Four identical thin, square metal sheets, \(S_1, S_2, S_3\) and \(S_4\), each of side \(a\) are kept parallel to each other with equal distance \(d(\ll a)\) between them, as shown in the figure. Let \(C_0=\varepsilon_0 a^2 / d\), where \(\varepsilon_0\) is the permittivity of free space.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=\pi /(2 \omega)\), then the value of \(4 b^2 / a^2\) will be ________ .

A conducting wire of parabolic shape, initially $\mathrm{y}={\mathrm{x}}^2,$ is moving with velocity $\overrightarrow{\mathrm{V}}={\mathrm{V}}_0\widehat{\mathrm{i}}$ in a non-uniform magnetic field $\overrightarrow{\mathrm{B}}={\mathrm{B}}_0\left(1+{\left(\frac{\mathrm{y}}{\mathrm{L}}\right)}^{\mathrm{\beta }}\right)\widehat{\mathrm{k}},$ as shown in figure. If ${\mathrm{V}}_0,{\mathrm{B}}_0,\mathrm{L}$ and $\mathrm{\beta }$ are positive constants and $\mathrm{\Delta }\mathrm{\varphi }$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

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 metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

The circular scale of a screw gauge has 50 divisions and pitch of \(0.5 \mathrm{~mm}\). Find the diameter of sphere. Main scale reading is 2 .

An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is

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The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
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\(\mathrm{XeF}_4\) and \(\mathrm{XeF}_6\) are expected to be

Consider the following compounds in the liquid form:
\(O_2, HF, H_2 O, NH_3, H_2 O_2, CCl_4, CHCl_3,\) \(C_6 H_6, C_6 H_5 Cl .\)
When a charged comb is brought near their flowing stream, how many of them show deflection as per the following figure?

A thermally isolated cylindrical closed vessel of height $8\mathrm{m}$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3\mathrm{kg}$. Thus the partition is held initially at a distance of $4\mathrm{m}$ from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains $0.1$ mole of an ideal gas at temperature $300\mathrm{K}$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in $\mathrm{m}$) will be ____________ (take the acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and the universal gas constant$=8.3\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$).

The substituents \(R_{1}\) and \(R_{2}\) for nine peptides are listed in the table given below. How many of these peptides are positively charged at \(\mathrm{pH}=7.0\) ?

\(\begin{array}{|c|c|c|}\hline Peptide & \mathbf{R}_{\mathbf{1}} & \mathbf{R}_{\mathbf{2}} \\\hline I & \mathrm{H} & \mathrm{H} \\\hline II & \mathrm{H} & \mathrm{CH}_{3} \\\hline III & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{H} \\\hline IV & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline V & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VI & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline VII & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VIII & \mathrm{CH}_{2} \mathrm{OH} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline IX & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \mathrm{CH}_{3} \\\hline\end{array}\)

Let the point $\mathrm{B}$ be the reflection of the point $\mathrm{A}\left(2,\mathrm{ }3\right)$ with respect to the line $8\mathrm{x}-6\mathrm{y}-23=0.$ Let ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ be circle of radii $2$ and $1$ with centres $\mathrm{A}$ and $\mathrm{B}$ respectively. Let $\mathrm{T}$ be a common tangent to the circle ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ such that both the circle are on the same side of $\mathrm{T}.$ If $\mathrm{C}$ is the point of intersection of $\mathrm{T}$ and the line passing through $\mathrm{A}$ and $\mathrm{B},$ then the length of the line segment $\mathrm{A}\mathrm{C}$ is ___________.