm{s}\) in the opposite directions, hits the ring at a height of \(0.75 \mathrm{~m}\) and goes vertically up with velocity 10 \(\mathrm{m} / \mathrm{s}\). Immediately after the collision

A spherical body of radius $\mathrm{R}$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $\mathrm{P}\left(\mathrm{r}\right)$ is the pressure at $\mathrm{r}(\mathrm{r}<\mathrm{R})$ , then the correct option(s) is (are)

Four solid spheres each of diameter \(\sqrt{5} \mathrm{~cm}\) and mass \(0.5 \mathrm{~kg}\) ar placed with their centres at the corners of a square of side \(4 \mathrm{~cm}\). The moment of inertia of the system about the diagonal of the square is \(N \times 10^{-4} \mathrm{~kg}-\mathrm{m}^2\), then \(N\) is

Statement 1 The plot of atomic number ( \(Y\)-axis) versus number of neutrons ( \(x\)-axis) for stable nuclei shows a curvature towards \(X\)-axis from the line of \(45^{\circ}\) slope as the atomic number is increased.
Statement 2 Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.

A planet of mass M, has two natural satellites with masses $m_1\mathrm{a}\mathrm{n}\mathrm{d}{m}_2$ . The radii of their circular orbits are $R_1 \mathrm{a}\mathrm{n}\mathrm{d} R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1,L_1,K_1 \mathrm{a}\mathrm{n}\mathrm{d} T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $1; \mathrm{a}\mathrm{n}\mathrm{d} v_2,L_2,K_2\mathrm{a}\mathrm{n}\mathrm{d} T_2$ and to be the corresponding quantities of satellite 2. Given $m_1/m_2=2 \mathrm{a}\mathrm{n}\mathrm{d} R_1/R_2=\frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\frac{v_1}{v_2}$	P)	$\frac{1}{8}$
B)	$\frac{L_1}{L_2}$	Q)	1
C)	$\frac{K_1}{K_2}$	R)	2
D)	$\frac{T_1}{T_2}$	S)	8

A signal which can be green or red with probability \(\frac{4}{5}\) and \(\frac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station B. The probability of each station receiving the signal correctly is \(\frac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal green is

\(\mathrm{MgSO}_4\) on reaction with \(\mathrm{NH}_4 \mathrm{OH}\) and \(\mathrm{Na}_2 \mathrm{HPO}_4\) forms a white crystalline precipitate. What is its formula?

In an electrochemicalcell, dichromate ions in aqueous acidic medium are reduced to \(\mathrm{Cr}^{3+}\). The current (in amperes) that flows through the cell for 48.25 minutes to produce 1 mole of \(\mathrm{Cr}^{3+}\) is _____ .
Use: 1 Faraday \(=96500 \mathrm{C} \mathrm{mol}^{-1}\)

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?