A rigid uniform bar $AB$ of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\mathrm{\theta }$. Which of the following statements about its motion is correct?

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $II$	Q)	Temperature of the gas remains unchanged
C)	In process $III$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $IV$	S)	Work done by the gas is $6P_0{V}_0$

Paragraph:
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

A hollow pipe of length \(0.8 \mathrm{~m}\) is closed at one end. At its open end a \(0.5 \mathrm{~m}\) long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is \(50 \mathrm{~N}\) and the speed of sound is \(320 \mathrm{~ms}^{-1}\), the mass of the string is

Consider a star of mass \(m_2 \mathrm{~kg}\) revolving in a circular orbit around another star of mass \(m_1 \mathrm{~kg}\) with \(m_1 \gg m_2\). The heavier star slowly acquires mass from the lighter star at a constant rate of \(\gamma \mathrm{kg} / \mathrm{s}\). In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is \(r\), then its relative rate of change \(\frac{1}{r} \frac{\mathrm{~d} r}{\mathrm{~d} t}\left(\right.\) in s \(\left.^{-1}\right)\) is given by:
[Note: The original question was awarded bonus marks in the JEE Advanced 2025 exam. We have changed the options to make the question valid and solvable.]

The left and right compartments of a thermally isolated container of length \(L\) are separated by a thermally conducting, movable piston of area \(A\). The left and right compartments are filled with \(\frac{3}{2}\) and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant k and natural length \(\frac{2 L}{5}\). In thermodynamic equilibrium, the piston is at a distance \(\frac{L}{2}\) from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is \(P=\frac{\mathrm{k} L}{\mathrm{~A}} \alpha\), then the value of \(\alpha\) is \(\qquad\)

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

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

An organic compound $\left({\mathrm{C}}_8{\mathrm{H}}_{10}{\mathrm{O}}_2\right)$ rotates plane-polarized light. It produces pink color with neutral ${\mathrm{FeCl}}_3$ solution. What is the total number of all the possible isomers for this compound?

In the following reaction sequence, the amount of $\text{D}$ (in g) formed from $10$ moles of acetophenone is ____.
(Atomic weights in $\text{g}\text{m}\text{o}{\text{l}}^{-1}:\text{H}=1,\text{C}=12,\text{N}=14,\text{O}=16,\text{B}\text{r}=80.$ The yield (%) corresponding to the product in each step is given in the parenthesis)

A straight line through the vertex \(P\) of a \(\triangle P Q R\) intersects the side \(Q R\) at the point \(S\) and the circumcircle of the \(\triangle P Q R\) at the point \(T\). If \(S\) is not the centre of the circumcircle, then

Match the following.
 

	List – I		List – II
(A)	Let $y\left(x\right)=\cos \left(3{\cos }^{-1}x\right), x ϵ \left[-1, 1\right], x\ne \pm \frac{\sqrt{3}}{2}.$ Then $\frac{1}{y\left(x\right)} \left\{\left(x^2-1\right)\frac{d^{2}y\left(x\right)}{d{x}^2}+x\frac{dy\left(x\right)}{dx}\right\}$ equals	(P)	1
(B)	Let $A_1, A_2, \dots , A_n\left(n>2\right)$ be the vertices of a regular polygon of n sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k, k=1, 2, \dots n.$ If $\left|{\sum }_{k=1}^{n-1}\left(\mathrm{ }\overrightarrow{a_k}\times \mathrm{ }\overrightarrow{a_{k+1}}\right)\right|=\left|{\sum }_{k=1}^{n-1}\left(\mathrm{ }\overrightarrow{a_k} \cdot \mathrm{ }\overrightarrow{a_{k+1}}\right)\right|$ , then the minimum value of n is	(Q)	2
(C)	If the normal from the point $P\left(h, 1\right)$ on the ellipse $\frac{x^2}{6}+\frac{y^2}{3}=1$ is perpendicular to the line $x+y=8,$ then the value of h is	(R)	8
(D)	Number of positive solutions satisfying the equation ${\tan }^{-1}\left(\frac{1}{2x+1}\right)+{\tan }^{-1}\left(\frac{1}{4x+1}\right)={\tan }^{-1}\left(\frac{2}{x^2}\right)$ is	(S)	9

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.