A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

The instantaneous voltages at three terminals marked $X, Y$ and $Z$ are given by
$V_X=V_0\sin \omega t$
$V_Y=V_0\sin \left(\omega t+\frac{2\pi }{3}\right)$ and
$V_Z=V_0\sin \left(\omega t+\frac{4\pi }{3}\right).$
An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$ . The reading(s) of the voltmeter will be

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	Column - I		Column - II
(a)	Work done by the system in process $1\to 2\to 3$	(p)	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
(b)	Change in internal energy in process $1\to 2\to 3$	(q)	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
(c)	Heat absorbed by the system in process $1\to 2\to 3$	(r)	$\mathrm{R}{\mathrm{T}}_0$
(d)	Heat absorbed by the system in process $1\to 2$	(s)	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		(t)	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		(u)	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process on one mole of monatomic ideal gas is as shown in the $\mathrm{P}\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is

A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

In the figure below, the switches $S_1\mathrm{a}\mathrm{n}\mathrm{d}{S}_2$ are closed simultaneously at $t=0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude $I_{max}$ time $t=\tau$. Which of the following statements is (are) true?

An AC voltage source of variable angular frequency \(\omega\) and fixed amplitude \(V_0\) is connected in series with a capacitance \(C\) and an electric bulb of resistance \(R\) (inductance zero). When \(\omega\) is increased

Match the following columns

A small particle of mass $\mathrm{m}$ moving inside a heavy, hollow and straight tube along the tube axis, undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $\mathrm{L}={\mathrm{L}}_0$ the particle speed is $\mathrm{v}={\mathrm{v}}_0.$ The piston is moved inward at a very low speed $\mathrm{V}$ such that $\mathrm{V}<<\frac{\mathrm{d}\mathrm{L}}{\mathrm{L}}{\mathrm{v}}_0,$ where $\mathrm{d}\mathrm{L}$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

Consider a reaction \(a G+b H \rightarrow\) products. When concentration of both the reactants \(G\) and \(H\) is doubled, the rate increases by eight times. However, when concentration of \(G\) is doubled keeping the concentration of \(H\) fixed, the rate is doubled. The overall order of the reaction is

Two ideal batteries of emf ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ and three resistances ${\mathrm{R}}_1,\mathrm{}{\mathrm{R}}_2$ and ${\mathrm{R}}_3$ are connected as shown in the figure. The current in resistance ${\mathrm{R}}_2$ would be zero if

Paragraph:
p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(X\), is

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $\mathrm{L},$ which of the following statement (s) is/are correct?

Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.

The mass of ink used to draw each of the two inner circles, and each of the two line segments is \(m\). The mass of the ink used to draw the outer circle is \(6 \mathrm{~m}\). The coordinates of the centres of the different parts are, outer circle 10 , \(0)\), left inner circle \((0,0)\), right inner circle \((a, a)\), vertical line \((0,0)\) and horizontal line \((0,-a)\). The \(y\)-coordinate of the centre of mass of the ink in this drawing is