An infinitely long uniform line charge distribution of charge per unit length $\mathrm{\lambda }$ lies parallel to the $\mathrm{y}$ -axis in the $\mathrm{y}$ - $\mathrm{z}$ plane at $\mathrm{z}=\frac{\sqrt{3}}{2}\mathrm{\alpha }$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ lying in the $\mathrm{x}$ - $\mathrm{y}$ plane with its centre at the origin is $\frac{\mathrm{\lambda }\mathrm{L}}{\mathrm{n}{\mathrm{\epsilon }}_0}$ ( ${\mathrm{\epsilon }}_0=$ permittivity of free space), then the value of $\mathrm{n}$ is

Some laws/processes are given in Column I. Match these with the physical phenomena given in Column II and indicate your answer by darkening appropriate bubbles in the \(4 \times 4\) matrix given in the ORS.

Two bodies, each of mass $M$, are kept fixed with a separation of $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centers, perpendicular to the line. The gravitational constant is $G$. The correct statement (s) is (are) :

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 _______.

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l.$ A point dipole of dipole moment $\overrightarrow{p}$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times \left(mgh\right)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ____________ (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{\mathrm{F}}{\sin \theta }$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is \(60^{\circ}\) ). In the position of minimum deviation, the angle of refraction will be

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Assume that two deuteron nuclei in the core of fusion reactor at temperature \(T\) are moving towards each other, each with kinetic energy \(1.5 \mathrm{kT}\), when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature \(T\) required for them to reach a separation of \(4 \times 10^{-15} \mathrm{~m}\) is in the range

\(29.2 \%(\mathrm{w} / \mathrm{w}) \mathrm{HCl}\) stock solution has a density of \(1.25 \mathrm{gmL}^{-1}\). The molecular weight of \(\mathrm{HCl}\) is \(36.5 \mathrm{~g} \mathrm{~mol}^{-1}\). The volume \((\mathrm{mL})\) of stock solution required to prepare a 200 \(\mathrm{mL}\) solution of \(0.4 \mathrm{M} \mathrm{HCl}\) is :

In a non-right-angled triangle $\mathrm{\Delta }PQR,$ let $p\mathit{,}q\mathit{,}r$ denote the lengths of the sides opposite to the angles at $P,Q,R$ respectively. The median from $R$ meets the side $PQ$ at $S,$ the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $0.$ If $p=\sqrt{3},q=1,$ and the radius of the circumcircle of the $\mathrm{\Delta }PQR$ equals $1,$ then which of the following options is/are correct?

An optical bench has $1.5\mathrm{ }\mathrm{m}$ long scale having four equal divisions in each $\mathrm{c}\mathrm{m}.$ While measuring the focal length of a convex lens, the lens is kept at $75\mathrm{ }\mathrm{c}\mathrm{m}$ mark of the scale and the object pin is kept at $45\mathrm{ }\mathrm{c}\mathrm{m}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135\mathrm{ }\mathrm{c}\mathrm{m}$ mark. In this experiment, the percentage error in the measurement of the focal length of the lens is_____

Extraction of metal from the ore cassiterite involves

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

Let \(A B C\) and \(A B C^{\prime}\) be two non-congruent triangles with sides \(A B=4\), \(A C=A C^{\prime}=2 \sqrt{2}\) and angle \(B=30^{\circ}\). The absolute value of the difference between the areas of these triangles is