An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$V\left(z\right)=\frac{\sigma }{2{ϵ}_0}\left(\sqrt{R^2+z^2}-z\right)$ A particle of positive charge $q$ is placed initially at rest at a point on the $z$-axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\overrightarrow{F}=-c\hat{k}$ with $c>0$. Let $\beta =\frac{2c{\epsilon }_0}{q\sigma }$. Which of the following statement(s) is(are) correct?

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement(s) is (are) true?

One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string is $100\mathrm{ }\mathrm{m}{\mathrm{s}}^{-1}$ . The other end of the string is vibrating in the y direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is (are)

Figure shows three resistor configurations \(R_1, R_2\) and \(R_3\) connected to \(3 \mathrm{~V}\) battery. If the power dissipated by the configuration \(R_1, R_2\) and \(R_3\) is \(P_1, P_2\) and \(P_3\), respectively, then

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Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\)\(\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)+2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=\) \(-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ;\) \(E_{\mathrm{red}}^{\circ}=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\) ]
Question:
Ammonia is always added in this reaction. Which of the following must be incorrect?

A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

A solution of $0.1\mathrm{M}$ weak base $\left(\mathrm{B}\right)$ is titrated with $0.1\mathrm{M}$ of a strong acid $\left(\mathrm{HA}\right)$. The variation of $\mathrm{pH}$ of the solution with the volume of $\mathrm{HA}$ added is shown in the figure below. What is the ${\mathrm{pK}}_{\mathrm{b}}$ of the base? The neutralization reaction is given by$\mathrm{B}+\mathrm{HA}\to {\mathrm{BH}}^{+}+{\mathrm{A}}^{-}$.

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match each entry in List-I with the appropriate entry in List-II and choose the correct options.

List - I	List - II
(P) Stephen reaction	(1)
(Q) Sandmeyer reaction	(2)
(R) Hoffmann bromamide degradation reaction	(3)
(S) Cannizzaro reaction	(4)
	(5)

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

The number of $3\times 3$ matrices $M$ with entries from $\{\mathrm{0,1},2\}$, such that the sum of the diagonal elements of $M^{T}M$ is $5$, are