Entrance Examination: M.Sc. Mathematics, 2011
Hall Ticket Number

Time 2 hours Part. A 25 marks Max. Marks. 75 T'a.1. fl 50 marks
instructions
1.
Write your Booklet Code llIld Hall Ticket Numher on the OMR Answer Sheet given to you. Also write t.he Hall Ticket Numher in the space provided abvye.

2.
There is negative marking. In PllJ t A a right snswer gets 1 mark llJId a wrong answer gets -0.33 mark. In part B a right answer gets 2 marks and a wrong answer g<:t.s -0.66 mark.

3.
Answers are to be marked on the OMR answer sheet following the


instructions provided there upon.
4.
Hand over the que,,.tion paper booklet and the OMR answer sheet 8t. the end of the exanlination.

5.
No additional sheets will be pr vided. Rough work can be done in the question paper itself/space provided 8t the end of the booklet.

6.
Calculators are not allowed.

7.
There are a tote.! of 50 questions in Part. A and Part. B together.

8.
The approlJriate answer should be coloured in either a blue or black ball point or sketch pen. DO NOT USE A PENCIL.


I


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Part A
1. Statement: All mathematicians axe intellectuals. Conclusions:

Raju is nnt a mathematicill.l1 so he is not an intellectual


All intelle<:t.uals are nIathematit:ill.l1s

A.
Only is correct

B.
Only is correct

C.
Both and are correct

D.
Neither nor is correct


2. For any natural numhern, the sum.
A. n2"


B. n2,-1
C. n2"+1
D. none of the above
3. Let f IR be defined as Ixl then
A. f is continuous and differentiable
B. f is continuous but nut differentiable
C. f differentiable hut discontinuous
D. f is discontinuous
4. Let be two convergent. s<'quences converging to I,m respec­'f' dd
tively. If we define en ban +117ll. IS0 then
. n+I fn. 18 even,
A. is a Cauchy ""'Iuem:e which is not convergent
B. is bounded but not. convergent
C. is a onnvergent sequence onnverging to 1+m
D. has only two onnvergent subsequences
5. Let G be a group and a. bEG. If a17 and a"" b"" then
A. a=b
B. ab ba and ora)
C. a b-I and ora)
D. ora) and a II
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6. Let be vector valued and scalar valued functions on R" respectively, then
A. curl(grad t/»
B. curl(curlJ) 0
C. grad(div 0
D. div(grad t/» 0
7. The number of points in the plane equidistant from P R is
A.O

D.
infinite


B.l
C.2
8. The numhpr of suhgroups of Z,o is
A.l
B.2
C.3
D.4
9. The number of nontrivial homomorphisms from thp c:yelie group ZI4 to a group of order 7 is
A.l
B.2
C.3
D.6
10. Let X Which of them admit group structure
A. only X
B. only Z
C. all of them
D. none of them
11. Two coins whose probabilities of heads showing up arc PI> P2 are tossed, the probability that. at least one tail shows up is
A. 2-P,-P2
B.
C.
D. 1 -111P2
12. 2 ones. 2 twos. I t.hree alld 1 are to be nrranged to get" 6 digit number. The number of different numbers thai be obtained this way is
A.6!


B. 2!

C.


D.
13. Let a., 'tn. E N then consider the st.atements: If E <In converges then E also converges S2) If E converges then E a" also converges
A. Doth and 82 are t rne
B. is t.rne but. 82 is f..lse
C. 82 is true but. SI is false
D. Both SI and S2 are false
14. Let x E JR, [xl denotes the greatest. integer less t.han or equal to t.hen consider the st.atement.s: Sll Ix2] 52) Ix21= Ixl2
A. Bot.h 51 and S2 are true
B. 51 is true but. 82 is false
C. S2 is t.rue but is false
D. Both 51 and S2 are false
15. LetxE R then t.he correct statement is:
A. If r EQ. thenr EQ
B. If E then r E Q
C. Ifr E Q and x4 E Q then EQ
D. Ifr E Q and EQthen x E Q
16. The number of solutions of X· =l{modl63) in Z'63 is
A.1
B.2
C.3
D.4
17. Let/ R Rbe a polynomial suth tlml leO) 0 and
I Ix E JR, then frO) is
A.I/4
B.I/3
C.1/2
D.l
18. LetJ III R be a polynumial and leI. be a sequence of mal numbers converging to 2. Then the se<1'IOnre converges to
A.
B.
C.
D.
19. Let V be the vector space of continuous functions on over R. Let E V defined as lxi, ...{x) xlxl t.hen
A. is linearly dependent
B. {u" U3, U4} is linearly dependent
C. {Il" U., ...} is linearly dependellt
D. none of the above
5



1 2 1-1
20. The rank of 1 I 5 2 1 4-1 is
4 5 1

A.I
B.2
C.3
D.4
21. i. a polynomial of degree 102011 lim z
:r.-.oo
A. i.O
B. is 1
C. is 00
D. docs not exist.
22. Let p dx, q /.1 IWd x dx. Then
A. r
B. 1


D. 1 p q
23. Let A be a 4 x 4 real mat.rix. Whieh of the following 4 conditions is not equivalent to t.he other
A. The mntrix A i.
B. The sysl.em of e'l"AI.ioIS 0 bas ollly trivial solulion.
C. Any two distinct. rows u and v of A are Iinel
1y independent.
D. The system of equabon. Ax b has a unique solution lib E
24. The set, of complex numbers satisfying I,he equation z is
A. an empty set.
B. a finite set.
C. un infinite set.
D. a line.

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25. Let A be a subset of reol numbers containing 011 the rationol numbers. Which of the following statements is true?
A. A is countable.
B. If A is uncountable, then A R.
C. If A is open, then A R.
D. None of the above statement is true.
PariB
26. Let X be the Ret of all nonempty finite subsets of N. Which one of the following is noi an equivalen... relation on
A. A if and only if min A min 8
B. A -8if and only if n haw same number of elements
C. A if and only if A B
D. A if and only if A n 11
27. Which of the following statementa is not true;
A. Every hounded oequen"" 01 r."ll1l1mbers hAS • l"Ol1ver lent

B. If subsequences aod of a sequence converges respectively to x aod 71 then 71
C. A monotone sequence of lea.! numbers is convergent if lind only if it is hounded
D. A sequence (xn of rea.! numbers is convergent if and only if the sequence (lxnll is convergent
28. The absolute maximum value of 1;Ixl 1 lion R is attained at
A. x 0 only
B. x only
C. x 0 and x 1 only
D. no point of R



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29. Which of the following fWlL'tions is uniformly continuous
.""·V1 +sinhx
A. I JR, 2
tan x .1
B. JR, X'SlD
. x
1
C. I: cos
x
D. none of the above
30. II I IR then pick up a true statement from lhe following:
A. II I is continuous then III is continuous
B. II I is differentiable then III is differenhable
C. If I is integrable then Ml is inr.egrable
D. If I is discontinuous then III is dis<:ontinuous
31. Let f h(x. be solenoidal where h f3 o.re scalar valued functions. Let 8 be '.he unit sphere in and it be unit outward
normal. Then Isxf,itdS
A.O
B.1<

C.41</3

D.41<
32. Let be the splICe of all 3 x 3 real matrices. Let V C be
the space of symmetric matrices with trace O. Then dimension of the .
quotIent l':ipace is
A.6
B.5
C.4
D.3
33. Let V be the vector space of continuous functions on over R. Which one of the following is a subspace of V
A. {J E VII vanishes at some point in II}
B. {f E VII(O)
c. {J E VII(x) 0 E II}
D. {J E


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34. Let be the space of all 2 x 2 real matrices, I be the identity matrix in M.(R). Pick up the correct statement
A. 3 two different matrices lJ e such that AB BA I
B. 3 two different matrices B e sud, that AB -BA I
C. 3 a singular matrix A e M.(Il) sud, t.hat 0
D. 3A e such that A3 but 0
35. Let G be the group under multiplication. Which of the following statements is true
A. identity map is the only bomomorphism from G to G
B.themap
Z is a homomorphism from to G

C.
the map Z is not a homomorphism from G to G

D.
none of the above


36. Each element of a 2 x 2 matrix A is selected randolnl)' from the set with "Qual probability. The probability that A is singular is
A.l/8
B. 2/8
C.3/8
D.4/8

37. General solution of 0 is
A. Cle" +C3 COS% sin%
B. Cle" C.",o-· C.. oosx+C,sin%
C. Cle" -C.e-· +c.cosh%
D. Cl%e" -C.e-· +C3 cos sin


38. Let An and Bn• n EN be non empty subsets of R such that A3 ... and B3 •..• Let the cardinality of An be a" and the cardinality of Bn be 1>". Then the cardinality of ao
A. nAnis lima"

....,
ao
B. nAn is

ao
C. UB" is lim
n-oo
n_1
00
D. UBn is
f1_1
39. The area bounded on lhe right by T on the left by y and below by x-axis is
A. 23/6
B.5/6
C.17/20

D.O
40. The distance of the point from the plane 0 measured along a line with direction ratios iR
A.Jl4
B.V24
C.J3ij


D.v'94
41. Consider the stHlernents: In the set or 2 x 2 real matrices . iI al
o 1 1B BlID ar to a lagon
matrix
If M iB a 2x2real matrix and M" for some 11 E N then M 1
A. Both and arc true
B. 5, is true but iB ralse
C. 5, is true but 51 is ralse
D. Both 51 and arc false





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42. Let then h..
A. all real roots are between 0 and 6
B. a real root between 0 and 1
C. a real root between 6 and 7
D. exactly two roots between 0 and 6
43. For a fixed 11 E the value of oj yJdx is twhere for a real
number It] is the greateo.'t integer less than or equal to
A.O
l
c. 2+11
0·11

44. Let V be the veetor space of all 2 x 3 real matrices and W be 11CCtor space of all 2 x 2 real matrices. Then
A. there is a one--one linf'ftr transformation from V W.
B. kernel of any linear transformat ion from l W is nontrivial.
C. there is !ill onto linear transformation from W V.
D. there is an isomorphism from V W.
45. Let A be a 2 x 2 real matrix. Which uf I he folluwing st.at,ements is t.rue?
A. All the entries of are non-negati,·•.
B. The determinant of ib non-negative.
C. the trace of is non-negatIve.
D. all the eigenvalues of are non-negative.
46. Let 9 R R he two differentiable functions. Suppose that 0 for x O. Then
A. for all x 0
B. for all x 0
c. for all x 0
D. for all x.
11

47. Let J R R be given by Ixl sin(x). Then
A. J is differentiable at 0 and O.
B. J is differentiable at 0 and I.
C. J is continuo"", at but not differentiable at O.
D. J is not continuous at but differentiable at O.
48. Consider tbe group z" x z" under addition. Tbe number of cyclic subgroups of order p is
A.l
B. p-l
C. p+l
D.
49. Let R be a ring witb unit.y. Then
A. Thp set of all non7Rro elrment.s in R forms a group under
multiplication
B. Tbe set of all nonzero invert.ible elements in R forms a group under multiplication
C. The set of all non zero divisors in R forms a group under
multiplicat.ion

D. none of tbe above
50. Let Be R and C E bE B}. Then the faJse statement. in the following is:
A. If B arc bounded set.s, then C is a bounded set
B. If C is a bounded set. then B are bounded sets
C. If R-R are bounded sets, then R is a bounded set
D. If R-C is a bounded set, then R IR are bounded sets