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GS-2015 (Mathematics)

TATA INSTITUTE OF FUNDAMENTAL RESEARCH

WrittenTestinMATHEMATICSDecember14,2014

Duration:Twohours(2hours)

Name:_______________________________________________Ref.Code:____________

Pleasereadallinstructionscarefullybeforeyouattemptthequestions.1. Pleasefillindetailsaboutname,referencecodeetc.ontheanswersheet.TheAnswerSheet

ismachinereadable.UseonlyBlack/Blueballpointpentofillintheanswersheet.2. Thereare thirty (30)multiple choicequestionsdivided into twoparts. Part I consistsof15

questionsandPart IIconsistsof15questions. Bachelorsstudentswhohaveappliedonly fortheIntegratedPh.D.programatTIFRCAM,BangalorewillonlybeevaluatedonPartI.Allotherstudents (includingBachelorsstudentsapplying to thePh.D.programsatbothTIFR,MumbaiandBangalore)willbeevaluatedonbothPartsIandII.

3. Indicate your answerON THEANSWER SHEETbyblackening theappropriate circle foreach

question. Each corect answer will get 1 mark. There is no negative marking forwronganswers.Aquestionnotansweredwillnotgetyouanymark.Donotmarkmorethanonecircleforanyquestion:thiswillbetreatedasawronganswer.

4. We advise you to first mark the correct answers on the QUESTION PAPER and then to

TRANSFERthesetotheANSWERSHEETonlywhenyouaresureofyourchoice.5. Roughworkmaybedoneonblankpagesofthequestionpaper. Ifneeded,youmayaskfor

extraroughsheetsfromanInvigilator.6. UseofcalculatorsisNOTpermitted.7. DoNOT ask for clarifications from the invigilators regarding the questions. They have been

instructed not to respond to any such inquiries from candidates. In case acorrection/clarificationisdeemednecessary,theinvigilatorswillannounceitpublicly.

8. NotationandConventionsusedinthistestaregivenonpage2ofthequestionpaper.

MTH

DarrylText BoxCorrect answers are ticked in green.

NOTATION AND CONVENTIONS

N := Set of natural numbers = {1, 2, 3, . . .}Z := Set of integersQ := Set of rational numbersR := Set of real numbersC := Set of complex numbersRn := n-dimensional vector space over R

(a, b) := {x R|a < x < b}(a, b] := {x R|a < x b}[a, b) := {x R|a x < b}[a, b] := {x R|a x b}

A sequence is always indexed by the set of natural numbers.The cyclic group with n elements is denoted by Zn.Unless stated otherwise, subsets of Rn carry the induced topology.For any set S, the cardinality of S is denoted by |S|.

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Part I

1. Let A be an invertible 1010 matrix with real entries such that the sumof each row is 1. Then

A. The sum of the entries of each row of the inverse of A is 1B. The sum of the entries of each column of the inverse of A is 1C. The trace of the inverse of A is non-zeroD. None of the above.

2. Let f : R R be a continuous function. Which one of the following setscannot be the image of (0, 1] under f ?

A. {0}B. (0, 1)C. [0, 1)D. [0, 1].

3. Let A be a 1010 matrix with complex entries such that all its eigenval-ues are non-negative real numbers, and at least one eigenvalue is positive.Which of the following statements is always false ?

A. There exists a matrix B such that AB BA = BB. There exists a matrix B such that AB BA = AC. There exists a matrix B such that AB +BA = AD. There exists a matrix B such that AB +BA = B.

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4. Let S be the collection of (isomorphism classes of) groups G which havethe property that every element of G commutes only with the identityelement and itself. Then

A. |S| = 1B. |S| = 2C. |S| 3 and is niteD. |S| = .

5. Let f : R R denote the function dened by f(x) = (1x2) 32 if |x| < 1,and f(x) = 0 if |x| 1. Which of the following statements is correct ?

A. f is not continuousB. f is continuous but not dierentiableC. f is dierentiable but f

is not continuous

D. f is dierentiable and fis continuous.

6. Let A be the 2 2 matrix(sin

18sin4

9

sin49

sin 18

). Then the smallest number

n N such that An = I is

A. 3B. 9C. 18D. 27.

7. Let f and g be two functions from [0, 1] to [0, 1] with f strictly increas-ing. Which of the following statements is always correct ?

A. If g is continuous, then f g is continuousB. If f is continuous, then f g is continuousC. If f and f g are continuous, then g is continuousD. If g and f g are continuous, then f is continuous.

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8. Let f(x) = e 1xx, where x (0, 1). Then, on (0, 1)

A. f is uniformly continuousB. f is continuous but not uniformly continuousC. f is unboundedD. f is not continuous.

9. Let {an} be a sequence of real numbers such that |an+1 an| n22n forall n N. Then

A. The sequence {an} may be unboundedB. The sequence {an} is bounded but may not convergeC. The sequence {an} has exactly two limit pointsD. The sequence {an} is convergent.

10. For a group G, let Aut(G) denote the group of automorphisms of G.Which of the following statements is true ?

A. Aut(Z) is isomorphic to Z2B. If G is cyclic, then Aut(G) is cyclicC. If Aut(G) is trivial, then G is trivialD. Aut(Z) is isomorphic to Z.

11. Let {an} be a sequence of real numbers. Which of the following is true ?

A. If

an converges, then so does

a4nB. If

|an| converges, then so does a2nC. If

an diverges, then so does

a3n

D. If |an| diverges, then so does a2n.

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12. Let f : R R be an innitely dierentiable function that vanishes at 10distinct points in R. Suppose f (n) denotes the n-th derivative of f , forn 1. Which of the following statements is always true ?

A. f (n) has at least 10 zeros, for 1 n 8B. f (n) has at least one zero, for 1 n 9C. f (n) has at least 10 zeros, for n 10D. f (n) has at least one zero, for n 9.

13. For a real number t > 0, lett denote the positive square root of t. For

a real number x > 0, let F (x) = 4x2x2

sint dt. If F is the derivative of

F , then

A. F(2) = 0

B. F(2) =

C. F(2) =

D. F(2) = 2.

14. Let n N be a six digit number whose base 10 expansion is of the formabcabc, where a, b, c are digits between 0 and 9 and a is non-zero. Then

A. n is divisible by 5B. n is divisible by 8C. n is divisible by 13D. n is divisible by 17.

15. The series

n=1cos(3nx)

2n

A. Diverges, for all rational x RB. Diverges, for some irrational x RC. Converges, for some but not all x RD. Converges, for all x R.

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Part II

16. Let X be a proper closed subset of [0, 1]. Which of the following state-ments is always true ?

A. The set X is countableB. There exists x X such that X \ {x} is closedC. The set X contains an open intervalD. None of the above.

17. In how many ways can the group Z5 act on the set {1, 2, 3, 4, 5} ?

A. 5B. 24C. 25D. 120.

18. Let f be a function from {1, 2, . . . , 10} to R such that(

10i=1

|f(i)|2i

)2=

(10i=1

|f(i)|2)(

10i=1

1

4i

).

Mark the correct statement.

A. There are uncountably many f with this propertyB. There are only countably innitely many f with this propertyC. There is exactly one such fD. There is no such f .

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19. Let U1 U2 be a decreasing sequence of open sets in Euclidean3-space R3. What can we say about the set Ui ?

A. It is inniteB. It is openC. It is non-emptyD. None of the above.

20. Let n 1 and let A be an n n matrix with real entries such thatAk = 0, for some k 1. Let I be the identity n n matrix. Then

A. I + A need not be invertibleB. Det(I + A) can be any non-zero real numberC. Det(I + A) = 1D. An is a non-zero matrix.

21. Let f : [0, 1] R be a xed continuous function such that f is dieren-tiable on (0, 1) and f(0) = f(1) = 0. Then the equation f(x) = f (x)admits

A. No solution x (0, 1)B. More than one solution x (0, 1)C. Exactly one solution x (0, 1)D. At least one solution x (0, 1).

22. A complex number C is called algebraic if there is a non-zero polyno-mial P (x) Q[x] with rational coecients such that P () = 0. Whichof the following statements is true ?

A. There are only nitely many algebraic numbersB. All complex numbers are algebraicC. sin(

3) + cos(

4) is algebraic

D. None of the above.

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23. For a group G, let F (G) denote the collection of all subgroups of G.Which one of the following situations can occur ?

A. G is nite but F (G) is inniteB. G is innite but F (G) is niteC. G is countable but F (G) is uncountableD. G is uncountable but F (G) is countable.

24. Let f : R R be a continuous function and A R be dened byA = {y R : y = lim

nf(xn), for some sequence xn +}.

Then the set A is necessarily

A. A connected setB. A compact setC. A singleton setD. None of the above.

25. How many nite sequences x1, x2, . . . , xm are there such that each xi = 1or 2, and

mi=1 xi = 10 ?

A. 89B. 91C. 92D. 120.

26. Let (X, d) be a path connected metric space with at least two elements,and let S = {d(x, y) : x, y X}. Which of the following statements isnot necessarily true ?

A. S is inniteB. S contains a non-zero rational numberC. S is connectedD. S is a closed subset of R.

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27. Let X R and let f, g : X X be continuous functions such thatf(X)g(X) = and f(X)g(X) = X. Which one of the following setscannot be equal to X ?

A. [0, 1]B. (0, 1)C. [0, 1)D. R.

28. Let X = {(x, y) R2 : 2x2 + 3y2 = 1}. Endow R2 with the discretetopology, and X with the subspace topology. Then

A. X is a compact subset of R2 in this topologyB. X is a connected subset of R2 in this topologyC. X is an open subset of R2 in this topologyD. None of the above.

29. Let G be a group. Suppose |G| = p2q, where p and q are distinct primenumbers satisfying q 1 mod p. Which of the following is always true ?

A. G has more than one p-Sylow subgroupB. G has a normal p-Sylow subgroupC. The number of q-Sylow subgroups of G is divisible by pD. G has a unique q-Sylow subgroup.

30. Let d(x, y) be the usual Euclidean metric on R2. Which of the followingmetric spaces is complete ?

A. Q2 R2 with the metric d(x, y)B. [0, 1] [0,) R2 with the metric d(x, y) = d(x,y)

1+d(x,y)

C. (0,) [0,) R2 with the metric d(x, y)D. [0, 1] [0, 1) R2 with the metric d(x, y) = min{1, d(x, y)}.

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