When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0 \mathrm{V}$. This potential drops to $0.6 \mathrm{V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take$\frac{hc}{e}=1.24\times {10}^{-6} \mathrm{J} {\mathrm{mC}}^{-1}$ .]

A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral.

A vibrating string of certain length \(l\) under a tension \(T\) resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length \(75 \mathrm{~cm}\) inside a tube closed at one end. The string also generates 4 beat/s when excited along with a tuning fork of frequency \(n\). Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be \(340 \mathrm{~m} / \mathrm{s}\), the frequency \(n\) of the tuning fork (in \(\mathrm{Hz}\) ) is

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	${10}^{-3} \mathrm{kg}$ of water $100 °\mathrm{C}$ is converted to steam at the same temperature, at a pressure of ${10}^5\mathrm{Pa}$. The volume of the system changes from ${10}^{-6}{\mathrm{m}}^3$ to ${10}^{-3}{\mathrm{m}}^3$ in the process. Latent heat of water $=2250 \mathrm{kJ} {\mathrm{kg}}^{-1}$	(P)	$2 \mathrm{kJ}$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{K}$ undergoes an isobaric expansion to volume $3 V$. Assume $8.0 \mathrm{J} {\mathrm{mol}}^{–1} {\mathrm{K}}^{–1}$	(Q)	$7 \mathrm{kJ}$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3}{\mathrm{m}}^3$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$.	(R)	$4 \mathrm{kJ}$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{kJ}$ of heat and undergoes isobaric expansion.	(S)	$5 \mathrm{kJ}$
		(T)	$3 \mathrm{kJ}$

Which one of the following options is correct?

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated ($P$) by the metal. The sensor has a scale that displays ${\log }_2\left(\frac{P}{P_0}\right)$, where $P_0$ is a constant. When the metal surface is at a temperature of $487 °\mathrm{C}$, the sensor shows a value $1$. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767 °\mathrm{C}$?

A double star system consists of two stars \(A\) and \(B\) which have time period \(T_A\) and \(T_B\). Radius \(R_A\) and \(R_B\) and mass \(M_A\) and \(M_B\). Choose the correct option.

On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{kg}$ is attached to a spring with natural length $l_0=0.1 \mathrm{m}$. The spring constant is $k_1=0.009 \mathrm{N} {\mathrm{m}}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016 \mathrm{N} {\mathrm{m}}^{-1}$ when $l<l_0$. Initially the bob is released from $l=0.15 \mathrm{m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=\left(n\pi \right)\mathrm{s}$, then the integer closest to $n$ is _______.

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Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g\left(\frac{1}{2}\right)\) is

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors \(\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}\) such that \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}=\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}=\frac{1}{2}\). Then, the volume of the parallelopiped is

\(\frac{d^2 x}{d y^2}\) equals

A \(0.1 \mathrm{~kg}\) mass is suspended from a wire of negligible mass. The length of the wire is \(1 \mathrm{~m}\) and its cross-sectional area is \(4.9 \times 10^{-7} \mathrm{~m}^2\). If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency \(140 \mathrm{rad} \mathrm{s}^{-1}\). If the Young's modulus of the material of the wire is \(n \times 10^9 \mathrm{Nm}^{-2}\), the value of \(n\) is

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.