1

HYDERABAD CENTRAL UNIVERSITY (HCU)

M.Sc. Mathematics Entrance - 2011

Time : 2 Hours Max. Marks: 75

Instructions:
(i) There are a total of 50 questions in Part-A and Part-B together.
(ii) There is negative marking. In Part-A a right answer gets 1 Mark and a wrong answer gets –0.33 Mark.
In Part-B a right answer gets 2 Marks and a wrong answer gets –0.66 Mark.

PART-A

1. Statement : All mathematicians are intellectuals

Conclusions :
(i) Raju is not a mathematician so he is not an intellectual
(ii) All intellectuals are mathematicians
(a) Only (i) is correct (b) Only (ii) is correct
(c) Both (i) and (ii) are correct (d) Neither (i) nor (ii) is correct

n n n n

2. For any natural number n, the sum,    2    3    ...  n   
 1  2  3 n

(a) n2 n (b) n2 n 1 (c) n2n 1 (d) none of the above

3.
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Let f :   {0}   be defined as f ( x)  | x | then
(a) f is continuous and differentiable (b) f is continuous but not differentiable
(c) f differentiable but discontinuous (d) f is discontinuous
4. Let (an ), (bn ) be two convergent sequences converging to l, m respectively. If we define

a  m if n is odd,
cn   n then
 bn  l if n is even,

(a) (cn ) is a Cauchy sequence which is not convergent

(b) (cn ) is bounded but not convergent

(c) (cn ) is a convergent sequence converging to l  m

(d) (cn ) has only two convergent subsequences

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5. Let G be a group and a, b  G . If a17  b17 and a30  b30 then

(a) a  b (b) ab  ba and o( a)  o(b)

(c) a  b 1 and o( a)  o(b) (d) o(a )  o(b) and a  b
6. Let f ,  be vector valued and scalar valued functions on 3 respectively, then

(a) curl(grad )  0 (b) curl(curl f )  0

(c) grad(div f )  0 (d) div(grad )  0

7. The number of points in the plane equidistant from P  (1, 0), Q  (1, 0), R  (0,1) is
(a) 0 (b) 1 (c) 2 (d) infinite
8. The number of subgroups of 10 is
(a) 1 (b) 2 (c) 3 (d) 4
9. The number of non-trivial homomorphisms from the cyclic group 14 to a group of order 7 is
(a) 1 (b) 2 (c) 3 (d) 6
10. Let X  {0, 1}, Y  {2, 7}, Z  {0, 2, 4} . Which of them admit group structure
(a) only X (b) only X, Z (c) all of them (d) none of them
11. Two coins whose probabilities of heads showing up are p1 , p2 are tossed, the probability that at least one
tail shows up is

(a) 2  p1  p2 (b) p1 p2 (c) p1 (1  p2 ) (d) 1  p1 p2

12. 2 ones, 2 twos, 1 three and 1 five are to be arranged to get a 6 digit number. The number of different
numbers that can be obtained this way is
6! 6! 6!
(a) 6! (b) (c) (d)
2! 2!2! 2!3!

13. Let an  0, n   then consider the statements:

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2
S1) If a n converges then a n also converges
2
S2) If a n converges then a n also converges
(a) Both S1 and S2 are true (b) S1 is true but S2 is false
(c) S2 is true but S1 is false (d) Both S1 and S2 are false
14. Let x  , [ x] denotes the greatest integer less than or equal to x, then consider the statements:

S1) [ x 2 ]  [ x]2

S2) [ x 2 ]  [ x]2
(a) Both S1 and S2 are true (b) S1 is true but S2 is false
(c) S2 is true but S1 is false (d) Both S1 and S2 are false

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15. Let x    {0} then the correct statement is:

(a) If x 2   , then x 3   (b) If x 3  , then x 2  

(c) If x 2   and x 4   then x 3   (d) If x 2   and x 5   then x  

16. The number of solutions of X 5  1(mod163) in 163 is

(a) 1 (b) 2 (c) 3 (d) 4
17. Let f :    be a polynomial such that f (0)  0 and f ( f ( x))  x  1x  , then f (0) is
(a) 1/4 (b) 1/3 (c) 1/2 (d) 1
18. Let f :    be a polynomial and let ( xn ) be a sequence of real numbers converging to 2. Then the

sequence ( f ( xn )) converges to
(a) f (2) (b) f (4) (c) f (8) (d) f (16)

19. Let V be the vector space of continuous functions on [–1, 1] over  . Let u1 , u2 , u3 , u4  V defined as

u1 ( x )  x , u2 ( x)  | x |, u3 ( x)  x 2 , u4 ( x)  x | x | then

(a) {u1 , u2 } is linearly dependent (b) {u1 , u3 , u4 } is linearly dependent

(c) {u1 , u2 , u4 } is linearly dependent (d) none of the above

1 2 1 1
 1 1 2 1 
20. The rank of  is
1 5 4 1
 
 1 4 5 1

(a) 1 (b) 2 (c) 3 (d) 4

21. If p ( x ) is a polynomial of degree 102011 then lim p ( x )e x

x 

(a) is 0 www.careerendeavour.com
(b) is 1 (c) is  (d) does not exist
1 1 1
22. Let p   1  x dx, q   1  x 2 dx and r   1  x dx
0 0 0

Then
(a) p  q  r  1 (b) p  q  1  r (c) q  p  1  r (d) 1  p  q  r
23. Let A be a 4 × 4 real matrix. Which of the following 4 conditions is not equivalent to the other 3?
(a) The matrix A is invertible
(b) The system of equations Ax  0 has only trivial solution
(c) Any two distinct rows u and v of A are linearly independent
(d) The system of equations Ax  b has a unique solution b   4

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24. The set of complex numbers satisfying the equation z  | z |2 is

(a) an empty set (b) a finite set (c) an infinite set (d) a line
25. Let A be a subset of real numbers containing all the rational numbers. Which of the following statements
is true?
(a) A is countable (b) If A is uncountable, then A  
(c) If A is open, then A   (d) None of the above statement is true

PART-B

26. Let X be the set of all non-empty finite subsets of  . Which one of the following is not an equivalence
relation on X:
(a) A ~ B if and only if min A = min B
(b) A ~ B if and only if A, B have same number of elements
(c) A ~ B if and only if A = B
(d) A ~ B if and only if A  B  
27. Which of the following statements is not true
(a) Every bounded sequence of real numbers has a convergent subsequence
(b) If subsequences ( x2 n ) and ( x3 n ) of a sequence ( xn ) converges respectively to x and y then x  y
(c) A monotone sequence of real numbers is convergent if and only if it is bounded
(d) A sequence ( xn ) of real numbers is convergent if and only if the sequence (| xn |) is convergent

1 1
28. The absolute maximum value of f ( x )   on  is attained at
1 | x | 1 | x  1|

(a) x  0 only (b) x  1 only

(c) x  0 and x  1 only (d) no point on 
29. www.careerendeavour.com
Which of the following functions is uniformly continuous

ecos x 1  sinh x
(a) f : (0,1)  , f ( x )  (b) f :   {0}  , f ( x)  x 2 sin  1 
2  tan 2 ( x 2 )  x

1
(c) f :   {0}  , f ( x )  cos   (d) none of the above
x
30. If f :    then pick up a true statement from the following
(a) If f is continuous then | f | is continuous (b) If f is differentiable then | f | is differentiable

(c) If f is integrable then f ( | x |) is integrable (d) If f is discontinuous then | f | is discontinuous

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31. Let f  (1, f 2 ( x, y, z ), f 3 ( x, y , z )) be solenoidal where f 2 , f3 are scalar valued functions. Let S be the unit

sphere in 3 and n̂ be unit outward normal. Then ˆ 

 xf  ndS
S

(a) 0 (b)  (c) 4 / 3 (d) 4

32. Let M 3 () be the space of all 3 × 3 real matrices. Let V  M 3 () be the space of symmetric matrices

M 3 ()
with trace 0. Then dimension of the quotient space is
V
(a) 6 (b) 5 (c) 4 (d) 3
33. Let V be the vector space of continuous functions on [–1, 1] over  . Which one of the following is a
subspace of V
(a) { f  V / f vanishes at some point in [–1, 1]} (b) { f  V / f (0)  0}
(c) { f  V / f ( x)  0x  [ 1,1]} (d) { f  V / f ( 1)  f (1)}

34. Let M 2 () be the space of all 2 × 2 real matrices, I be the identity matrix in M 2 () . Pick up the correct
statement
(a)  two different matrices A, B  M 2 () such that AB  BA  I

(b)  two different matrices A, B  M 2 () such that AB  BA  I

(c)  a singular matrix A  M 2 () such that A2  I  0

(d) A  M 2 () such that A3  0 but A  0

4

35. Let G  {1, 1, i, i} be the group under multiplication. Which of the following statements is true
(a) identity map is the only homomorphism from G to G
(b) the map z  z is a homomorphism from G to G
(c) the map z  z 2 is not a homomorphism from G to G
(d) none of the abovewww.careerendeavour.com
36. Each element of a 2 × 2 matrix A is selected randomly from the set {–1, 1} with equal probability. The
probability that A is singular is
(a) 1/8 (b) 2/8 (c) 3/8 (d) 4/8

d4y
37. General solution of 4  y  0 is
dx

(a) C1e x  C2 e x  C3 cos x  C4 sin x (b) C1e x  C2 xe  x  C3 cos x  C4 sin x

(c) C1e x  C2 e  x  C3 cos h x  C4 sin h x (d) C1 xe x  C2 e  x  C3 cos x  C4 sin x

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38. Let An and Bn, n   be non empty subsets of  such that A1  A2  A3  ... and B1  B2  B3  ....
Let the cardinality of An be an and the cardinality of Bn be bn. Then the cardinality of

 

(a)  An is lim
n 
an (b) A
n 1
n is min an
n
n 1

 

(c)  Bn is lim
n 
bn (d) B n is max bn
n
n 1 n 1

2
39. The area bounded on the right by x  y  2 , on the left by y  x and below by x-axis is
(a) 23/6 (b) 5/6 (c) 17/20 (d) 0
40. The distance of the point (3, –4, 5) from the plane 2 x  4 y  6 z  6  0 measured along a line with
direction ratios 2, 1, 3 is

(a) 14 (b) 24 (c) 30 (d) 94

41. Consider the statements:

 1 1
S1) In the set of 2 × 2 real matrices   is similar to a diagonal matrix
 0 1
S2) If M is a 2 × 2 real matrix and Mn = I for some n   then M  I
(a) Both S1 and S2 are true (b) S1 is true but S2 is false
(c) S2 is true but S1 is false (d) Both S1 and S2 are false
42. Let f ( x)  ( x  2)( x  4)( x  6)  2 then f has
(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
1
43. For a fixed y  [0,1], the value of  [ x  y ]dx is (where for a real number t, [t] is the greatest integer less
0

than or equal to t) www.careerendeavour.com

1
(a) 0 (b) 1 (c) y (d) y
2
44. Let V be the vector space of all 2 × 3 real matrices and W be vector space of all 2 × 2 real matrices.
Then
(a) there is a one-one linear transformation from V  W
(b) kernel of any linear transformation from V  W is nontrivial.
(c) there is an onto linear transformation from W  V
(d) there is an isomorphism from V  W

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45. Let A be a 2 × 2 real matrix. Which of the following statements is true?
(a) A ll the entries of A 2 are non-negative (b) The determinant of A2 is non-negative
(c) The trace of A2 is non-negative (d) all the eigenvalues of A2 are non-negative
46. Let f , g :    be two differentiable functions. Suppose that f ( x)  g ( x )  0 for x  0 . Then
(a) f ( x)  g ( x) for all x  0 (b) f ( x)  g ( x) for all x  0
(c) f ( x)  f (0)  g ( x)  g (0) for all x  0 (d) f ( x)  f (0)  g ( x)  g (0) for all x
47. Let f :    be given by f ( x )  | x | sin( x) . Then
(a) f is differentiable at 0 and f (0)  0 (b) f is differentiable at 0 and f (0)  1
(c) f is continuous at 0, but not differentiable at 0 (d) f is not continuous at 0, but differentiable at 0
48. Consider the group  p   p under addition. The number of cyclic subgroups of order p is
2
(a) 1 (b) p  1 (c) p  1 (d) p  1
49. Let R be a ring with unity. Then
(a) The set of all non zero elements in R forms a group under multiplication
(b) The set of all non zero invertible elements in R forms a group under multiplication
(c) The set of all non zero divisors in R forms a group under multiplication
(d) none of the above
50. Let A, B   and C  {a  b / a  A, b  B} . Then the false statement in the following is
(a) If A, B are bounded sets, then C is a bounded sed
(b) If C is a bounded set, then A, B are bounded sets
(c) If   A,   B are bounded sets, then   C is a bounded set
(d) If   C is a bounded set, then   A,   B are bounded sets

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 8

HYDERABAD CENTRAL UNIVERSITY (HCU)

M.Sc. Mathematics Entrance - 2011

ANSWER KEY

PART-A

1. (d) 2. (b) 3. (a) 4. (c) 5. (a)

6. (a) 7. (b) 8. (d) 9. (d) 10. (b)
11. (d) 12. (c) 13. (b) 14. (c) 15. (d)
16. (a) 17. (b) 18. (a) 19. (d) 20. (b)
21. (a) 22. (b) 23. (c) 24. (b) 25. (d)

PART-B

26. (d) 27. (d) 28. (c) 29. (b) 30. (a)
31. (c) 32. (c) 33. (b) 34. (a) 35. (b)
36. (c) 37. (__) 38. (c) 39. (b) 40. (a)
41. (b) 42. (a) 43. (d) 44. (b) 45. (b)
46. (c) 47. (a) 48. (c) 49. (b) 50. (d)

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