JEE (Advanced) 2009 Paper-1

For the following questions answer them individually

Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, $$_{1}^{2}H$$, known as deuteron and denoted by D, can be thought of as a candidate for fusion reactor. The D-D reaction is $$_{1}^{2}H + _{1}^{2}H \rightarrow _{2}^{3}He + n +$$ energy. In the core of fusion reactor, a gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of $$_{1}^{2}H$$ 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 deuterons, the product nt0 is called Lawson number. In one of the criteria, a reactor is termed successful if Lawson number is greater than $$5 \times 10^{14}$$ s/cm$$^3$$.

It may be helpful to use the following. Boltzmann constant $$k = 8.6 \times 10^{-5}$$eV/K; $$\frac{e^2}{4 \pi \varepsilon_0} = 1.44 \times 10^{-9}$$ eVm.

When a particle is restricted to move along x-axis between x = 0 and x = a, where a is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends x = 0 and x = a. The wavelength of this standing wave is related to the linear momentum p of the particle according to the de Broglie relation. The energy of the particle of mass m is related to its linear momentum as $$E = \frac{p^2}{2m}$$. Thus, the energy of the particle can be denoted by a quantum number 'n' taking values 1, 2, 3, . . . . . (n = 1, called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line x = 0 to
$$x = a$$. Take $$h = 6.6 \times 10^{-34}$$J-s and $$e = 1.6 \times 10^{-19}$$ C.

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example.
If the correct matches are A - p, s and t; B - q and r; C - p and q; and D - s and t; then the correct darkening of bubbles will look like the following.