The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons \(N \approx 4 \times 10^{27} \mathrm{~m}^{-1}\). Take \(\varepsilon_0=10^{-11}\) and \(m=10^{-30}\), where these quantities are in proper SI units.

${ }^{131}I is$ an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of $8 days$ . A small amount of a serum labelled with ${ }^{131}I$ is injected into the blood of a person. The activity of the amount of ${ }^{131}I$ injected was $2.4\times {10}^5$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of $115 Bq$ . The total volume of blood in the person's body, in litres is approximately (you may use \(\mathrm{e}^{\mathrm{x}} \approx 1+\mathrm{x}\) for \(|\mathrm{x}|<<1\) and \(\ln 2 \approx 0.7\)).

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question :

Which one of the following statements is correct?

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{Q}{R{T}_0}$ is:

A mixture of 2 moles of helium gas (atomic mass \(=4 \mathrm{amu}\) ) and 1 mole of argon gas (atomic mass \(=40 \mathrm{amu}\) ) is kept at \(300 \mathrm{~K}\) in a container. The ratio of the rms speeds \(\left(\frac{v_{\mathrm{rms}}(\text { helium })}{v_{\mathrm{rms}}(\operatorname{argon})}\right)\) is

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures \(2 T\) and \(3 T\) respectively. The temperature of the middle (i.e. second) plate under steady state condition is

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $\mathrm{V}\mathrm{ }\mathrm{m}{\mathrm{m}}^3$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is 10mm, then the value of $\frac{\mathrm{V}}{250\mathrm{\pi }}\mathrm{ }$ is

The difference in the oxidation numbers of the two types of sulphur atoms in \(\mathrm{Na}_2 \mathrm{~S}_4 \mathrm{O}_6\) is

Let \(z\) be a complex number such that the imaginary part of \(z\) is non-zero and \(a=z^{2}+z+1\) is real. Then a cannot take the value

In the Newman projection for 2,2-dimethylbutane

\(X\) and \(Y\) can respectively be

Among the species given below, the total number of diamagnetic species is _____________.
\(H\) atom, \(N O_2\) monomer, \(O_2^{-}\)(superoxide), dimeric sulphur in vapour phase,
\(Mn_3 O_4, \left(NH_4\right)_2\left[FeCl_4\right], \left(NH_4\right)_2\left[NiCl_4\right],\) \(K_2 MnO_4, K_2 CrO_4\)