# Joint Entrance Screening Test (JEST) Past Year Question Papers 2017 Part 3

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Q19. Suppose that we toss two fair coins hundred times each. The probability that the same number of heads occur for both coins at the end of the experiment is

(a)

(b)

(c)

(d)

Q20. What is the equation of the plane, which is tangent to the surface *xyz* = 4 at the point (1,2, 2) ?

(a)

(b)

(c)

(d)

Q21. If the ground state wave function of a particle moving in a one-dimensional potential is proportional to , then the potential in suitable units such that , is proportional to

(a)

(b)

(c)

(d)

Q22. A possible Lagrangian for a free particle is

(a)

(b)

(c)

(d)

Q23. A rod of mass *m* and length is suspended from two massless vertical springs with a spring constants and . What is the Lagrangian for the system, if and be the displacements from equilibrium position of the two ends of the rod?

(a)

(b)

(c)

(d)

Q24. Two equal positive charges of magnitude + *q* separated by a distance *d* are surrounded by a uniformly charged thin spherical shell of radius 2*d* bearing a total charge -2*q* and centred at the midpoint between the two positive charges. The net electric field at distance from the midpoint is

(a) Zero

(b) Proportional to *d*

(c) Proportional to

(d) Proportional to

Q25. If the Hamiltonian of classical particles is , then at temperature *T* is equal to

(a)

*(*b)

(c)

(d)

## Part-B: 3-Mark Questions

Q1. A solid, insulating sphere of radius 1*cm* has charge distributed uniformly over its volume. It is surrounded concentrically by a conducting thick spherical shell of inner radius 2*cm*, outer radius 2.5*cm* and is charged with . What is the electrostatic potential in Volts on the surface of the sphere?

Q2. A particle is described by the following Hamiltonian

where the quartic term can be treated perturbatively. If and denote the energy correction of to the ground state and the first excited state respectively, what is the fraction ?

Q3. A simple pendulum has a bob of mass 1 *kg* and charge 1 Coulomb. It is suspended by a massless string of length 13 *m*. The time period of small oscillations of this pendulum is . If an electric field is applied, the time period becomes *T*. What is the value of *?*

Q4. Let a particle of mass *kg*, constrained to have one-dimensional motion, be initially at the origin . The particle is in equilibrium with a thermal bath . What is of the particle after a time ?

Q5. For the circuit shown below, what is the ratio ?

Q6. A ball of mass 0.1*kg* and density 2000 kg/ is suspended by a massless string of length 0.5*m* under water having density 1000 *kg*/ . The ball experiences a drag force, , where *b* and *w* are the velocities of the ball and water respectively. What will be the frequency of small oscillations for the motion of pendulum, if the water is at rest?

Q7. Suppose that the number of microstates available to a system of *N* particles depends on *N* and the combined variable , where *U* is the internal energy and *V* is the volume of the system. The system initially has volume and energy 200*J*. It undergoes an isentropic expansion to volume . What is the final pressure of the system in SI units?

Q8. The temperature in a rectangular plate bounded by the lines, *x =* 0, *y* = 0, *x* = 3 and *y* = 5 is . What is the maximum temperature difference between two points on the plate?

Q9. A sphere of inner radius 1*cm* and outer radius 2*cm*, centered at origin has a volume charge density , where *K* is a nonzero constant and *r* is the radial distance. A point charge of magnitude is placed at the origin. For what value of *K* in units of the electric field inside shell is constant.