Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.

Paragraph:
When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The allowed energy for the particle for a particular value of \(n\) is proportional to

A drop of liquid of radius $R={10}^{-2}m$ having surface tension $S=\frac{0.1}{4\pi }N{m}^{-1}$ divides itself into K identical drops. In the process the total change in the surface energy $\mathrm{\Delta }U={10}^{-3}J$ . If $K={10}^{\alpha }$ then the value of $\alpha$ is

Consider a concave mirror and a convex lens (refractive index = $1.5$) of focal length $10$ $\mathrm{cm}$ each, separated by a distance of $50$ $\mathrm{cm}$ in air (refractive index = $1$), as shown in the figure. An object is placed at a distance of $15$ $\mathrm{cm}$ from the mirror. Its erect image, formed by this combination, has magnification ${\mathrm{M}}_1$ . When the set-up is kept in a medium of refractive index $\frac{7}{6}$ , the magnification becomes ${\mathrm{M}}_2$. The magnitude $\left|\frac{{\mathrm{M}}_2}{{\mathrm{M}}_1}\right|$ is

A proton is fired from very far away towards a nucleus with charge \(Q=120 e\), where \(e\) is the electronic charge. It makes a closest approach of \(10 \mathrm{fm}\) to the nucleus. The de Broglie wavelength (in units of \(\mathrm{fm}\) ) of the proton at its start is: (take the proton mass, \(m_{p}=(5 / 3) \times 10^{-27} \mathrm{~kg} ; h / e=4.2 \times\) \(10^{-15} \mathrm{~J} . \mathrm{S} / \mathrm{C}\);

\(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{~m} / \mathrm{F} ; 1 \mathrm{fm}=10^{-15} \mathrm{~m}\)

Paragraph :
Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature \(T_C(0)\). An interesting property of superconductors is that their critical temperature becomes smaller than \(T_C(0)\) if they are placed in a magnetic field i.e., the critical temperature \(T_C(B)\) is a function of the magnetic field strength \(B\). The dependence of \(T_C(B)\) on \(B\) is shown in the figure.

Question :
A superconductor has \(T_C(0)=100 \mathrm{~K}\). When a magnetic field of \(7.5\) Tesla is applied, its \(T_C\) decreases to \(75 \mathrm{~K}\). For this material one can definitely say that when (Note : \(\mathrm{T}\) = Tesla)

In the given $P-V$ diagram, a monoatomic gas $\left(\gamma =\frac{5}{3}\right)$ is first compressed adiabatically from state $A$ state $B$. Then it expands isothermally from state $B$ to state $C$. [Given : ${\left(\frac{1}{3}\right)}^{0.6}=0.5,\ln 2\approx 0.7$].

Which of the following statement(s) is(are) correct?

The colour of light absorbed by an aqueous solution of \(\mathrm{CuSO}_{4}\) is:

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

Two spherical planets \(P\) and \(Q\) have the same uniform density \(\rho\), masses \(M_{P}\) and \(M_{Q}\) and surface areas \(A\) and 4A respectively. A spherical planet \(R\) also has uniform density \(\rho\) and its mass is \(\left(M_{P}+M_{Q}\right)\). The escape velocities from the planets \(P, Q\) and \(R\) are \(V_{P}, V_{Q}\) and \(V_{R}\), respectively. Then

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$ , centred at $P$ with distance $OP=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\overrightarrow{r}$ is $\overrightarrow{E}\left(\overrightarrow{r}\right)$ , then the correct statement(s) is (are)

Different possible thermal decomposition pathways for peroxyesters are shown below. Match each pathway from List I with an appropriate structure from List II and select the correct answer using the code given below the lists.

	List – I		List – II
A	Pathway P	P.
B.	Pathway Q	Q.
C.	Pathway R	R.
D.	Pathway S	S.

The solubility of a salt of weak acid $\left(\mathrm{AB}\right)\mathrm{ }\mathrm{at}\mathrm{ }\mathrm{pH}3\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{Y}\times {10}^{-3}\mathrm{ }\mathrm{mol}\mathrm{ }{\mathrm{L}}^{-1}$ . The value of $\mathrm{Y}$ is ____. (Given that the value of solubility product of $\mathrm{AB}\left({\mathrm{K}}_{\mathrm{sp}}\right)=2\times {10}^{-10}\mathrm{ }$, and the value of ionization constant of $\mathrm{HB}\left(\mathrm{Ka}\right)=1\times {10}^{-8}$)