A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct?

A parallel plate capacitor of capacitance $\mathrm{C}$ has spacing d between two plates having area $\mathrm{A}.$ The region between the plates is filled with $\mathrm{N}$ dielectric layers, parallel to its plates, each with thickness $\mathrm{\delta }=\frac{\mathrm{d}}{\mathrm{N}}.$ The dielectric constant of the ${\mathrm{m}}^{\mathrm{t}\mathrm{h}}$ layer is ${\mathrm{K}}_{\mathrm{m}}=\mathrm{K}\left(1+\frac{\mathrm{m}}{\mathrm{N}}\right).$ For a very large $\mathrm{N}\left(>{10}^3\right),$ the capacitance $\mathrm{C}$ is $\mathrm{\alpha }\left(\frac{\mathrm{K}{\in }_0\mathrm{A}}{\mathrm{d}\mathrm{l}\mathrm{n}2}\right).$ The value of $\mathrm{\alpha }$ will be____
$[{\in }_0$ is the permittivity of free space]

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

If the tension in each string is ${\mathrm{T}}_0,$ the correct match for the highest fundamental frequency in ${\mathrm{f}}_0$ units will be,

Paragraph :
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.

A string of length $1 \mathrm{m}$ and mass $2\times {10}^{-5} \mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{Hz}$ and $1000 \mathrm{Hz}$. The value of tension $T$ is _____ newton.

In an $X$-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2},$ the potential difference $V$ is increased to $2V,$ and the separation distance $d$ is reduced to $\frac{d}{2},$ then

Paragraph:

A small block of mass \(1 \mathrm{~kg}\) is released from rest at the top of a rough track. The track is a circular arc of radius \(40 \mathrm{~m}\). The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point \(Q\), as shown in the figure below, is \(150 \mathrm{~J}\). (Take the acceleration due to gravity, \(g=10 \mathrm{~m} \mathrm{~s}^{-2}\) ).





Question:

The speed of the block when it reaches the point \(Q\) is

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is

For the reaction sequence given below, the correct statement(s) is(are)

Perpendiculars are drawn from points on the line $\frac{\text{x}+2}{2}=\frac{\text{y}+1}{-1}=\frac{\text{z}}{3}$  to the plane $x+y+z=3$. The feet of perpendiculars lie on the line

The number of resonance structures for N is

Paragraph :
Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential \(\left(E^{\circ}\right)\) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their \(E^{\circ}(V\) with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.
\(\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} E^{\circ}=0.54 \)
\( \mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} E^{\circ}=1.36 \)
\( \mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} E^{\circ}=1.50 \)
\( \mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} E^{\circ}=0.77 \)
\( \mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} E^{\circ}=1.23\)
Question :
Sodium fusion extract, obtained from aniline, on treatment with iron (II) sulphate and \(\mathrm{H}_2 \mathrm{SO}_4\) in presence of air gives a prussian blue precipitate. The blue colour is due to the formation of

Upon treatment with ammoniacal ${\text{H}}_2\text{S}$, the metal ion that precipitates as a sulphide is