paper maths june2011

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Part-B

June – 2011

1. Let W be the vector space of all real polynomials of degree at most

3. Define :T W W by (Tp) (x) = p' (x) where p' is the derivative

of p. The matrix of T in the basis {1, x, x2, x3}, considered as

column vectors, is given by

1.

0 0 0 00 1 0 00 0 2 00 0 0 3

2.

0 0 0 01 0 0 00 2 2 00 0 3 3

3.

0 1 0 01 0 2 00 0 0 30 0 0 0

4.

0 1 2 31 0 2 00 0 0 00 0 0 0

2. Using the fact that

1

1 1

1 1log2, equals

1

n n

n n n

1. 1-2 log2 2. 1+log 2

3. (log2)2 4. –(log2)2

3. The reaction time to a stimulus X (in seconds) is distributed

normally in

Group 1 with mean 2 and variance 8;

Group 2 with mean 4 and variance 1.



The two groups appear in equal proportions. x is an observable

value of X. The best discriminant function (in the sense of

minimizing misclassification probabilities) is to classify into group

1. 2 if x > 3; otherwise in group 1

2. 1 if x > 3; otherwise in group 2

3. 2 if 0 x 83

; otherwise in group 1

4. 1 if 0 x 83

; otherwise in group 2

4. For 21 2,V V V ¡ and 2

1 2, ,W W W ¡ consider the

determinant map det: 2 2× ¡ ¡ ¡ defined by

det (V, W) = V1W2 –V2 W1 Then the derivative of the determinant

map at 2 2, ×V W ¡ ¡ evaluated on (H, K) 2 2× ¡ ¡ is

1. det (H, W) + det (V, K)

2. det (H, K)

3. det (H, V) +det (W, K)

4. det (V, H)+ det (K,W)

5. Consider the LP problem maximize x1+x2

Subject to

1 2

2 2

1 2

2 102 10

, 0

x xx xx x

Then



1. The LP problem admits an optimal solution

2. The LP problem is unbounded

3. The LP problem admits no feasible solution

4. The LP problem admits a unique feasible solution

6. A system of 5 identical units consists of two part A and B which

are connected in series. Part A has 2 units connected in parallel and

part B has 3 units connected in parallel. All the 5 units function

independently with probability of failure 12

. Then the reliability of

the system is

1. 3132

2. 1132

3. 132

4. 2132

7. Consider a group G. Let Z(G) be its centre, i.e., Z(G) =

: for all .g G gh hg h G For n¥ , the set of positive

integers, define

1 1,..., ×...× : ,...,n n nJ g g Z G Z G g g e .

1. not necessarily a subgroup.

2. a subgroup but not necessarily a normal subgroup.

3. a normal subgroup.

4. isomorphic to the direct product Z(G) ×…×Z(G) ((n–1)

times).



8. The number of elements in the set

:1 1000, and 1000 are relatively primem m m is

1. 100 2. 250

3. 300 4. 400

9. Let 1 2, ,..., ,nX X X 2,n be i.i.d. observations from 20,N

distribution, where 20,N distribution, where 0 < 2 < is an

unknown parameter. Then the uniformly minimum variance

unbiased estimate for 2 is

1. 2

1

1 n

ii

Xn

2. 2

1

11

n

ii

Xn

3. 2

1

1 n

ii

X Xn

4. 2

1

11

n

ii

X Xn

10. Consider an aperiodic Markov chain with state space S and with

stationary transition probability matrix , .ijP p ij S Let the n-

step transition probability matrix be denoted by

, , .n nijP p i j S Then which of the following statements is

true?

1. lim 0nn iip only if i is transient.



2. lim 0nn iip if and only if i is recurrent.

3. lim lim0n nn ii n iip p if i and j are in the same communicating

class.

4. lim lim0n nn ii n iip p if i and j are in the same communicating

class.

11. Suppose that we have i.i.d. observations

1 1 2, 2 ,, , ,..., , 3,n nX Y X Y Y Y n where Xi and Yi are independent

normal random variables. Consider = the sample Kendall's rank

correlation coefficient computed from this data. Then which of the

following is correct?

1. P 102

2. P 102

3. E 0

4. E 0

12. Let :f £ £ be a complex valued function given by

, , .f z u x y iv x y Suppose that v(x, y) = 3xy2. Then

1. f cannot be holomorphic on £ for any choice of u.

2. f is holomorphic on £ for a suitable choice of u.

3. f is holomorphic on £ for all choices of u.

4. v is not differentiable as a function of x and y.



13. The power series 2

02 n nz

converges if

1. |z| 2

2. |z| > 2

3. |z| 2

4. |z| < 2

14. Batteries for torch lights are packed in boxes 10 and a lot contains

10 boxes. A quail inspector randomly chooses a box and then

checks two batteries selected randomly without replacement from

that box. The lot will rejected if any one of the two chosen batteries

turns out to be defective. Suppose that 9 of the 10 boxes in the lot

contain no defective batteries and only one box contains 2

defective ones. What is the probability that the lot will NOT be

passed by the Inspector?

1. 1974950

2. 982475

3. 8225

4. 17450

15. Let X(t) be the number of customers in are M/M/1 queuing system

with arrival rate 3 and service rate 6. Which of the following is

true?

1. lim 5 0t P X t

2. lim 1532t P X t



3. lim 31532t P X t

4. lim 5 1t P X t

16. Which of the following is/are correct.

1. A free particle in 3¡ can have infinite degrees of freedom.

2. The number of degree of freedom on N particles is greater

than 3N.

3. A system of N particles with k constant has 3N+k degrees of

freedom

4. A system consisting of three point masse connected by three

rigid massless rods has six degrees of freedom.

17. The unit digit of 2100 is

1. 2 2. 4

3. 6 4. 8

18. Let G be a group of order 77. Then the center of G is isomorphic to

1. 1¢ 2. 7¢

3. 11¢ 4. 77¢

19. Let 1/ for 0,1 .nnf x x x Then

1. limt nf x exists for all 0,1 .x

2. limt nf x defines a continuous function on [0, 1].

3. {fn} converges uniformly on [0, 1].



4. lim 0t nf x for all 0,1 .x

20. To examine whether two different skin creams, A and B, have

different effect on the human body n randomly chosen persons

were enrolled in a clinical trial. Then cream A was applied to one

of the randomly chosen arms of each person, cream B to the other.

What kind of a design is this?

1. Completely Randomized Design

2. Balanced Incomplete Block Design

3. Randomized Block Design

4. Latin Square Design

21. Consider the ODE

" ' 0,1u t P t u t Q t u t Rt

There exist continuous function P, Q and R defined on [0, 1] and

two solution u1 and u2 of this ODE such that the Wronskian W of

u1 and u2 is

1. 2 1, 0 1W t t t

2. sin 2 , 0 1W t t t

3. cos2 , 0 1W t t t

4. 1, 0 1W t t ,

22. Let I1 be the ideal be the ideal generated by 4 23 2x x and I2 be

the ideal generated by 3 1x in .x¤ If 1 1/1F x ¤ and

2 2,/F x I ¤ then,



1. F1 and F2 are fields.

2. F1 is a field, but F2 is not a field.

3. F1 is not a field while F2 is a field.

4. Neither F1 nor F2 is a field.

23. For the Volterra type linear integral equation

0

2 ,x

xx x e d the resolvent kernel R(x, ; 2) of the

kernel xe is

1. 2 2 xx e

2. xx e

3. 3 xe

4. xe

24. A general solution of the second order equation 4 0xx yyu u is of

the form ,u x y

1. f x g y

2. 2 2f x y g x y

3. 4 4f x y g x y

4. 4 4f x y g x y

where f and g are twice differentiable functions.



25. The dimension of the vector space of all symmetric matrices of

order n×n (n 2) with real entries and trace equal to zero is

1. 2 / 2 1n n

2. 2 2 / 2 1n n

3. 2 / 2 1n n

4. 2 2 / 2 1n n

26. Let D be a non-zero n×n real matrix with n 2. Which of the

following implications is valid?

1. det (D) = 0 implies rank (D) = 0

2. det (D) =1 implies rank (D) 1

3. rank (D) = 1 implies det (D) 0

4. rank (D) = n implies det (D) 1

27. The number of characteristics curves of the PDE

2 3 22 1 3 0xx yy xy xx y u y y x u x y u u u

passing through the point 1, 1x y is

1. 0 2. 1

3. 2 4. 3

28. Let 1 2 .I U ¡ For ,x ¡ let

φ x dist , I inf | |: I .x x y y Then

1. φ is discontinuous somewhere on ¡ .



2. φ is continuous on ¡ but not differentiable only at x =1.

3. φ is continuous on ¡ but not differentiable only at x = 1 and

2.

4. φ is continuous on ¡ but not differentiable only at x=1, 3/2

and 2.

29. Let 5×5 '

: 0 1 , , 1 .ij ij ijj

S A A a a or i j a i

Then the number of elements in S is

1. 52 2. 55

3. 5! 4. 55

30. Suppose X is a random variable with E(X) = Var (X). Then the

distribution of X.

1. is necessarily Poisson.

2. is necessarily Exponential.

3. is necessarily Normal.

4. cannot be identified from the given data.

31. Let 2 : 0 1A x x and 3 :1 2 .B x x Which of the

following statements is true?

1. There is a one to one, onto function from A to B.

2. There is no one to one, onto function from A to B taking

rationals to rationals.

3. There is no one to one function from A to B which is onto.



4. There is no onto function from A to B which is one to one.

32. Let P be a polynomial of degree N, with 2N . Then the initial

value problem u'(t)=P(u(t)), u(0) = 1 has always.

1. a unique solution in ¡ .

2. N number of distinct solution in ¡ .

3. no solution in any interval containing 0 for some P.

4. a unique solution in an interval containing 0.

33. Let be a primitive fifth root of unity. Define

2

1

2

0 0 0 00 0 0 00 0 1 0 00 0 0 00 0 0 0

A

For a vector 51, 2, 3, 4, 5,v v v v v v ¡ , def | | | |T

Av vAv where vT is

transpose of If w = (1, –1, 1, 1, –1), then |w|A equals

1. 0 2. 1

3. –1 4. 2

34. The variational problem of extreminzing functional

2

2 2

0, 0 1, 2

x dI y x y y dx y zdx

has

1. a unique solution

2. exactly two solutions



3. an infinite number of solutions

4. no solution

35. The set 1 1sin : nn n

¥ has

1. one limit point and it is 0

2. one limit point and it is 1

3. one limit point it is –1

4. three limit points and these are –1 and 1



PART C

Unit-I

36. Let M be the vector space of all 3 × 3 real matrices and let

2 1 00 2 00 0 3

A

Which of the following are subspaces of M?

1. :X M XA AX

2. :X M X A A X

3. : trace 0X M AX

4. : det 0X M AX

37. Let N be a 3 × 3 nonzero matrix with the property N3 = 0. Which of

the following is/are true?

1. N is not similar to a diagonal matrix.

2. N is similar to a diagonal matrix.

3. N has one non-zero eigenvector.

4. N has three linearly independent eigenvector.

38. Let {fn} be a sequence of integrable functions defined on an

interval [a, b]. Then

1. If b

a

0 . ., then 0n nf x a e f x dx



2. If b

a

0,nf x dx and each fn is a bounded function, then

b

a

0nf x dx

3. If 0nf x a.e. and each fn is a bounded function, then

b

a

0nf x dx

4. If 0nf x a.e. and the 'snf are uniformly bounded, then

b

a

0nf x dx

39. Let W = {p(B) = {p(B) : p is a polynomial with real coefficients},

where 0 1 00 0 11 0 0

B

The dimension d of the vector space W

satisfies

1. 4 6d 2. 6 9d

3. 3 8d 4. 3 4d

40. Let X denote the two-point set {0, 1} and write 0,1jX for

every j = 1, 2, 3,… Let 1

n

jj

Y X

. Which of the following is/are

true?

1. Y is a countable set.

2. Card Y = card [0,1].



3. 1 1

n

jn j

X

U is uncountable.

4. Y is uncountable.

41. Which of the following is/are metrics on ¡ ?

1. , mind x y x , y

2. , | |d x y x y

3. 2 2, | |d x y x y

4. 3 3, | |d x y x y

42. For 1, 2,...,d

dx x x x ¡ , and p 1, define 1/

1|| || | |

pp

p jj

x x

and || || max | |: 1,2,.... .jx x j d Which of the following

inequalities hold for all dx ¡ ?

1. 1 2|| || || || || ||x x x

2. 1|| || || ||x d x

3. 1|| || || ||x d x

4. 1 2|| || || ||x d x

43. Which of the following is/are correct?

1. 1log 1 as 1

n nn



2. 11 log 1 1as n nn

3. 2 1log 1 1as n nn

4. 21log 1 1as n nn

44. Let T be a linear transformation on the real vector space n¡ over

¡ such that 2T T for some ¡

1. || || | | || ||Tx x for all nx ¡

2. If || || || ||Tx x for some non-zero vector nx ¡ , then 1

3. T I where I is the identity transformation on n¡ .

4. If || || || ||Tx x for a nonzero vector nx ¡ , then T is

necessarily singular.

45. Let ,1 , ,ij i ja a a i j b where 1,......., na a are real number. Let

ijA a be the n × n matrix ija . Then

1. It is possible to choose a1……, an so as to make the matrix A

non-singular.

2. The matrix A is positive definite if (a1….,an) is a nonzero

vector

3. The Matrix A is positive semi-definite for all (a1…,an),

4. For all (a1,…,an), zero is an eigenvalue of A.



46. Consider the map 2 2:f ¡ ¡ defined by

2 2, 3 2 ,4 5f x y x y x x y y Then

1. f is discontinuous at (0, 0).

2. f is continuous at (0, 0) and all directional derivatives exist at

(0, 0).

3. f is differentiable at (0, 0) but the derivative Df (0, 0) is not

invertible.

4. f is differentiable at (0, 0) and the derivative Df (0, 0) is

invertible

47. Let 2 2, 3 2 ,4 5f x y x y x x y y Then

1. f is discontinuous at (0, 0).

2. f is continuous at (0, 0) and all

3. f is differentiable at (0, 0) but the derivative Df (0, 0) is not

invertible.

4. f is differentiable at (0, 0) and the derivative Df (0, 0) is

invertible

48. Let x, y, n£ . Consider

2,φ

, sup || || , ,i

qf x y e x e y ¡

Which of the following is/are correct?

1. 2 2, || || || || 2Re | , |f x y x y x y

2. 2 2, || || || || 2Re | , |f x y x y x y



3. 2 2, || || || || 2 | , |f x y x y x y

4. 2 2, || || || || 2Re | , |f x y x y x y

49. Which of the following subsets of 2¡ are convex?

1. , :| | 5 | | 10x y x y

2. 2 2, : 1x y x y

3. 2, :x y y x

4. 2, :x y y x

50. Consider the function | cos | | sin 2 | .f x x x At which of the

following points is f not differentiable?

1. 2 1 :2

n n

¢

2. :n n ¢

3. 2 :n n ¢

4. :2

n n

¢

51. Let : :| | | |aF f f x f y K x y ¡ ¡ for all ,x y ¡

and for some 0 and some K > 0. Which of the following is/are

true?

1. every f F is continuous

2. every Ff is uniformly continuous



3. every differentiable function f is in F

4. every Ff is differentiable

52. Suppose A, B, are n × n positive definite matrices and I be the n ×

n identity matrix. Then which of the following are positive definite.

1. A + B

2. ABA

3. A2+I

4. AB

53. If {xn} {yn} are sequences of real numbers, which of the following

is/are true?

1. limsup limsup 1limsupn n n nn n nx y x y

2. sup sup supn n n nn n nlim x y lim x lim y

3. inf inf infn n n nn n nlim x y lim x lim y

4. inf inf infn n n nn n nlim x y lim x lim y

54. Which of the following sets are dense in ¡ with respect to the

usual topology.

1. 2, :x y x ¡ ¥

2. 2, : is a rational numberx y x y ¡

3. 2 2, : =5x y x y ¡



4. 2, : 0x y xy ¡

Unit II

55. Let , 1,2 3,4H e and

, 1,2 3,4 , 1,3 2,4 , 1,4 2,3K e be subgroups of S4, where

e denotes the identify element of S4. Then

1. H and K are normal subgroups of S4.

2. H is normal in K and K is norm in A4.

3. H is normal in A4 but not normal in S4.

4. K is normal in S4, but H is not

56. Let :f D D be holomorphic with 102

f and 1 0,2

f

where = :| | 1 .z z D= Which of the following is correct?

1. | ' 0 3/ 4f

2. | ' 1/ 2 | 4 /3f

3. | ' 0 | 3/ 4 and | ' 1/ 2 4 /3f f

4. ,f z z z D

57. Let <p(x)> denote the ideal generated by the polynomial p(x) in

.x¤ If 3 2 1f x x x x and 3 2 1g x x x x , then

1. 3f x g x x x

2. .f x g x f x g x



3. 2 1f x g x x

4. 4 1f x g x x

58. At z = 0 the function 11

z

zef ze

has

1. a removable singularity.

2. a pole.

3. an essential singularity.

4. the residue of at 0f z z is 2.

59. Define

: 0H z y £

: 0H z y £

: 0L z x £

: 0L z x £

The function 3 1

zf zz

1. maps H+ onto H+ and H– onto H–

2. maps H+ onto H– and H– onto H+

3. maps H+ onto L+ and H– onto L–

4. maps H+ onto L– and H– onto L+

60. Consider three subsets of 2 ,¡ namely



2 21 , : 1A x y x y

2 1, :A y y ¡

3 0,2A

Then there always exists a continuous real valued function f on 2¡

such that

jf x a for , 1,2,3jx A j

1. if an only if at least two of the numbers 1, 2, 3a a a are equal

2. if 1 2 3a a a

3. for all real values of 1 2 3, ,a a a

4. if and only if 1 2a a

NOTE :- Questions after this are for statistics students only and therefore aren,t included here.

paper maths june2011

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