A train is moving along a straight line with a constant acceleration a. A boy standing in the train throws a ball forward with a speed of \(10 \mathrm{~m} / \mathrm{s}\), at an angle of \(60^{\circ}\) to the horizontal. The boy has to move forward by \(1.15 \mathrm{~m}\) inside the train to catch the ball back at the initial height. The acceleration of the train in \(\mathrm{m} / \mathrm{s}^2\), is

When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0 \mathrm{V}$. This potential drops to $0.6 \mathrm{V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take$\frac{hc}{e}=1.24\times {10}^{-6} \mathrm{J} {\mathrm{mC}}^{-1}$ .]

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

Planck' constant $\mathrm{h}$ , speed of light $\mathrm{c}$ and gravitational constant $\mathrm{G}$ are used to form a unit of length $\mathrm{L}$ and a unit of mass $\mathrm{M}$ . Then the correct option(s) is (are)

A spherical soap bubble inside an air chamber at pressure \(P_0=10^5 \mathrm{~Pa}\) has a certain radius so that the excess pressure inside the bubble is \(\Delta P=144 \mathrm{~Pa}\). Now, the chamber pressure is reduced to \(8 P_0 / 27\) so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure \(\Delta P\) in both the cases to be much smaller than the chamber pressure. The new excess pressure \(\Delta P\) in \(\mathrm{Pa}\) is ________ .

Function \(x=A \sin ^2 \omega t+B \cos ^2 \omega t+C\) \(\sin \omega t \cos \omega t\) represents SHM.

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

The spin only magnetic moment value (in Bohr magneton units) of \(\mathrm{Cr}(\mathrm{CO})_6\) is

A small electric dipole \(\vec{p}_0\), having a moment of inertia \(I\) about its center, is kept at a distance \(r\) from the center of a spherical shell of radius \(R\). The surface charge density \(\sigma\) is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle \(\theta\) as shown in the figure. While staying at a distance \(r\), the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is(are) correct?
[ \(\varepsilon_0\) is the permittivity of free space.]

For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

Let S be the set of all twice differentiable functions f from $\mathrm{ℝ}$ to $\mathrm{ℝ}$ such that $\frac{d^{2}f}{d{x}^2}\left(x\right)>0$ for all $x\in \left(-1,1\right)$. For $f\in S$, let $X_f$ be the number of points $x\in \left(-1,1\right)$ for which $f\left(x\right)=x$. Then which of the following statements is(are) true?

A small block is connected to one end of a massless spring of un-stretched length \(4.9 \mathrm{~m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2 \mathrm{~m}\) and released from rest at \(t=0\). It then executes simple harmonic motion with angular frequency \(\omega=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously at \(t=0\), a small pebble is projected with speed \(v\) form point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. Point \(P\) is at a horizontal distance of \(10 \mathrm{~m}\) from \(O\). If the pebble hits the block at \(t=1 \mathrm{~s}\), the value of \(v\) is (take \(g\) \(=10 \mathrm{~m} / \mathrm{s}^{2}\) )