On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{kg}$ is attached to a spring with natural length $l_0=0.1 \mathrm{m}$. The spring constant is $k_1=0.009 \mathrm{N} {\mathrm{m}}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016 \mathrm{N} {\mathrm{m}}^{-1}$ when $l<l_0$. Initially the bob is released from $l=0.15 \mathrm{m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=\left(n\pi \right)\mathrm{s}$, then the integer closest to $n$ is _______.

An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in:

A source of constant voltage $\mathrm{V}$ is connected to a resistance$\mathrm{R}$ and two ideal inductors ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ through a switch $\mathrm{S}$ as shown. There is no mutual inductance between the two inductors. The switch $\mathrm{S}$ is initially open. At $\mathrm{t}=0$ , the switch is closed and current begins to flow. Which of the following options is/are correct?

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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?

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

A positive, singly ionized atom of mass number \(A_{\mathrm{M}}\) is accelerated from rest by the voltage \(192 \mathrm{~V}\). Thereafter, it enters a rectangular region of width \(w\) with magnetic field \(\vec{B}_0=0.1 \hat{k}\) Tesla, as shown in the figure. The ion finally hits a detector at the distance \(x\) below its starting trajectory.
[Given: Mass of neutron/proton \(=(5 / 3) \times 10^{-27} \mathrm{~kg}\), charge of the electron \(=1.6 \times 10^{-19} \mathrm{C}\).]

Which of the following option(s) is(are) correct?

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by \(\vec{E}=30(2 \hat{x}+\hat{y}) \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right]\)\(\text{ V m} ^{-1}\). Which of the following option(s) is(are) correct?
[Given: The speed of light in vacuum, \(\text c=3 \times 10^8\text{ m s} ^{-1}\) ]

Let $b$ be a nonzero real number. Suppose $f:\mathrm{ℝ}\to \mathrm{ℝ}$ is a differentiable function such that $f\left(0\right)=1$. If the derivative $f^{\text{'}}$ of $f$ satisfies the equation $f^{\text{'}}\left(x\right)=\frac{f\left(x\right)}{b^2+x^2}$ for all $x\in \mathrm{ℝ},$ then which of the following statements is/are TRUE?

Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,....\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,....\right\}$ of rectangles as follows:
${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1;$
${\mathrm{R}}_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_1;$
${\mathrm{E}}_{\mathrm{n}}:$ ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}_{\mathrm{n}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}_{\mathrm{n}}^2}=1$ of largest area inscribed in ${\mathrm{R}}_{\mathrm{n}-1},\mathrm{n}>1;$
${\mathrm{R}}_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_{\mathrm{n}},\mathrm{n}>1.$
Then which of the following options is/are correct?

Four fair dice \(D_{1}, D_{2}, D_{3}\) and \(D_{4}\); each having six faces numbered \(1,2,3,4,5\) and 6 are rolled simultaneously. The probability that \(D_{4}\) shows a number appearing on one of \(D_{1}, D_{2}\) and \(\mathrm{D}_{3}\) is

The number of hexagonal faces that are present in a truncated octahedron is

The species having bond order different from that in \(\mathrm{CO}\) is

Let \((x, y, z)\) be points with integer coordinates satisfying the system of homogeneous equations \(3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0\). Then, the number of such points for which \(x^2+y^2+z^2 \leq 100\) is