Young’s modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck’s constant $h$ and the speed of light $c$, as $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option?

As shown in the figures, a uniform rod \(O O^{\prime}\) of length \(l\) is hinged at the point \(O\) and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end \(\left(O^{\prime}\right)\) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is \(f_1\). On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is \(f_2\). Ignoring gravity and assuming motion only in the plane of the diagram, the value of \(\frac{f_1}{f_2}\) is:

A ball of mass \(0.2 \mathrm{~kg}\) rests on a vertical post of height \(5 \mathrm{~m}\). A bullet of mass \(0.01 \mathrm{~kg}\), travelling with a velocity \(v \mathrm{~m} / \mathrm{s}\) in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of \(20 \mathrm{~m}\) and the bullet at a distance of \(100 \mathrm{~m}\) from the foot of the post. The initial velocity \(v\) of the bullet is

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\mathrm{\lambda }$ are kept parallel to each other. In their resulting electric field, point charges $\mathrm{q}$ and $-\mathrm{q}$ are kept in equilibrium between them. The point charges are confined to move in the $\mathrm{x}$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statements(s) is (are)

A particle, of mass ${10}^{-3}\mathrm{ kg}$ and charge $1.0 \mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\overrightarrow{E}\left(t\right)=E_0\sin \left(\omega t\right) \hat{\mathrm{i}}$ where $E_0=1.0\mathrm{ N} {\mathrm{C}}^{-1}$ and $\omega ={10}^3\mathrm{ rad} {\mathrm{s}}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m} {\mathrm{s}}^{-1}$, attained by the particle at subsequent times is ____________.

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance d of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the lists.
 

	List I		List II
A.	Lift is accelerating vertically up.	$P.$	$\mathrm{d}=1.2\mathrm{ }\mathrm{m}$
B.	Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.	$Q.$	$\mathrm{d}>1.2\mathrm{ }\mathrm{m}$
C.	Lift is moving vertically up with constant speed.	$R.$	$\mathrm{d}<1.2\mathrm{ }\mathrm{m}$
D.	Lift is falling freely.	$S.$	No water leaks out of the jar

Starting at time $t=0$ from the origin with speed $1 \mathrm{m} {\mathrm{s}}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\frac{x^2}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_x$ and $a_y$, respectively. Then

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

The graph between \(1 / \lambda\) and stopping potential \((V)\) of three metals having work function \(\phi_1, \phi_2\) and \(\phi_3\) in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here \(\lambda\) is the wavelength of the incident ray].

A charged shell of radius $\mathrm{R}$ carries a total charge $\mathrm{Q}.$ Given $\mathrm{\Phi }$ as the flux of electric field through a closed cylindrical surface of height $\mathrm{h},$ radius $\mathrm{r}$ and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ ${\in }_0$ is the permittivity of free space]

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

A piece of wire is bent in the shape of a parabola \(y=k x^2\) (y-axis vertical) with a bead of mass \(m\) on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the \(x\)-axis with the constant acceleration \(a\). The distance of the new equilibrium position of the bead, where the bead can stays at rest with respect to the wire, from the \(y\)-axis is

Paragraph:
\(RCONH _2\) is converted into \(RNH _2\) by means of Hofmann bromamide degradation.

In this reaction, \(R \mathrm{CONHBr}\) is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.
Question:
How can the conversion of (i) to (ii) be brought about?