Consider a configuration of $n$ identical units, each consisting of three layers. The first layer is a column of air of height $h=\frac{1}{3} \mathrm{cm}$, and the second and third layers are of equal thickness $d=\frac{\sqrt{3}-1}{2} \mathrm{cm}$, and refractive indices ${\mu }_1=\sqrt{\frac{3}{2}}$ and ${\mu }_2=\sqrt{3}$, respectively. A light source $O$ is placed on the top of the first unit, as shown in the figure. A ray of light from $O$ is incident on the second layer of the first unit at an angle of $\theta =60°$ to the normal. For a specific value of $n$, the ray of light emerges from the bottom of the configuration at a distance $l=\frac{8}{\sqrt{3}} \mathrm{cm}$, as shown in the figure. The value of $n$ is _______.

The graph between object distance \(u\) and image distance \(v\) for a lens is given below. The focal length of the lens is

Assume that the nuclear binding energy per nucleon \((B / A)\) versus mass number \((A)\) is as shown in the figure. Use this plot to choose the correct choice (s) given below.

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

A \(20 \mathrm{~cm}\) long string, having a mass of \(1.0 \mathrm{~g}\), is fixed at both the ends. The tension in the string is \(0.5 \mathrm{~N}\). The string is set into vibration using an external vibrator of frequency 100 \(\mathrm{Hz}\). Find the separation (in \(\mathrm{cm}\) ) between the successive nodes on the string.

Match the following for the given process

A field line is shown in the figure. This field cannot represent

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A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.


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If \(1\) ball is drawn from each of the boxes \(B_{1}, B_{2}\) and \(B_{3}\), the probability that all \(3\) drawn balls are of the same colour is

A signal which can be green or red with probability \(\frac{4}{5}\) and \(\frac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station B. The probability of each station receiving the signal correctly is \(\frac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal green is

The total number of distinct naturally occurring amino acids obtained by complete acidic hydrolysis of the peptide shown below is

If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are

Consider the reaction sequence from $\mathrm{P}$ to $\mathrm{Q}$ shown below. The overall yield of the major product $\mathrm{Q}$ from $\mathrm{P}$ $75\%.$ What is the amount in grams of $\mathrm{Q}$ obtained from $9.3 \mathrm{mL}$ $\mathrm{P}$? (Use density of $=1.00{\text{\hspace{0.33em}g\hspace{0.33em}mL}}^{-1};$ Molar mass of $\mathrm{C}=12.0,\mathrm{H}=1.0,\mathrm{O}=16.0$ and $\mathrm{N}=14.0 \mathrm{g} {\mathrm{mol}}^{-1}$)

The Henry's law constant for the solubility of \(\mathrm{N}_2\) gas in water at \(298 \mathrm{~K}\) is \(1.0 \times 105 \mathrm{~atm}\). The mole fraction of \(\mathrm{N}_2\) in air is \(0.8\). The number of moles of \(\mathrm{N}_2\) from air dissolved in 10 moles of water of \(298 \mathrm{~K}\) and \(5 \mathrm{~atm}\) pressure is