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Re: Joint Entrance Screening Test PhD entrance exam modal question papers
Joint Entrance Screening Test is conducted on a joint basis by many institutes for admission in to the PhD courses. The discipline in which PhD is offered is mainly Physics and theoretical Computer Science. And therefore I am providing you here both the subjects papers of this exam that are: Physics Theoretical Computer Science JEST Physics Paper1. Black-body radiation, at temperature Ti fills a volume V. The system expands adiabatically and reversibly to a volume 8V. The final temperature Tf = xTi, where the factor x is equal to (a) 0.5 (b) 2.8 (c) 0.25 (d) 1 2. A particle of mass m, constrained to move along the x-axis. The potential energy is given by, V(x)=a + bx +cx2, where a, b and c are positive constants. If the particle is disturbed slightly from its equilibrium position, then it follows that (a) it performs simple harmonic motion with period 2 pÖ(m/2c) (b) it performs simple harmonic motion with period 2 pÖ(ma/2b2) (c) it moves with constant velocity (d) it moves with constant acceleration 3. Consider a square ABCD, of side a, with charges +q, -q, +q, -q placed at the vertices, A, B, C, D respectively in a clockwise manner. The electrostatic potential at some point located at distances r (where r >> a) is proportional to (a) a constant (b) 1/r (c) 1/r2 (d) 1/r3 4. The general solution of dy/dx – y = 2ex is (C is an arbitrary constant) (a) e2x+Cex (b) 2xex+Cex (c) 2xex+C (d) ex2+C 5. As q®0, lim + q q sin ) sin 1 ln( is (a) ¥ (b) -¥ (c) 1 (d) 0 6. If P^ is the momentum operator, and s^ are the three Pauli spin matrices, the eigenvalues of (s^.P^) are (a) px and pz (b) px ± ipy (c) ± |p| (d) ± (px + py +pz) 7. Two parallel infinitely long wires separated by a distance D carry steady currents I1 and I2 (I1 > I2) flowing in the same direction. A positive point charge moves between the wires parallel to the currents with a speed v at a distance D/2 from either wire. The magnitude of an electric field that must be turned on to maintain the trajectory of the particle is proportional to (a) (I1-I2)v/D (b) (I1+I2)v/D (c) (I1-I2)v2/D2 (d) (I1+I2)v2/D2 8. An ideal gas of non-relativistic fermions in three dimensions is at a temperature of 0 K. When both the mass of the particles and the number density are doubled, the energy per particle is multiplied by a factor, (a) Ö2 (b) 1 (c) 21/3 (d) 1/21/3 9. The rotational part of the Hamiltonian of a diatomic molecule is (1/2 µ1)(Lx 2+Ly 2) + (1/2 µ2) Lz 2 where µ1 and µ2 are moments of inertia. If µ1 = 2µ2, the three lowest energy levels (in units of h2/4 µ2) are given by (a) 0, 2, 3 (b) 0, 1, 2 (c) 1, 2, 3 (d) 0, 2, 4 10. A particle of mass 1 gm starts from rest and moves under the action of a force of 30 Newtons defined in the rest frame. It will reach 99% the velocity of light in time (a) 9.9 x 103 sec (b) 7 x 104 sec (c) 0.999 sec (d) 0.7 sec JEST Theoretical Computer Paper 1. Select the correct alternative in each of the following: (a) Let a and b be positive integers such that a > b and a2 - b2 is a prime number. Then a2 - b2 is equal to (A) a - b (B) a + b (C) a × b (D) none of the above (b) When is the following statement true? (A [ B) \ C = A \ C (A) If ¯ A \ B \ C = _ (B) If A \ B \ C = _ (C) always (D) never (c) If a fair die (with 6 faces) is cast twice, what is the probability that the two numbers obtained di_er by 2? (A) 1/12 (B) 1/6 (C) 2/9 (D) 1/2 (d) T(n) = T(n/2) + 2; T(1) = 1. When n is a power of 2, the correct expression for T(n) is: (A) 2(log n + 1) (B) 2 log n (C) log n + 1 (D) 2 log n + 1 2. Consider the following function, defined by a recursive program: function AP(x,y: integer) returns integer; if x = 0 then return y+1 else if y = 0 then return AP(x-1,1) else return AP(x-1, AP(x,y-1)) (a) Show that on all nonnegative arguments x and y, the function AP terminates. (b) Show that for any x, AP(x, y) > y. 3. How many subsets of even cardinality does an n-element set have ? Justify answer. 4. A tournament is a directed graph in which there is exactly one directed edge between every pair of vertices. Let Tn be a tournament on n vertices. (a) Use induction to prove the following statement: Tn has a directed hamiltonian path (a directed path that visits all vertices). (b) Describe an algorithm that finds a directed hamiltonian path in a given tournament. Do not write whole programs; pseudocode, or a simple description of the steps in the algorithm, will suffice. What is the worst case time complexity of your algorithm? 5. Describe two different data structures to represent a graph. For each such representation, specify a simple property about the graph that can be more efficiently checked in that representation than in the other representation. Indicate the worst case time required for verifying both of your properties in either representation. 6. Two gamblers have an argument. The first one claims that if a fair coin is tossed repeatedly, getting two consecutive heads is very unlikely. The second, naturally, is denying this. They decide to settle this by an actual trial; if, within n coin tosses, no two consecutive heads turn up, the first gambler wins. (a) What value of n should the second gambler insist on to have more than a 50% chance of winning? (b) In general, let P(n) denote the probability that two consecutive heads show up within n trials. Write a recurrence relation for P(n). (c) Implicit in the second gambler’s stand is the claim that for all sufficiently large n, there is a good chance of getting two consecutive heads in n trials; i.e. P(n) > 1/2. In the first part of this question, one such n has been demonstrated. What happens for larger values of n? Is it true that P(n) only increases with n? Justify your answer. 7. Consider the following program: function mu(a,b:integer) returns integer; var i,y: integer; begin ---------P---------- i = 0; y = 0; while (i < a) do begin --------Q------------ y := y + b ; i = i + 1 end return y end Write a condition P such that the program terminates, and a condition Q which is true whenever program execution reaches the place marked Q above.
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