univ.tifr.res.in : GS Graduate School Admissions Exam Computer Science Previous Question Papers

Name of the Centre : Tata Institute of Fundamental Research
Document type : Previous Question Papers
Name Of The Exam : GS Graduate School Admissions Exam
Name Of The Subject : Computer Science
Year : 2019
Website : http://univ.tifr.res.in/gs2019/Prev_QP/Prev_QP.htm

GS Admissions Exam Computer Science Previous Question

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Instructions

Please Read These Instructions Carefully Before Attempting the Questions :
1. Please fill in details about name, reference code etc. on the answer sheet and question paper. Use only blue/black ballpoint pen. The Answer Sheet is machine –readable and will not read other colours.

2. This test consists of three parts: Section A, Section B and Section C. You must answer questions according to the programme you are applying for.

3. Indicate your Answer On The Answer Sheet as follows.
Multiple choice questions have four options (a), (b), (c) and (d), of which only one option is correct. Indicate the answers by filling up the bubble on the Answer Sheet corresponding to the correct option. If more than one bubble is filled in, it will be treated as not answered.

** Numerical questions have answers which are 3 (three) digit integers. Indicate the answers by filling in the corresponding bubbles on the Answer Sheet.
** Unless all three bubbles for a given question are filled, it will be treated as not answered. (See inside for details.)

** Symbolic questions have answers which are a number, a short formula or a word. Indicate the answers by writing in the boxes on the Answer Sheet next to the appropriate question numbers. (See inside for details.)

4. The marking for these questions shall be as follows.

5. Candidates are advised to mark the Answer Sheet only when they are sure of the answer. Till then, they may mark the answers on the question paper.

6. Rough work may be done on blank pages of the question paper. If needed, you may ask for extra sheets from the invigilators.

7. Use of scientific, non-programmable calculators is permitted. Calculators which plot graphs are NOT allowed. Multiple-use devices, such as cell phones, smartphones, etc. CANNOT be used as calculators.

8. Do NOT ask the invigilators for clarifications regarding the questions. They have been instructed not to respond to any such queries. In case a correction/clarification is deemed necessary, it will be announced in the examination hall.

9. A list of useful physical constants is given on the next page. Make sure to use only these values in answering the questions, especially those of numeric type.

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2018 :
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2019 :
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Common Part

1. Let X be a set with n elements. How many subsets of X have odd cardinality?
(a) n
(b) 2n
(c) 2n/2
(d) 2n-1
(e) Cannot be determined without knowing whether n is odd or even

2. How many proper divisors (that is, divisors other than 1 or 7200) does 7200 have?
(a) 18
(b) 20
(c) 52
(d) 54
(e) 60

3. A is an n × n square matrix for which the entries in every row sum to 1. Consider the following statements:
(i) The column vector [1,1,…,1]T is an eigenvector of A.
‘(ii) det(A – I)=0. (iii) det(A)=0. Which of the above statements must be TRUE?
(a) Only (i)
(b) Only (ii)
(c) Only (i) and (ii)
(d) Only (i) and (iii)
(e) (i), (ii) and (iii)

4. What is the probability that a point P = (a, ß) picked uniformly at random from the disk x2 + y2 = 1 satisfies a + ß = 1?
(a) 1p
(b) 34 + 14 · 1p
(c) 34 + 14 · 2p
(d) 1 (e) 2p

5. Asha and Lata play a game in which Lata first thinks of a natural number between 1 and 1000. Asha must find out that number by asking Lata questions, but Lata can only reply by saying “yes” or “no”. Assume that Lata always tells the truth. What is the least number of questions that Asha needs to ask within which she can always find out the number Lata has thought of? (a) 10 (b) 32 (c) 100 (d) 999 (e) None of the above

6. A function f : R ? R is said to be convex if for all x, y ? R and ? such that 0 = ? = 1,
f(?x + (1 – ?)y) = ?f(x) + (1 – ?)f(y).
Let f : R ? R be a convex function, and define the following functions: p(x) = f(-x), q(x) = -f(-x), and r(x) = f(1 – x).
Which of the functions p, q and r must be convex? (a) Only p (b) Only q (c) Only r (d) Only p and r (e) Only q and r

7. What are the last two digits of 1! + 2! + ··· + 100!?
(a) 00
(b) 13
(c) 30
(d) 33
(e) 73

8. Consider the following toy model of traffic on a straight, single lane, highway. We think of cars as points, which move at the maximum speed v that satisfies the fol- lowing constraints:
1. The speed is no more than the speed limit vmax mandated for the highway. 2. The speed is such that when travelling at this speed, it takes at least time t0 (where t0 is a fixed time representing the reaction time of drivers) to reach the car ahead, in case the car ahead stops suddenly.

9. Avni and Badal alternately choose numbers from the set {1,2,3,4,5,6,7,8,9} with- out replacement (starting with Avni). The first person to choose numbers of which any 3 sum to 15 wins the game (for example, Avni wins if she chooses the numbers 8, 3, 5, 2, since 8+5+2=15). A player is said to have a winning strategy if the player can always win the game, no matter what the other player does. Which of the following statements is TRUE? As a hint, there are exactly 8 ways in which 3 numbers from the set {1,2,3,4,5,6,7,8,9} can sum up to 15, shown as the three rows, the three columns, and the two diagonals in the following square:
(a) Avni has a winning strategy
(b) Badal has a winning strategy
(c) Both of them have a winning strategy
(d) Neither of them has a winning strategy
(e) The player that picks 9 has a winning strategy

14. A drawer contains 9 pens, of which 3 are red, 3 are blue, and 3 are green. The nine pens are drawn from the drawer one at a time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is the probability that two red pens are drawn in succession?
(a) 7/12
(b) 1/6
(c) 1/12
(d) 1/81
(e) None of the above

Computer Science

1. Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits?
(a) 0.1
(b) 0.2
(c) 0.4
(d) 0.5
(e) All the above

2. How many distinct minimum weight spanning trees does the following undirected, weighted graph have?
a) 8
(b) 16
(c) 32
(d) 64
(e) None of the above

3. A graph is d-regular if every vertex has degree d. For a d-regular graph on n vertices, which of the following must be TRUE? (a) d divides n (b) Both d and n are even (c) Both d and n are odd (d) At least one of d and n is odd (e) At least one of d and n is

4. A formula is said to be a 3-CF-formula if it is a conjunction (i.e., an AND) of clauses, and each clause has at most 3 literals. Analogously, a formula is said to be a 3-DF- formula if it is a disjunction (i.e., an OR) of clauses of at most 3 literals each. Define the languages 3-CF-SAT and 3-DF-SAT as follows:
3-CF-SAT = {F|F is a satisfiable 3-CF formula} 3-DF-SAT = {F|F is a satisfiable 3-DF formula}
Which of the following best represents our current knowledge of these languages? (a) Both 3-CF-SAT and 3-DF-SAT are in NP but only 3-CF-SAT is NP-
complete (b) Both 3-CF-SAT and 3-DF-SAT are NP-complete (c) Both 3-CF-SAT and 3-DF-SAT are in P (d) Both 3-CF-SAT and 3-DF-SAT are in NP but only 3-DF-SAT is NP-complete (e) Neither 3-CF-SAT nor 3-DF-SAT are in P

5. Consider the following program fragment:
var a, b : integer ; procedure G (c, d : integer) ; beginc := c – d ; d := c + d ; c := d – c end ; a := 2 ; b := 3 ; G (a, b); If both parameters to G are passed by reference, what are the values of a and b at the end of the above program fragment? (a) a = 0 and b = 2 (b) a = 3 and b = 2 (c) a = 2 and b = 3 (d) a = 1 and b = 5 (e) None of the above

9. Consider the following program fragment:
var x, y: integer ; x := 1; y := 0 ; while y < x do
CSS 2019 Computer Science Page 10 of 17
beginx := 2 * x ;
y := y + 1 end ;
For the above fragment, which of the following is a loop invariant? (a) x = y + 1 (b) x = (y + 1)2 (c) x = (y + 1)2y (d) x = 2y
(e) None of the above, since the loop does not terminate

10. Let the language D be defined on the binary alphabet {0,1} as follows:
D := {w ? {0,1}*| substrings 01 and 10 occur an equal number of times in w}.
For example, 101 ? D while 1010 /? D. Which of the following must be TRUE of the language D? (a) D is regular (b) D is context-free but not regular (c) D is decidable but not context-free (d) D is decidable but not in NP (e) D is undecidable

System Science

1. Consider a discrete-time system which in response to input sequence x[n] (n integer)
outputs the sequence y[n] such that
y[n] =
{ 0, n = -1,-2,-3,…,
ay[2n – 1] + ßy[n – 1] + ?x[n – 1] + x[n]+1, n = 0,1,2,…
Which of the below makes the system linear and time-invariant?
(a) Only a = ß = ? = 0
(b) Only a = ß = 0 (parameter ? can take any value)
(c) Only a = 0 (parameters ß and ? can take arbitrary values)
(d) Always non-linear, but time-invariant only if a =

2. Let A and B be two square matrices that have full rank. Let ?A be an eignevalue of
A and ?B an eigenvalue of B. Which of the following is always TRUE? (a) AB has full rank (b) A – B has full rank (c) ?A?B is an eigenvalue of AB (d) A + B has full rank (e) At least one of ?A or ?B is an eigenvalue of AB

3. Consider a function f : R ? R such that f(x)=1 if x is rational, and f(x)=1 -e,
where 0 <e< 1, if x is irrational. Which of the following is TRUE?
(a) limx?8 f(x)=1 (b) limx?8 f(x)=1 – e (c) limx?8 f(x) exists, but is neither 1 nor 1 – e (d) maxx=1 f(x)=1
(e) None of the above 4. Let f(x) = vx2 – 4x + 4, for x ? (-8, 8). Here, vy denotes the non-negative square root of y when y is non-negative. Then, which of the following is TRUE? (a) f(x) is not continuous but differentiable (b) f(x) is continuous and differentiable (c) f(x) is continuous but not differentiable (d) f(x) is neither continuous nor differentiable (e) None of the above

5. Consider the function f(x) = ex2 – 8×2 for all x on the real line. For how many
distinct values of x do we have f(x)=0?
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(a) 1 (b) 4 (c) 2 (d) 3 (e) 5

6. Suppose that X1 and X2 denote the random outcomes of independent rolls of two dice. Each of the dice takes each of the six values 1,2,3,4,5, and 6 with equal probability. What is the value of the conditional expectation
E[max(X1,X2)|min(X1,X2) = 3]?
(a) 33/7 (b) 4 (c) 5 (d) 9/2 (e) 19/4