SUBJECT CODE BOOKLET CODE

2012 (11) MATHEMATICAL SCIENCES

TEST BOOKLET Time : 3:00 Hours Maximum Marks: 200

INSTRUCTIONS

You have opted for English as medium of Question Paper. This Test Booklet contains one hundred and twenty (20 Part'AY+40 Part 'B' +60 Part 'C') Multiple Choice Questions (MCQs). You are required to answer a maximum of 15,25 and 20 questions from part 'A' 'B' and 'C' respectively. If more than required number of questions are answered, only first 15, 25 and 20 questions in Parts 'A' 'B' and 'C' respectively, will be taken up for evaluation. Answer sheet has been provided separately. Before you start filling up your particulars, please ensure that the booklet contains requisite number of pages and that these are not tom or mutilated. If it is so, you may request the Inviglator to change the booklet. Likewise, check the answer sheet also. Sheets for rough work have been appended to the test booklet. Write your Roll No., Name, your address and Serial Number of this Test Booklet on the Answer sheet in the space provided on the side 1 of Answer sheet. Also put your signatures in the space identified. You must darken the appropriate circles with a pencil related to Roll Number, Subiect Code, Booklet Code and Centre Code on the OMR answer sheet. It is the sole resmnsibilitv of the candidate to meticulouslv follow the instructions given on the Answer Sheet, failing which, the computer shall not be able to decipher the correct details which may ultimatelv result in loss. including rejection of the OMR answer sheet. Each question in Part 'A' canies 2 marks, Part 'B' 3 marks and Part 'C' 4.75 marks respectively. There will be negative marking @ 0.5marks in Part 'A' and @ '0.75 in Part 'B' for each wrong answer and no negative marking for Part 'C'. Below each question in Part 'A' and 'B', four alternatives or responses are given. Only one of these alternatives is the "correct" option to the question. You have to find, for each question, the correct or the best answer. In Part 'C' each question may have 'ONE' or 'MORE' correct options. Credit in a question shall be given only on identification of 'ALL' the correct options in Part 'C'. No credit shall be allowed in a question if ani incorrect option is marked as correct answer. Candidates found copying or resorting to any unfair means are liable to be disaualified from this and future examinations. Candidate should not write anything anywhere except on answer sheet or sheets for rough work. After the test is over, you MUST hand over the Test Booklet and answer sheet (OMR) to the invigilator. Use of calculator is not permitted.

Roll No. ....................... I have verified all the informati01 filled in by the candidate.

Name ...........................

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Which of the following numbers is the largest?

234, 243, 324, 342, 423, 432.

2. k $ w A E K D E F G H m~JrSoTa & m i ' / S f f $ c A , C m F & & $ V m h & The cube ABCDEFGH in the figure has each edge equal to a. The area of the triangle with vertices at A, C and F is

3. i j ~ m UGCCSIR &h&lif;;rfc);97i9)&*~~2 & U M I W-W=I&W ,d? What is the number of distinct arrangements of the letters of the word UGCCSIR so that U and I cannot come together?

4. W ? f & m E T ? T T r T r n d w m 2 1 i l / ~ ~ & d $ m m - ~ ~ m 9 ~ ? r G W il? Suppose the sum of the seven positive numbers is 21. What is the minim possible value of the average of the sqaures of these numbers?

113 +213 +313 113 + 313 + 513 + . . . + 9913 213 + 413 + 613 + . . . + 10013 A = , B = , C =

100 50 5 0 f 3 9 $ 3 $ m & 8 ? Which of the following is true?

XYmz)Im5 $ ~ d & ~ & v ~ ~ m r 5 7 ' ; i 7 ~ w & r f l 8 / ~ x - - m m ) ~ 8 ~ y-m7?6 5 ~ y i m m w w c d ~ / d ~ + i ~ & ~ Q :

A circle of radius 5 units in the XY plane has its centre in the first quadrant, touches the x-axis and has a chord of length 6 units on the y-axis. The coordinates of its centre are

6 # L d r n + w + ~ l Ff ) m ~ ~ ~ m a ? r ; n m $ ~ ~ ~ & ~ ~ #

~ ; n m ~ ~ ~ m 8 / ~ ~ ~ 4 m m m i ? M i + m ~ 8 : A wire of length 6m is used to make a tetrahedron of each edge lm, using only one strand of wire for each edge. The minimum number of times the wire has to be cut is

8. 4 R f 3 9 - & & d & m r n x 8i=ltlog2x mmmi??

If the sum of the next two terms of the series below is x, what is the value of log2x ?

2, -4, 8, -16, 32, -64, 128,.. ... .. ..

A conical vessel with semi-vertical angle 30" and height 10.5 cm has a thrn lid. A sphere kept inside it touches the lid. The radius of the sphere in cm is

mar, Akbar and Anthony are three fiends, one of whom is a doctor, another is an engineer and the third is a professor. Amar is not an engineer. Akbar is the shortest. The tallest person is a doctor. The engineer's height is the geometric mean of the heights of the other two. Then which of the following is true?

1 . Amar is a doctor and he is the tallest 2. Akbar is a professor and he is the tallest 3. Anthony is an engineer and he is shortest 4. Anthony is a doctor and he is the tallest

11. ~ 1 0 0 m i 0 0 mr)'100 $ m 8 & 7 m d 7 em ,d+m~ h? If 100 cats catch 100 mice in 100 minutes, then how long will it take for 7 cats to catch 7 mice?

12. m ~ ~ m m a T 8 ? What does this diagram demonstrate?

13. ~ T ~ & W & + N N ~ $ ~ ~ I M & N ~ ~ ~ ~ ~ I ~ J R ~ W ~ ~ W # * m z w . * w i f t M m d m ~ ~ & & m & ? Suppose there are socks of N different colors in a box. If you take out one sock at a time, what is the maximum number of socks that you have to take out before a matching pair is found? Assume that N is an even number.

At what time after 4 0' clock, the hour and the minute hands will lie opposite to each . other?

Which of the following curves just touches the x axis?

~ A B 7 C D & ~ i 9 ~ A 0 , 2 0 D $ ~ 8 ~ E J z + J T O A B B ~ T & ~ v O C D . $ *drn3;rrw8? If AB is paralled to CD and AO=20D7 then the area of triangle OAB is bigger than the area of triangle OCD by a factor of

A semi-circular arch of radius R has a vertical pole put on the ground together with one of its legs. An ant on the top of the arch finds the angular height of the tip of the pole to be 45". The height of the pole is

Suppose we make Nidentical smaller spheres from a big sphere. The total surface area of the smaller spheres is X times the total surface area of the big sphere, where X is

19. -241 301 331 391 511 ................ &Mwkim$? What is the next number in the sequence 24,30,33,39,51,------?

20. w m + m M # i i # 8 ) W ~ ; r * - G ) m 8 ; 7 z + $ & m 8 1 m m ~ ~ r n ~ \ u ) m ) ~ ~ ~ ) ~ r n r n ~ ~ ~ ~ ~ ~ - ~ ~ d ) m ~ ~ -8 Four lines are drawn on a plane with no two parallel and no three concurrent. Lines are drawn joining the points of intersection of the previous four lines. The number of new lines obtained this way is

21. Let f be a twice differentiable function on W . Given that f " ( x ) > 0 for all XEW,

1. f ( x ) = 0 has exactly two solutions on W

2. f ( x ) = 0 has a positive solution if f (0 ) = 0 and f ' ( 0 ) = 0 .

3. f ( x ) = 0 has no positive solution if f (0 ) = 0 and f ' ( 0 ) > 0 .

4. f ( x ) = 0 has no positive solution if f (0) = 0 and f ' ( 0 ) c 0 .

22. Let f be a continuously differentiable real-valued function on [a. b] such that 1 f '(x)l i K for

all x ~ [ a , b ] . ~ o r a ~ a r t i t i o n ~ = { a = a ~ <al < . - - < a , =b],let ~ ( f , P ) a n d ~ ( f . ~ ) d e n o t e

the upper and lower Riernann sums of f with respect to P . Then

1. / ~ ( f , ~ ) l ~ ~ ( b - a ) < l ~ ( f . ~ ) \ .

2. u ( ~ . P ) - ' L ( ~ , P ) s ~ ( b - a ) .

3. ~(f,~)-~(f,~)I~~~~~~,where~~~~~=max(a~+~-a,)isthenomofthe o<isn-l partition.

4. u ( ~ . P ) - L ( ~ , P ) ~ K I ~ P N ( ~ - u ) .

2. s R a c b , irt lirn f ( x ) i l i m f ( x ) 81 x+a+ x+b-

23. Let f be a monotone nondecreasing real-valued function on W. Then

1. lirn f (x) exists at each point a. x+a

2. If a c b , then lirn f (x)< lirn f (x) x+a+ x+b-

3. f is an unbounded function. 4. The function g(x) = e-f '"' is a bounded function.

24. Let f be a real-valued hct ion on @ satisfjmg (for a fixed a EW) f (n) = ra f ( x ) for any

r>O and X E @ .

1. If f ( x ) = f ( y ) whenever llxll= llyll = P for a p > 0, then f ( x ) = flllxlr .

2. If f (x) = f ( y ) whenever llxll= 11 yll = 1 , then f ( x ) = 1 1 x 11" . 3 . If f ( x ) = f ( y ) whenever Ilxll= l l y l l = 1 , then f ( x ) = c 11 x IT, for some constant c.

4. If f ( x ) = f ( y ) whenever llxIl= Ilyll, then f must be a constant function.

sin x 2 . f ( x ) = - .

X

COS X 4 . f ( x ) = - .

X

25. Which of the following real-valued functions on (0,l) is uniformly continuous?

1 1 . f ( x ) = - .

X

1 3 . f ( x ) =sin-.

X

sin x 2. f ( x ) = - .

X

COS X 4. f ( x ) = - . X

26. Let Xbe a metric space and A c X be a connected set with at least two distinct points. Then the number of distinct points in A is

1. 2 . 3. countably infinite.

2. more than 2, but finite. 4. uncountable.

n ( n + l ) n(n - 1 ) 1. 2. n. 4. 0.

2 3.

2

27. Let n be a positive integer and let M, (W) denote the space of all n x n real matrices. If

T : M, ( w ) -+ M, (R) is a linear transformation such that T ( A ) = 0 whenever A E M, (W) is

symmetric or skew-symmetric, then the rank of T is

28. Let S: @+A? and T.. A?+& be linear transformations such that ToS is the identity map of @. Then

1 . SAT is the identity map of @. 2. SOT is one-one, but not onto.

3. SOT is onto, but not one-one. 4. SOT is neither one-one nor onto.

29. Let V be a 3-dimensional vector space over the field F3=Z/ 3 2 of 3 elements. The number of

distinct ldimensional subspaces of V is

( p j q ) = j p ( ~ ) ~ ( x ) d r t w?w%8 1 Y ww- & m d : 0

1. {I, XI. 2. {l? x&}. 3. 1 2 - 1 4. (1, x - f } .

30. Let V be the inner product space consisting of linear polynomials, p: [0, 11 4%' (i.e., V

consists of polynomialsp of the form p ( x ) = ax + b, a, b E R), with the inner product

defined by I

An orthonormal basis of V is

1. {I, x ) . 2. {I, x&}. 3. {I, (2x -I)&} . 4. {I, x -9

31. Let f (x) be the minimal polynomial of the 4 x 4 matrix

Then the rank of the 4x 4 rnatnx f (A) is

1. 0. 2. 1. 3. 2.

32. Let a, b, c be positive real numbers such that b2 + c2 < a < 1. Consider the 3 x 3 matrix

1. All the eigenvalues o&A are negative real numbers. 2. All the eigenvalues of A are positive real numbers. 3. A can have a positive as well as a negative eigenvalue. 4. Eigenvalues of A can be non-real complex numbers.

1. f ~ m * m ~ h c r ; l 8 1 2. @ w g w * m 8 1

3. S mf**/ 4. S w g *8l

33. Consider the functions f ,g : C + C defined by f ( z ) = e z , g ( r ) = eiz . Let

S={ZE@:R~ZE[-A,?]) . Then

1. f is an onto entire function. 2. g is a bounded function on C. *

3. f is bounded on S. 4. g is bounded on S.

34. Let f :D 4 D be a holomorphic function with f (0) = 0, where D is the open unit disc

{ z E @: Iz( < 1) . Then

m

35. Consider the power series zn! . The radius of convergence of this series is n=I

1. 0. 2. a. 3. 1. 4. a real number greater than 1.

36. In a group of 265 persons, 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is

37. The last two digits of 781 are

1. 07. 2. 17.

1. & F 3 , d l = r d $ t : ~ 2. & I F 7 , m3nmtdml

3 . & I F , , , i%JFaTm?~t: / 4. * M - Q ~ T & Q I ,

38. In which of the following fields, the polynomial

is irreducible in P [x] ?

1. the field F3 with 3 elements. 2. the field F, with 7 elements.

3. the field IFl',, with 13 elements. 4. the field Q of rational numbers.

39. Let o be a complex number such that 0 3 = l and o *l. Suppose L is the field ~ ( f i , o )

gene&ed by f i and o over the field Q of rational numbers. Then the number of subfields K of L such that Q 5 K 5 L is

1. artfa E R a20 $GT [-a, a ] . 2 . [0, 11.

3. artfa E R a20 $ & [ O , a ] . 4. * a , b ~ R 0 1 a 4 $ & [ a , b l .

40. Let f : Wn + R' be a linear map with f (0, - - - , 0) = 0 . Then the set

1. [-a, a] for some a E R, a2O. 2. [0, 11.

3. [O, a] for some a E $ aTO. 4. [a, b] for some a, b E & O i a<b.

dY 41. Let 3 ( x ) and y, ( x ) be the solutions of the differential equation - = y + 17 with initial

dx

conditions y, (0) = 0, y2 (0) = 1. Then

1. yl andy, will never intersect. 2. y, and y, will intersect at x=17.

3. y, andy, will intersect at x=e. 4. y, and y, will intersect at x = l .

42. & n;i y ( t ) = [:;!I

42. Let y ( t ) = (i: satis@

where A is a 2 x 2 constaht matrix with real entries satisfyng trace A = 0 and

det A > 0. Then y, (t ) and Y2 ( t ) both are

1. monotonkally decreasing hnctions of t . 2. monotonically increasing functions of t . 3. oscillating functions of t . 4. constant functions oft.

43. The partial differential equation

a2u d2u y-+x-=o at2 ay2

is hyperbolic in

1. 9 e second and fourth quadrants. 2. the fist and second quadrants. 3. the second and third quadrants. 4. the first and third quadrants.

44. A bounded harmonic fimction in the unit disc centered at origin and taking the value sin28 on the boundary is

45. The system of equations

where ~ E R , has an infinite number of solutions for

46. Let

where u(x) is a smooth function defined on [0,1] satisfying u(O)=O and u(l)=l. Which of the functions minimizes J ?

1

47. For the homogeneous Fredholm integral equation # ( x ) = A je"# ( t )dt , a non-trivial solution 0

exists, when X has the value

48. The total number of vibrational degrees of freedom of a molecule containing n collinear atoms is

1. 3n-6 . 2. 3n-5. ; 3. 3n-2 . 4. 3n.

x,=+.-.+xn2 49. Let XI, X,, . . . be i.i.d. standard normal random variables and let T, = . Then

n

1. The limiting distribution of Tn- 1 is X2 with 1 degree of freedom. 1, - 1

2. The limiting distribution of - is normal with mean 0 and variance 2. &

3. The limiting distribution of &(T, -1) is X2 with 1 degree of fieedorn.

4. The limiting distribution of &(T, - 1) is normal with mean 0 and variance 2.

- 50. Let {pn, n 2 0) be a sequence of numbers, such that for all n 2 0 , pn > 0, pn =l and

n=O

m

npn < m. Consider a Markov chain on the state space (0.1~2, - - .} with transition

probability matrix

Po P, P2 " '

Then

1. the chain is not irreducible. 2. the chain is irreducible and transient. 3. the chain is irreducible and null recurrent. 4. the chain is irreducible and positive recurrent.

51. Suppose X I , X2 , X3 , X, are i.i.d. random variables taking values 1 and -1 with probability !4

each. Then E (xi + X, + X, + X , equals

1. 4. 2. 76. 3. 16. 4. 12.

52. Suppose that has Normal (h,, , Z,, ) distribution, where C2,, is nonsingular. Let

X3 = -W2. Then whlch of the following has a singular normal distribution?

53. The radius of a circle is measured with an error of measurement which is normally distributed with mean 0 and variance d. Let X,;.., X , be n measurements on the radius. Let T and

2 s =l ,, z ( x i - f r b e the sample mean and sample variance respectively. Which of the

following represents an unbiased estimate of the area of the circle?

54. 973 I% & A @ m B H,: p=2 a?m HI: p 2 $ & 15 M20 d rn- @ N(~,c?) , 02>0 ~ 81 GM w? + fi ~l;rr-w v - . m m &m=r i ~ ~ i 8 3 ? r ~ $ ~ : X , = 3 = 1 . 8 , s , = s 2 = s / ~ h ? m t - * m ~ & d $ *

p - @ p A q p B T T F W T G W ~ ? ? / i + 7 $ + ) 9 ~ F L ) 8 ?

54. Suppose person A and person B draw random samples of sizes 15 and 20 respectively from N ( ~ , & ) , &>0 for testing H,: p=2 against HI: p>2. In both the cases, the observed sample means

and sample standard deviations are same with values Y, = = 1.8 , s, = s2 = s . Both of them

use the usual t-test & state the p-values pA and p, respectively. Then which of the following is correct?

1. PA ' P B .

2. P A = P B .

3- PA < PB 4. Relation between pA and p~ depends on the value of s.

{ P ( Y =1Ix,,x2 + 1 )

2. log ~ ( ~ = l I x ~ . x ~ ) 1 -

~ ( ~ = l I x , . x ~ + l ) p ( y = o I x , , x 2 ) 4. log X

P(Y =0Ix,,x2 +I) P ( Y =11x,,x2) i

55. In logistic regression model involving a binary response Y and two explanatory variables X, and X2, the coefficient of X2 is

i P(Y = 1(x,,x2 + I ) 2. log

~ ( ~ = l I x , . x ~ ) 1.

i ~ ( ~ = l I x , . x ~ + l ) P ( Y = O I X , . X ~ ) 4. log X

P ( Y =0Ix,,x2 +I) P ( Y =1(x,,x2). ; 1.

56. The failure rate of a parallel system of two components, where the component lifetimes are independent and have the exponential distribution with mean 2, is

1. aconstant. 2. a monotone and bounded function. 3. a monotone and unbounded function. 4. a non-monotone function.

57. Consider the linear model

where yi are observations, Oi are parameters and E[ are uncorrelated random variables with mean zero and constant variance for i=l, 2,3. Then which of the following is true?

1 . 2y, - y, - y3 is an unbiased estimator of 8,- 4 03. 2. 2y, - y, - y3 is the BLUE of 81- 4 83. 3. y, -3y3 is the BLUE of 81- 4 B3. 4. y, - 4y3 is an unbiased estimator of 81 - 4 83.

58. Consider the following design with 6 treatments 1, 2,. . . ,6 and 4 blocks as follows: {1,2,2}, {2,3,3), (3,441 and {5;6,6}. Which of the following is true?

1. The design is orthogonal and all treatment contrasts are estimable. 2. The design is non-orthogonal and all treatment contrasts are estimable. 3. The design is orthogonal and not all treatment contrasts are estimable. 4. The design is non-orthogonal and not all treatment contrasts are estimable.

59. Consider a population of 50 units {1,2, ..., 50) and suppose that 50 possible samples are listed as : {I) , {1,2), {1,2,3), {1,2,3,4), ..., {1,2,3 ,..., 50). One of these samples is chosen at rapdom. Let n; be the probability that unit i is in the selected sample. Then which of the following is necessarily true?

1. The expected sample size is 25. 2. The expected sample size is 25.5.

60. Men arrive in a queue according to a Poisson process with rate hl and women arrive in the same queue according to another Poisson process with rate h2. The arrivals of men and women are independent. The probability that the first arrival in the queue is a man is

61. Let sf.) be a sequence of continuous real-valued functions defined on [ ~ , m ) . ~ u ~ ~ o s e

fn ( x ) -+ f (x ) for all x E [O, m ) and that f is integrable. Then

* 2. i f f , + f m i f d y o n [O,m), then i L ( x ) d x + j f ( x ) d r . 0 0

m m

3. if f , + f un i fodyon [0,rn), then i f , ( x ) & + / f ( x ) d r . 0 0

1

4. if i ( f , ( x ) - f ( x ) I d x + o , then f,--t f uniformlyon[O, I]. 0

1. 2. 4R f mi? d G v.81 3. sR f mzt a t G i+Tqv8/ 4. sR f ww%zvimm8 m 8 1

62. Let X be a compact topological space and let f : X +R be a function. The graph of f is the

set G = { ( x , f (x)) : x E X ) E X x 8. Which of the follbwing are necessarily true?

1. G is a closed set if and only i f f is continuous. 2. I f f is continuous, then G is closed. 3. I f f is continuous, then G is connected. 4. If f is a bounded continuous function, then G is compact.

63. Suppose f :R + R is a differentiable function. Then which of the following statements are necessarily true?

1. If f'(x) l r < 1 for all x E W , then f has at least one fxed point.

2. Iff has a unique fixed point, then f'(x) < r < 1 for all x E 173 . . 3. Iff has a unique fixed point, then f (x) 2 r > -1 for all x E W . 4. 1f f ( x ) r < 1 for all x E W , then f has a unique fixed point.

64. Which of the conditions below imply that a hc t ion f :[0,1] + W is necessarily of bounded variation?

1. f is a monotone function on [0, 11. 2. f is a continuous and monotone fimction on [0, 11. 3. f has a derivative at each x ~ ( 0 , l ) .

4. f has a bounded derivative on the interval (0,l).

x3 . x2 65. Let f (x) = sin x - x +- and g(x) = cos x - 1 + - for x s R . Which of the following

3! 2! statements are correct?

1. f(x)20forallx>O. 2. g is an increasing function on [0,m).

3. g is a decreasing function on [0, a) . 4. f is a decreasing function on [o. a) .

1 66. Let f : A u E + W2 be differentiable, where A = (x, y) E R2 :- < x2 + y2 < 1

2

(x. y ) E R2 :(x - 2)2 + ( y - 2)2 . Let Df be the derivative of the function$ Wluch

of the following are necessarily correct?

1. If (Df )(x, y) = 0 for all (x, y) E A u E , then f is constant.

2. If (Df )(x, y) = 0 for all (x, y) E A , then f is constant on A .

3. If (Df) (x, y ) = 0 for all (x, y ) E E , then f is constant on E.

4. If(Df)(x.y)=O forall ( x , y ) ~ A ~ E , t h e n , f o r s o m e ( X ~ , ~ ~ ) , ( X , , ~ , ) E ~ ~ ,

f (x, y) = (xo, yo) for all (x, y) E A and f (x. y) = (x,, y,) for all (x. y ) E E .

67. Let L:Rn + 8 be the function L(x) = (x, y), where (.;) is some inner product on Wn

and y is a fixed vector in Rn . Further denote by DL, the derivative of L. Which of the following are necessarily correct?

1. DL(U)=DL(V) f o r a l l u , v ~ R " . 2. DL(O,O;..,O)=L .

3. DL (x) = llx(12 for all x E Wn . 4. ~ ~ ( 1 , 1 , - - , l ) = 0 .

68. f:[:ln,2x]+W2 w f ( t ) = ( c o s t , sint) 8 1 R v 3 ) ' d a ; h - d ~ : & 8 ?

3. t o ~ [ n , 2 n ] ~ ~ d * llf(2x)-f(n)ll~nIIf'(t0)ll dl

4. & t 4% 2x1 $ W f '(t) = (-sint, cost) 81

68. Let f : [n, 2n ]+R2 be the function f (t ) = (cost, sin t ) . &Which of the following are

necessarily correct?

1 1. There exists to E [n, 2x1 such that f '(to) = -( f (2n) - f (n)) .

n 1

2. There does not exist any to E [n,2n] such that f '(to) = -( f (2x) - f (n)) n

3. Thereexists to ~[n ,2n]suchtha t 1 1 f (2n)- f(n)ll <nl l f1( t0) l l .

4. f '(t) = (-sin t, cost) for all t E [n, 2x1 .

69. LetX=[-l.l]x[-l,l], A = { ( x . ~ ) E x : x ~ + ~ ~ = ~ ] , ~ = { ( ~ , ~ ) ~ X : l ~ l + l ~ l = l ) ,

C = { ( X , ~ ) E X : X ~ = O ) and D = ( ( X . ~ ) E X : X = ~ ~ ) . Then

1. A is homeomorphic to B.' 3. C is homeomorphic to D.

2. B is homeomorphic to C. 4. D is homeomorphic to A.

70. Let n be an integer, n 2 3, and let ul, 2.42, ..., u, be n linearly independent elements in a vector space over R . Set uo = 0 and u,+l = ul. Define vi = ui + ui+, and wi = u,, + ui for i = 1,2;- . ,n . Then

1. v, , v, , . . , v,, are linearly independent, if n = 20 10.

2. v, , v, , . . , vn are linearly independent, if n = 20 1 1.

3. w,, w2, ., w, are linearly independent, if n = 20 10.

4. w, , w2, . . ., wn are linearly independent, if n = 20 1 I.

71. Let V and W be finite-dimensional vector spaces over R and let Tl : V + V and T2: W + W be linear transformations whose minimal polynomials are even by

f i (x )=x3+x2+x+l and fi(x)=x4-x2-2.

Let T : V@ W i V@ W be the linear transformation defined by

T (v, w) = (Tl(v), T~(w)) for (v, W)E V@ W

and let f (x) be the minimal polynomial of T. Then

1. deg f (x)=7 . 3. nullity (7) = 1.

2. deg f(x)=5. 4. nullity (7) = 0.

72. Let a, b,c,d E R and let T:R2 T) R2 be the linear transformation defined by

Let S: C -+ C be the corresponding map defined by

~ ( x + i y ) = ( r n + b ~ ) + i ( c x + d ~ ) for x, y E 19. Then

1. Sisalways C-linear,thatis ~ ( z , + z ~ ) = ~ ( z ~ ) + ~ ( z ~ ) f o r a l l z , , z , ~ C a n d

~ ( a z ) = a ~ ( z ) f o r all a E C and z E C . 2. S is (27 -linear if b = -c and d = a. 3. S is C -linear only if b = - c and d = a. 4. S is C -linear if and only if T is the identity transformation.

73. Let A= (a& be an n x n complex matrix and let A* denote the conjugate transpose of A. Which of the following statements are necessarily true?

1. If A is invertible, then t r ( ~ * A ) z 0, i.e., the trace of A*A is nonzero. 2. Iftr(A.A)*O,thenAisinvertible. 3. If I t r ( ~ * A ) I C n2, then (avl < 1 for some i, j. 4. 1f tr(~2) = 0, then A is the zero matrix.

74. Let n be a positive integer and V be an (n + 1) - dimensional vector space over R . If { e , , e,, ..., en,,) is a basis of V and T: V -t V is the linear transformation satisfjmg

T(ej) = ei+1 for i = 1, 2 ,..., nand T(e,+l)= 0.

Then . .

1. trace of T is nonzero. 2. rank of T is n. 3. nullity of T is 1. 4. T n = T 0 T o . . . o T (ntimes)isthezeromap.

75. & ' & A ~ . ' B n x n e m 8 ' & A B = B A = O W . A + B . V . ~ ~ ~ ~ I ~ ~ ~ ~ ) . ~ *-a * i¶ $7

1. 37777 (A)= 37777(B) 81 2. & (A)+ 37777 (B)=n 81 3. ?WI (A)+ - (B)=n 81 4. A-B 3-h 81

75. Let A and B be n x n real matrices such that AB = BA = 0 and A+B is invertible. Which of the following are always true?

1. rank (A) = rank (B). 2. rank (A) + rank (B)= n. 3. nullity (A) + nullity (B)= n. 4. A-B is invertible.

76. Let n be an integer t 2 and let Mn (R ) denote the vector space of n x n real matrices.

Let B E Mn (R) be an orthogonal matrix and let B' denote the transpose of B. Consider

W, = {B'AB : A E M,, ( I )} . Which of the following are necessarily true?

1. W, is a subspace of M, (R) and dim W, S rank (B).

2. WB is a subspace of Mn (R) and dim WB = rank (B) rank (B').

3. w,=M,(R).

4. WB is not a subspace of M,, ( I ) .

77. Let A be a 5 x 5 skew-symmetric matrix with entries in R and B be the 5 x 5 symmetric

matrix whose (i, j)lh entry is the binomial coefficient ) for 1 Si 5 j 5 5 . Consider the

10 x 10 matrix, given in block form by

Then

2. det C=O. 4. trace of C is 5.

78. Suppose A is a 3 x 3 symmetric matrix such that

Letp be the number of positive eigenvalues ofA and let q = rank (A) - p . Then

1 . p = l . 2. p = 2 . 3 . q = 2 . 4 . q = 1.

79. Which of the following functions f are entire functions and have simple zeros at z = ik for all kcZ.

2. f(z) = a sin 2ni z, for some a&. :

3 . f(z) = b cos 2z(iz - %), for some bcC.

4 . f(z) = em, for some ~ E C .

80. Let y, = {ke": 0 5 B 5 2n) for k = 1.2.3. Which of the following are necessarily correct?

1 1 . - j L d z = 0 for k=1,2.3.

2ni yk z

81. Let f be an analyhc hmction defined on D = {r c C: lrl< 1) such that the range o f f is

contained in the set C \ (-m, 01. Then

1. f is necessarily a constant function. 2. there exists an analyhc fhction g on D such that g(z) is a square root of f (z) for each

Z E D .

3. there exists an analytic fhction g on D such that Re g(z) 2 0 and g(z) is a square root of f (z) for each z E D .

4. there exists an analyhc function g on D such that Re g(z). 5 0 and g(z) is a square root of f (z) for each z E D .

1. *Kc f(Q) 2 i h p r > l $ / M D , c f ( ~ ) b ~

2. &+Kc f(S2) i?. d p r > l & W D r = f ( ~ )

3. eqc f ( ~ ) i?. i h p r > l & f % Kc f ( ~ )

4. f (Q) m $1

82. Let f : Q -+ @ be an analytic function on an open set R c C. For r > 0, let ID,=

{ZEC : /zI < r } and let & be its closure. Which of the following are necessarily true?

1 . If Kc f(R),then D r c f ( R ) forsorne r > 1 .

2. If c f (a), then IDr = f (R) for some r > 1 .

3. I f & = f ( R ) , t h e n F c f ( R ) forsome r > 1 .

4. f (R) is open.

1 83. Let f (z)= z +- for z E C with z # 0 . Which of the following are always true?

z 1 . f is an analytic function on C \ (0) . 2. f is a conformal map on C \ (0) . 3. f maps the unit circle to a subset of the real axis. 4. The image of any circle in C \ (0) is again a circle.

84. ~ m w m & ~ & & q ( m ) ~ ~ ~ ~ ~ ~ i f & l l k l m ~ w (k, m)=li?/ F/? h mpn) r)' 3) 3)-3) mqr~chcl.. ~8 8.7

1 . f l m @ n &f?dq(n),n W - W ~ I

2. ~ ~ f f m p % a a n & E i + p ( a ~ - l ) t n ~ r n b I

3. K J t m p i % a ~ n d M 9 ( a n - l ) & n ~ i m m d z f & m m m

(a, n)=1 &I

4. i M f t @ a a n d E i + p ( a n - 1 ) t a m r n 8 M m ~ m (a, n)=l &I

84. For a positive integer m, let ~ ( m ) denote the number of integers k such that 1 I k I m and GCD(k, m)=l . Then which of the following statements are necessarily true?

1 . q(n) divides n for every positive integer n.

2. n divides e ( a n - 1) for all positive integers o and n.

3. n divides p(on - 1 ) for all positive integers a and n such that GCD(a, n)=l

4. a divides q(an - 1) for all positive integers n and n such that GCD(a, n)=l .

85. For a positive integer n 2 4 and a prime number p S n , let Upen denote the union of all p-Sylow

subgroups of the alternating group A, on n letters. Also let K, , denote the subgroup of A,

generated by Up,, and let I K,, ( denote the order of Kp,n . Then

4. n x w x & p W m - + e e m Q [ X ] + f P ( x p e - ' ) w a m m ~ d I .

86. For a positive integer n, let f , ( x ) = xn-' + xn" + . . . + x + 1. Then #

1. f, ( x ) is an irreducible polynomial in ~ [ x ] for every positive integer n.

2 f , ( x ) is an irreducible polynomial in Q [ x ] for every prime number p.

3 . fpe ( x ) is an irreducible polynormal in Q [ x ] for every prime number p and every

positive integer e.

4. f , ( x p I-') is an irreducible polynomial in Q [ x ] for every prime number p and every

positive integer e.

87. Consider the ring R = z[&] = { a + b G : a , 6 E Z) and the element a = 3 + f i of R.

Then

1. a is prime. 2. a is irreducible. 3. R is not a unique factorization domain. 4. R is not an integral domain.

88. Consider the polynomial f ( x ) = x4 - x3 + 14x2 + 5 x + 1 6 . Also for a prime number p, let IF,, denote the field withp elements. Which of the following are always true?

1. Considering f as a polynomial with coefficients in IF3, it has no roots in F3.

2. Considering f as a polynomial with coefficients in IF3, it is a product of two irreducible

factors of degree 2 over F3.

3. Considering f as a polynomial with coefficients in IF7, it has an irreducible factor of degree 3 %

over IF7.

4. f is a product of two polynomials of degree 2 over Z.

89. For a positive integer m, let a,,, denote the number of distinct prime ideals of the ring a x 1 (xm - 1)

Then

90. Let z be the topology on R for which the intervals [a,b), -m < a < b < oo, form a base. Let o be a topology on R such that a 2 z . Then

1. either a = z or a is the discrete topology. 2. if, moreover, the map x H -x is continuous for 0 , then o is the discrete topology. 3. if, moreover, the map x H -x is a homeomorphism for o , then a is the discrete topology. 4. if, moreover, the map x HI x I is a homeomorphism for a, then o is the discrete topology.

91. The differential equation

has

1. a unique solution. 3. no solution.

2. two solutions. 4. infinite number of solutions.

1

92. ~TFFT W&ST += ( x ) ; ~ ~ ( 0 . 1 ) y(0) = y( l ) = 0 in ~7 ; r ( x ) = ~ ~ ( x . { ) f ( t ) d t $ dx o

92. The solution of the differential equation

is given by

0

where

2. u ( x . t ) = e-'e-* . 4. u(x,t)=x-e-' .

A bounded solution to the partial differential equation

2. u ( x , t ) = e-xe-* .

4. u ( x , t ) = x - e - ' .

If u ( x , t ) satisfy the partial differential equation

then u ( x , t ) can be of the form

Where f and g are non-trivial smooth functions.

Let f be a continuous map from the interval [0,1] into itself and consider the iteration

xn+, = f (x$.

Which of the following maps will yield a fixed point for f ?

96. Consider the ordinary difkrential equation

and the Euler scheme with step size h

Which of the following are necessarily true for Yl whch approximates Y ( h ) = e"" ?

1. Yl is a polynomial approximation. 2. Yl is a rational function approximation. 3. Yl is a trigonometric function approximation. 4. YI is a truncation of infinite series.

1 . y(x) = A sin(x).

3 . y(x) = A sinh(Ax + B),.

97. Consider the functional b

J= ~ F ( X , Y , Y ' ) ~ ~

where F (x, y, yf)=(l + y2) / y"

for admissible functions y(x). Which of the following are extremals for J?

1. y(x) = A sin(x).

3. y(x) = A sinh(Ax + B).

98. The initial value problem

is equivalent to the Volterra integral equation

99. The integral equation

1

a ( ~ ) - I J ~ ~ ~ [ x ( ~ - t ) ] ~ ( t ) d t = f (x) -1

has

1. a unique solution for. $# 1 when f ( x ) = x .

2. no solution for I #l when f ( x ) = 1 . 3. no solution for 1=1 when f ( x ) = x .

4. infinite number of solutions for 1 = 1 when f ( x ) = 1 .

100. Let f ( u ) = u 3 - u - 1 .

1 . Starting with the initial guess u(O) = 1.5, the fixed point iterates of the equation 3 u = g ( u ) , where g ( u ) = u - 1 converge.

2. Starting with the initial guess u(O) = 1.5, the fixed point iterates of the equation

u = g ( u ) , where g ( u ) = converge.

3. If u* is a root of the equation f ( u ) = 0 and u* > 1, then u* is a stable fixed point

of the equation u = g ( u ) . 4. f ( u ) = 0 has a root between 1 and 2.

101. To compute the value of et in the interval [0,1], pick tl = 0, t2 = 0.5 and t3 = 1. Let p be

the qyadratic polynomial that interpolates e' , that is, (ti ) = e4, i = 1.2.3. Then

1. the polynomial p can be written in the form L, (t) + ell2& ( t ) + eL, ( t ) for some

choice of quadratic polynomials 4,4, L, . 2. if the polynomial p is written in the form 4 ( t ) + ell2 ( t ) + eL, ( t ), where 4,h and

L, are polynomials, then 4,4, and Lj are uniquely determined.

3. i fp is written in the form L, ( t ) + el"& ( t ) + eL, ( t ) , then one of L, . 4 , or L, must

be linear. 4. the polynomial p is uniquely determined.

102. The energy of a pendulum executing small oscillations in a Cartesian coordinate system is proportional to

1. square of the amplitude. 2. square of the frequency . 3. its mass. 4. inverse of the product of mass, frequency and amplitude.

103. Let {x, : n t 0) and X be random variables defined on a common probability space. Further

assume that Xn's are nonnegative and X takes values 0 and 1 with probability p and 1 - p respectively, where 0 I p I 1. Which of the following statements are necessarily true?

1. Ifp = 0 and X, converges to X in distribution, then Xn converges to Xin probability. 2. Ifp = 1 and X, converges to X in distribution, then Xn converges to X in probability. 3. If 0 < p < 1 and Xn converges to X in distribution, then Xn converges to X in probability. 4. If Xn converges to Xin probability, then X, converges to X almost surely.

P ( x = ~ ) &T it?

104. Let X be a binomial random variable with parameters 1 1, - . At which value(s) of k is ( 3 P ( X = k ) maximized?

105. d B - I E : Y,Z,N(O,I) ( w m ~ ) & r n & m % ; 7 ~ ~ 6 1 &B- f:B+R&

x>O X. irf f (x)=ld d 4Rx<O X. ~ f (x )=-1 rt W'&L@V 81 M B U, E W ,

u=lxl- f ( Y ) , V = I Y I . f (x). w=lzl. f (x) ?? - 81 at

105. A', Y , Z are independent random variables with N(0,l) (standard normal) distribution. Let

f:R+R be defined by f (x)=l, if x2O and f (x)=-1, if x<O. Let U, Y, Wbe defined

by u = I x I . f l y ) . v = I Y I . ~ ( x ) , w = I z I - ~ ( x ) . Then

1. U and V are independent each having a N(O, 1) distribution. 2. U and Ware independent each having N(0,l) distribution. 3. V and Ware independent each having N(0,l) distribution. 4. U, V and Ware independent random variables.

106. One ~hooses letters with replacement from the set {a, b, c, d ) as follows:

After one letter is observed at step n, in (n + l)th step another letter will be chosen from the other three with equal probability 113. Let Xn denote the letter chosen at the nth step. Which of the following are necessarily true for the Markov chain X,?

1. P [X, = a] converges to 1/3. 2. P [X, = b] converges to 114.

1 3. The average proportion of times c is observed converges to - .

4 4. The chain is @ irreducible.

107. Consider a Markov chain {X,, : n 2 0) on the state space (0, 1) with transition probability

matrix P. Which of the following statements are necessarily true?

1 0 I . when P=( ), lim [Xn =i] converges for i = 0,1, but the limits depend on the

0 1 n+m

initial distribution v.

2. When P = [O '1, lime [x,, = 1] exists and is positive for all choices of the initial 1 0

distribution v. 1/2 1 /2

3. When P = ( ) , lim < [x" = 01 does not exist for any choice of the initial n+m

distribution v. 1/2 1 / 2

4. When P= ( ), lim P [x,, =o] always exists, but may be 0 for some n+m

choice of the initial distribution v.

108. Let XI and 4 be two independent random variables with 4- binomial (m,i) and 4 - binomial

(n , t ) , m t n . Which of the following are always true?

2. X2 -XI + rn - binomial (m + n , i ) . 3. Conditional distribution of X2 given (XI+ X2) is hypergeometric. 4. Distribution of XI - X2 is symmetric about 0.

109. 9 ~ 3 & x , , x z ,... ,xn, ( n 2 3 ) pm m m ( 8 - 5 , 8 - 3 ) i+ f%m-r w * 81 * @ X , , ) @ x,, m n f m 4 f & ~ 4 M ~ 9 T ; T m ) f 3 P G a d 8 l a t f 3 9 z ) ' i + * i +

* m 8 ?

109. Let X I , X2,. . . ,Xn, (n 2 3) be a random sample from uniform ( 8 - 5, 8- 3). Let X ( l ) and X(,, denote the smallest and largest of the sample values. Then which of the following are always true?

1 . (X(,,, X(,)) is complete sufficient for 8. 2. X I + X 2 - 2 X 3 is an ancillary statistic. 3. X(,) + 3 is unbiased for 8. 4. X( l )+ 5 is consistent for 8.

(x + 1) log, 2 3. m 97EV

1 + log, 2 8 /

110. Let X - Poisson (9, the prior distribution of 6 be exponential with median 1 0 ~ 2 and the . loss function be squared error. Then which of the following are necessarily true?

1. The posterior distribution is Gamma. 2. The prior is a conjugate prior. (x + 1) loge2 X+l

3. The posterior mean is 4. The Bayes estimator of 0 is - 1 +log, 2 2

111. Let XI, X2 and X3 be independent with XI - N(l, I), X2 - N(-1, 1) and X3 -N(O, 1). Let

Then which of the following statements are always true?

1. ql has a central chi-square distribution. 2. 92 has a central chi-square distribution. 3. gl + 92 has a central cii-square distribution. 4. q, and q2 are independent.

112. The lifetimes (measured in hours) of a batch of lithium batteries are independent and identically distributed with density f (t) = ~ ~ t e - ~ , t 2 0, A > 0. One battery from this batch is put to test and is observed to fail afier 3 hours, while another battery is observed for 2 hours with no failure. Whlch of the following statements are necessarily correct?

1. The censoring involved here is of type 11. 2. The likelihood for R is proportional to R2(1+3 A)e-" ,

3. The maximum likelihood estimate of A is 2 . 1

4. The maximum likelihood estimate of -is 2. A

113. w a i w m ( ~ ) F i w m ~ ( x ) $ J ) e T O 7 ) & ~ G ? ~ + T F ? & W ? G V ? ? & V ~ m)*&~mw~z)'w~m3l%f39&~)'*-)'3nqrusc~ it?

113. In the linear regression fit involving independent and identically distributed pairs of observations of a response variable (Y) and an explanatory variable (3, indicate which of the following statements are necessarily correct?

1. The residuals of the regression of Yon X have the same variance. 2. The multiple R~ computed from the regression fit of Yon x and 2 cannot be

smaller than that computed from the regression fit of Yon X alone. 3. The adjusted R* computed from the regression fit of Yon x and 3 cannot be

smaller than that computed from the regression fit of Yon X alone. 4. The variance of the estimated regression coefficient of X computed from the

regression of Yon X and 2 is larger than that computed from the regression of Yon X alone.

114.. Consider a linear model E ( Y ) = x,O where Y is the vector of n observations and /3 is the - - vector of n parameters. The (i, jyhelement of the design matrix X is of the form:

1 + ij + i2 j 2 , 1 5 i, j I n . Then P is estimable for -

115. For which of the following set of values will a Balanced Incomplete Block design with parameters v, b, r, k, h not exist?

Block 1: ab ac be abc

116. Consider a Z3 factorial design laid out in 2 blocks, each of size 4, as follows

Block 1: Block 2: ab ac bc abc

Here the treatment combinations are written in Yates' notation. Then which of the following are always true?

1. Main effect A is confounded. 2. Main effect B is unconfounded. 3. Interactions AB, BC, AC are all unconfounded. 4. Interaction ABC is confounded.

117. Let ÿÿÿÿÿÿ denote the multivariate normal distribution with mean vector pand - - dispersion matrix C . Which of the following are correct?

1. The Bayes classifier between N ( ~ , Z , ) and N ( p 2 . Z , ) consists of comparing a

linear b c t i o n of the classifiable vector with a threshold. 2. The rnisclassification probability of the Bayes classifier between the populations

N (?I 1 x) and N ( ? ~ , Z ) ~ S more than that of the Bayes classifier between the

populations N ( z ~ , 2Z) and A.' (2&, 2 2 ) .

3. When ,q and p2are unknown, the maximum likelihood classifier between - - ( 1 and N ( ~ ~ , I ) , based on sampled data from the two populations,

depends on the data only through the respective sample means. 4. The Bayes classifier between N (1,0) and N (1 0.4) would classify the data point 2 as

belonging to the first population.

118. ~ 3 ) 5 0 ~ ~ ~ w ~ ~ ~ ~ ~ - ~ ~ ~ ~ ~ $ ~ &J ?& & 81 $7 50 &#%$ d f?id (I) ''m W */&!? dfi.) i?? " (ii) "CW 31~9.

*T*dm$?y7 I' $ w f 3 9 2 ~ 2 m & & i ; ) & g 1

Lung Cancer

118. To study the relationship between smoking and lung cancer, 50 adult individuals are randomly selected fiom the population. For these 50 individuals, the responses to the questions (i) "Do you smoke?" (ii) "Do you have lung cancer?" lead to the following 2 x 2 table.

Lung Cancer

- Then which of the following statements are correct at 5% level of significance? (Note: 95th percentile of the chi-square distribution with 1 d.f. is 3.841).

1. Lung cancer and smoking habits are significantly related. 2. Lung cancer and smoking habits are independent. 3. The proportions of individuals with lung cancer are significantly different in the

smoking and non-smoking populations. 4. The proportions of smokers in the populations of individuals with or without lung

cancer are significantly different.

119. Let X be a discrete random variable with probability mass function p ( x ) , x E (-1,0,1,2,3) . The hypotheses to be tested are

H, : ( x ) = ( x ) versus H, : p ( x ) = p, ( x ) ,

where po ( x ) and p, ( x ) are as given below.

Then

1. (1) is the critical region of an MP test at level a = 0.05. 2. (-1, 0) is tlie critical region of an MP test at level a = 0.03. 3. (-1, 1) is the critical region of an MP test at level a = 0.06. 4. {-I, 0, I ) is the critical region of an MP test at level a = 0.08.

120. A line?, programming problem

max ~ ' 5 , subjecttoA5 54 , 52 (2, x

attains the optimum value at two distinct feasible solutions. Which of the following :wst be true?

1. There must be infinitely many feasible solutions where the optimal value is attained. 2. All feasible solutions must be optimal solutions. 3. The dual problem has unbounded feasible region.

I. Rank of (5) = Rank of A .

56

Trii WIROUGH WORK

MATHEMATICAL SCIFNCFS Roll Number BI

m ANSWER SHE -

d Code Medium Subject Code

ANSWER COLUMNS

WI I"..., I ..... 2

00

00 00

0 . Q . o e e o 0 8 0 8 8 0 9 0 o e e o 0 . 0 8 0 9 . e 0 0 8 8 e o e 0 0 e . o 0 8 . 0 o e ! o o 0 a . 0 d @ Q O 0 8 8 0 0 . 0 0 0 . 0 0 0 0 0 0 . 0 0 0 . 0 6 0 . . Q . 8 0 0 . 9 . Q 8 Q O O O * . O @ o o o e 0 . 0 @ 8 8 @ 0 @ @ O O 0 8 @ 0 a m 0 0 0 0 . 0 ( b o o 0 0 6 . 0