A thin and uniform rod of mass $\mathrm{M}$ and length $\mathrm{L}$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^{\mathrm{o}}$ with vertical?
[ $\mathrm{g}$ is the acceleration due to gravity]

Two solid spheres \(A\) and \(B\) of equal volumes but of different densities \(d_A\) and \(d_B\) are connected by a string. They are fully immersed in a fluid of density \(d_F\). They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

Paragraph:
When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
If the mass of the particle is \(m=1.0 \times 10^{-30} \mathrm{~kg}\) and \(a=6.6 \mathrm{~nm}\), the energy of the particle in its ground state is closest to

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then

Electrons with de-Broglie wavelength \(\lambda\) fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

A thermally isolated cylindrical closed vessel of height $8\mathrm{m}$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3\mathrm{kg}$. Thus the partition is held initially at a distance of $4\mathrm{m}$ from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains $0.1$ mole of an ideal gas at temperature $300\mathrm{K}$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in $\mathrm{m}$) will be ____________ (take the acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and the universal gas constant$=8.3\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$).

Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
Now level of the liquid starts decreasing slowly. When the level of liquid is at a height \(h_1\) above the cylinder, the block starts moving up. At what value of \(h_1\), will the block rise?

In thermodynamics, the $\mathrm{P}-\mathrm{V}$ work done is given by $\mathrm{w}=-\int {\mathrm{dVP}}_{\mathrm{ext}}$. For a system undergoing a particular process, the work done is,
$\mathrm{w}=-\int \mathrm{dV}\left(\frac{\mathrm{RT}}{\mathrm{V}-\mathrm{b}}-\frac{\mathrm{a}}{{\mathrm{V}}^2}\right)$. This equation is applicable to a

Four solid spheres each of diameter \(\sqrt{5} \mathrm{~cm}\) and mass \(0.5 \mathrm{~kg}\) ar placed with their centres at the corners of a square of side \(4 \mathrm{~cm}\). The moment of inertia of the system about the diagonal of the square is \(N \times 10^{-4} \mathrm{~kg}-\mathrm{m}^2\), then \(N\) 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.

A particle of mass $m$ is attached to one end of a massless spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5u_0$, it collides elastically with a rigid wall. After this collision :

The function \(f:[0,3] \rightarrow[1,29]\), defined by \(f(x)=2 x^{3}-15 x^{2}+36 x+1\), is

Let $f_1: \mathbb{R}\to \mathbb{R}, f_2\left(-\frac{\pi }{2},\frac{\pi }{2} \right)\to \mathbb{R}, f_3:\left(-1, e^{\frac{\pi }{2}}-2\right)\to \mathbb{R}$ and $f_4: \mathrm{ℝ}\to \mathrm{ℝ}$ be functions defined by
(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$
(ii) $f_2\left(x\right)=\left\{\begin{array}{ll}\frac{\left|\sin x\right|}{{\tan }^{-1}\left(x\right)}& , x\ne 0\\ 1& , x=0\end{array}\right.$ , where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
(iii) $f_3\left(x\right)=\left[\sin \left({\log }_{\text{e}}\left(x+2\right)\right)\right]$ , where, for $t\in R, \left[t\right]$ denotes the greatest integer less than or equal to $t$ ,
(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right) & , x\ne 0\\ 0 & , x=0\end{array}\right.$

LIST-I	LIST-II
A. the function $f_1$ is	P. NOT continuous at $x=0$
B. The function $f_2$ is	Q. continuous at $x = 0$ and NOT differentiable at $x=0$
C. The function $f_3$ is	R. differentiable at $x=0$ and its derivative is NOT continuous at $x=0$
D. The function $f_4$ is	S. differentiable at $x=0$ and its derivative is continuous at $x=0$

The correct option is :