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]

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

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Two plane harmonic sound waves are expressed by the equations.
\(
\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}
\)
(All parameters are in MKS)
Question :
What is the speed of the sound?

There are two vernier calipers both of which have $1\mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are

Consider a system of three charges \(\frac{q}{3}, \frac{q}{3}\) and \(-\frac{2 q}{3}\) placed at points \(A, B\) and \(C\), respectively, as shown in the figure. Take \(O\) to be the centre of the circle of radius \(R\) and angle \(C A B=60^{\circ}\).

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Two plane harmonic sound waves are expressed by the equations.
\(\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}\)
(All parameters are in MKS)
Question :
How many times does an observer hear maximum intensity in one second?

Function \(x=A \sin ^2 \omega t+B \cos ^2 \omega t+C\) \(\sin \omega t \cos \omega t\) represents SHM.

For positive integer $n$, define $f\left(n\right)=n+\frac{16+5n-3n^2}{4n+3n^2}+\frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-3n^2}{12n+3n^2}+\dots +\frac{25n-7n^2}{7n^2}$. Then, the value of $\underset{n\to \infty }{\lim }f\left(n\right)$ is equal to

A \(2 \mu \mathrm{F}\) capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch \(S\) is turned to position 2 , is

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A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the density of air is \(\rho_{a}\) and that of the liquid \(\rho_{\ell}\), then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

A series \(R\) - \(C\) circuit is connected to \(A C\) voltage source. Consider two cases; \((A)\) when \(C\) is without a dielectric medium and \((B)\) when \(C\) is filled with dielectric of constant 4. The current \(I_R\) through the resistor and voltage \(V_C\) across the capacitor are compared in the two cases. Which of the following is/are true?

Let the function $f:R\to R$ be defined by $f\left(x\right)=x^3-x^2+\left(x-1\right)\sin x$ and let $g:R\to R$ be an arbitrary function. Let $fg:R\to R$ be the product function defined by $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$. Then which of the following statements is/are TRUE?

Extraction of copper from copper pyrite $\left(\mathrm{C}\mathrm{u}\mathrm{F}\mathrm{e}{\mathrm{S}}_2\right)$ involves