The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}_7{}^{16}\mathrm{N}+{}_2{}^4\mathrm{He}\to {}_1{}^1\mathrm{H}+{}_8{}^{19}\mathrm{O}$ in a laboratory frame is $n$ (in $\mathrm{MeV}$). Assume that ${}_7{}^{16}\mathrm{N}$ is at rest in the laboratory frame. The masses of ${}_7{}^{16}\mathrm{N}, {}_2{}^4\mathrm{He}, {}_1{}^1\mathrm{H}$ and ${}_8{}^{19}\mathrm{O}$ can be taken to be $16.006 \mathrm{u}, 4.003 \mathrm{u}$, $1.008 \mathrm{u}$ and $19.003 \mathrm{u}$, respectively, where $1 \mathrm{u}=930 \mathrm{MeV} {\mathrm{c}}^{-2}$. The value of $n$is
If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Two spherical bodies \(A\) (radius \(6 \mathrm{~cm}\) ) and \(B\) (radius \(18 \mathrm{~cm}\) ) are at temperatures \(T_1\) and \(T_2\), respectively. The maximum intensity in the emission spectrum of \(A\) is at \(500 \mathrm{~nm}\) and in that of \(B\) is at \(1500 \mathrm{~nm}\). Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by \(A\) to that of \(B\) ?

In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma =\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a1}$, the radius of the bubble is found to be $r_1$ and the temperature of the enclosed gas is $T_1$. When the atmospheric pressure is $P_{\mathrm{a}2}$, the radius of the bubble and the temperature of the enclosed gas are $r_2$ and $T_2$, respectively. Which of the following statement(s) is(are) correct?

Steel wire of length \(L\) at \(40^{\circ} \mathrm{C}\) is suspended from the ceiling and then a mass \(m\) is hung from its free end. The wire is cooled down from \(40^{\circ} \mathrm{C}\) to \(30^{\circ} \mathrm{C}\) to regain its original length \(L\). The coefficient of linear thermal expansion of the steel is \(10^{-5} /{ }^{\circ} \mathrm{C}\), Young's modulus of steel is \(10^{11} \mathrm{~N} / \mathrm{m}^2\) and radius of the wire is \(1 \mathrm{~mm}\). Assume that \(L>\) diameter of the wire. Then, the value of \(m\) in \(\mathrm{kg}\) is nearly

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 ____________.

Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.

Match List I with List II and select the correct answer using the codes given below the lists:

	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)

Let $f\left(x\right)= \underset{n\to \infty }{\lim }{\left(\frac{n^n\left(x+n\right)\left(x+\frac{n}{2}\right)\dots ..\left(x+\frac{n}{n}\right)}{n!\left(x^2+n^2\right) \left(x^2+\frac{n^2}{4}\right)\dots ..\left(x^2+\frac{n^2}{n^2}\right)}\right)}^{\frac{x}{n}}$ , for all $x>0$ . Then

At $143\mathrm{ }\mathrm{K},$ the reaction of $\mathrm{X}\mathrm{e}{\mathrm{F}}_4$ with ${\mathrm{O}}_2{\mathrm{F}}_2$ produces a xenon compound $\mathrm{Y}.$ The total number of lone pair(s) of electrons present on the whole molecule of $\mathrm{Y}$ is __________

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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:
Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

Among the following, intensive property is (properties are)

Consider a system of three connected strings, \(S_1, S_2\) and \(S_3\) with uniform linear mass densities \(\mu \mathrm{kg} / \mathrm{m}\), \(4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_1\) and \(S_2\) are connected at the point \(P\), whereas \(S_2\) and \(S_3\) are connected at the point \(Q\), and the other end of \(S_3\) is connected to a wall. A wave generator O is connected to the free end of \(S_1\). The wave from the generator is represented by \(y=y_0\) \(\cos (\omega t-k x) \mathrm{cm}\), where \(y_0, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct: