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In a mixture of \(\mathrm{H}-\mathrm{H}^{+}\)gas \(\left(\mathrm{He}^{+}\right.\)is singly ionised \(\mathrm{He}\) atom), \(\mathrm{H}\) atoms and \(\mathrm{He}^{+}\)ions are excited to their respective first excited states. Subsequently, \(\mathrm{H}\) atoms transfer their total excitation energy to \(\mathrm{He}^{+}\)ions (by collisions). Assume that the Bohr model of atom is exactly valid.
Question:
The quantum number \(n\) of the state finally populated in \(\mathrm{He}^{+}\)ions is

A ray of light travelling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).

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 \(C A B=60^{\circ}\).

On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{kg}$ is attached to a spring with natural length $l_0=0.1 \mathrm{m}$. The spring constant is $k_1=0.009 \mathrm{N} {\mathrm{m}}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016 \mathrm{N} {\mathrm{m}}^{-1}$ when $l<l_0$. Initially the bob is released from $l=0.15 \mathrm{m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=\left(n\pi \right)\mathrm{s}$, then the integer closest to $n$ is _______.

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

Consider both sources emitting sound. When $S_2$ is at $R$ and $S_1$ approaches the detector with a speed $4 \mathrm{m} {\mathrm{s}}^{-1}$, the beat frequency measured by the detector is _____ $\mathrm{Hz}$.

Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

A mixture of two salts is used to prepare a solution

The correct option (s) for the salt mixture is(are)

A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $\theta ={30}^o$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are:

If \(x^2-10 a x-11 b=0\) have roots \(c\) and \(d\) and \(x^2-10 c x-11 d=0\) have roots \(a\) and \(b\), then, find \(a+b+c+d\).

Total number of hydroxyl groups present in a molecule of the major product $\mathrm{P}$ is ______

All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II.