# Joint Entrance Screening Test (JEST) Past Year Question Papers 2019 Part 4

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Q9. A hoop of diameter *D* is pivoted at the topmost point on the circumference as shown in the figure. The acceleration due to gravity *g* is acting downwards. What is the time period of small oscillations in the plane of the hoop?

(a)

(b)

(c)

(d)

Q10. For a spin particle placed in a magnetic field *B*, the Hamiltonian is where is the *y* -component of the spin operator. The state of the system at time is , where . At a later time *t*, if *z S* measured then what is the probability to get a value ?

(a)

(b)

(c) 0

(d)

Q11. In a fixed target, elastic scattering experiment, a projectile of mass *m*, having initial velocity , and impact parameter *b*, approaches the scatterer. It experiences a central repulsive force . What is the distance of the closest approach *d?*

(a)

(b)

(c)

(d)

Q12. Consider a quantum particle in a one-dimensional box of length *L*. The coordinates of the leftmost wall of the box is at and that of the rightmost wall is at *x* = *L*. The particle is in the ground state at . At , we suddenly change the length of the box to 3*L* by moving the right wall. What is the probability that the particle is in the ground state of the new system immediately after the change?

(a) 0.36

(b)

(c)

(d)

Q13. Consider a function in the domain , where is any polynomial of degree *k*. What is the maximum possible number of extrema of the function?

(a)

(b)

(c)

(d)

Q14. The energy spectrum of a particle consists of four states with energies . Let and denote the canonical partition functions for four non-interacting particles at temperature *T*. The subscripts *B*, *F*, and *C* correspond to bosons, fermions, and distinguishable classical particles, respectively. Let . Which one of the following statements is true about and *?*

(a) They are polynomials in *y* of degree 12,6, and 12, respectively.

(b) They are polynomials in *y* of degree 16,10, and 16, respectively

(c) They are polynomials in *y* of degree 9,6, and 12, respectively.

(d) They are polynomials in *y* of degree 12,10, and 16, respectively.

Q15. Consider a quantum particle of mass *m* and a charge *e* moving in a two dimensional potential given as:

The particle is also subject to an external electric field , where is a constant and corresponds to unit vectors along *x* and *y* directions, respectively. Let and be the energies of the first excited state and ground state, respectively. What is the value of *?*

(a)

(b)

(c)

(d)