Filename: December 2014 question & solution.pdf

1. What is the 94th term of the following sequence?1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,...1. 8 2. 93. 10 4. 11

2. Which of the following numbers is a perfectsquare?1. 1022121 2. 20421223. 3063126 4. 4083128

3. The equation 2 33 1 0,m n where m & n areintegers, has1. no solution2. exactly one solution3. exactly two solutions4. infinitely many solutions

4. The following graphs depict variation in the valueof Dollar and Euro in terms of the Rupee over sixmonths.

J a n F e b M a r A p r M a y

7 97 77 5

5 95 75 5

Which of the folllowing statements is true?1. Values of Dollar and Euro rose steadily from January to June2. Values of Dollar and Euro rose by equal rate between January to March3. The rise in the value of Dollar from April to May is three times the fall in Euro during the same period4. Values of Dollar and Euro rose euqally between May and June

5. What is the maximum number of whole laddooshaving diameter of 6 cm that can be packed in abox whose inner dimensions are 24 18 17 3cm ?1. 24 2. 303. 33 4. 36

6. Which of the following figures best shows that y isinversely proportional to x?

1.Y

X(0 , 0 )

2.

(0 , 0)Y

X

3.(0, 0) X

Y

4.

(0, 0)

Y

X

7. What is the next term in the following sequence?7, 11, 13, 17, 19, 23, 29,...1. 37 2. 353. 31 4. 33

8. What is the area of the triangle bounded by thelines 2 , 2y x y x and 6y ?1. 36 2. 183. 12 4. 24

9. Three volumes of a Hindi book, identical in shapeand size are next each other in a shelf, all upright,so that their spines are visible, left to right: I, II andIII. A worm starts eating from the outside frontcover of volume, I, and eats its way horizontally tothe outside back cover of volume III. What is thedistance travelled by the worm, if each volume is6 cm thick?1. 6cm 2. 12cm3. 18cm 4. a little more than 18cm

10. A cubical piece of wood was filed to make it intothe largest possible sphere. What fraction of theoriginal volume was removed?1. More than 3/42. 1/23. Slightly less than 1/24. Slightly more than 1/2

11. Two platforms are separated horizontally bydistance A and vertically by distance B . They areto be connected by a staircase having identicalsteps. If the minimum permissible step lenght is a,and the maximum permissible step height is a , andthe maximum permissible step height is b , thenumber of steps the staircase can have is1. /B b2. /A a

CSIR 2014 DECEMBER QUESTION BVHNGGJGJGHJKHKJHKGFGFG

3. /B b and A / a4. B / b and A / a

12. Ajay, Bunty, Chinu and Deb were agent, baker,compounder and designer, but not necessarily inthat order. Deb told the baker that Chinu is on hisway, Ajay is sitting across the designer and next tothe compounder. The designer didnt say anything.What is each persons occupation?1. Ajay-compunder; Bunty-designer; Chinu-

baker; Deb-agent2. Ajay-compounder; Bunty-baker;

Chinu-agent; Deb-designer3. Ajay-baker; Bunty-agent; Chinu-designer;

Deb-compounder4. Ajay-baker;Bunty-designer; Chinu-agent;

Deb-compounder13. Every month the price of a particular commodity

falls in this order: 1024, 640, 400, 250,...What is the next value?1. 156.25 2. Approximately 393. 64 4. 40

14. We define a function f N = sum of digits of N ,expressed as decimal number e.g. 137f = 1 + 3+ 7 = 11. Evaluate 7 5 62 3 5f .1. 10 2. 183. 28 4. 11

15. A certain day, which is x days before 17th August,is such that 50 days prior to that day it was 4 x dayssince March 30th of the same year. What is x?1. 18 2. 303. 22 4. 16

16. A mouse has to go from point A and B withoutretracing any part of the path, and never movingbackwards. What is the total number of distinctpaths that the mouse take to go from A to B?

A B

1. 11 2. 483. 72 4. 24

17. The sum of the first n natural number with one ofthem missed is 42. What is the number that wasmissed?1. 1 2. 23. 3 4. 4

18. A 2.2 m wide rectangular steel plate is corrugatedas shown in the diagram. Each corrugated is asemi-circle in cross section having a diameter of7 cm. What will be the width of steel sheet after it

is corrugated?

2.2 m

1. 1.4 m 2. 1.6 m3. 0.7 m 4. 1.1 m

19. If N, E and T are distinct positive integers such thatN E T = 2013, then which of the following isthe maximum possible sum of N, E and T?1. 39 2. 20153. 675 4. 671

20. The areas of the inner circle and the shaded ringare equal. The radii 1r and 2r are related by

1

2

1. 1 2r r 2. 1 2 2r r3. 1 2 3r r 4. 1 22r r

PART BUnit -121. Let n be an open set and :f be

a differentiable function such that 0Df x for all x .Then which of the following is true?1. f must be a constant function2. f must be constant on connected

components of 3. 0f x or 1 for x 4. The range of the function f is a subset

of 22. Let : 1na n be a sequence of real numbers such

that 1 nn a is convergent. and 1 | |nn a

is

divergent. Let R be the radius of convergence of

the power series 1 .n

nna x

Then we can

conclude that

CSIR 2014 DECEMBER QUESTION B

1. 0 1R 2. R = 13. 1 R 4. R =

23. Let { : 0 1},G xx f x be the graphof a real valued differentiable function f . Assumethat .1, 0 G Suppose that the tangent vector to G at any point is perpendicular tothe radius vector at that point. Then which of thefollowing is true?1. G is the arc of an ellipse2. G is the arc of a circle3. G is a line segment4. G is the arc of a parabola

24. Let A, B be n n matrices such that2 2BA B I BA where I is the n n identity

matrix. Which of the following is always true?1. A is nonsingular 2. B is nonsingular3. A + B is nonsingular 4. AB is nonsingular

25. Suppose p is a polynomial with real coefficients.Then which of the following statements isnecessarialy true?1. There is no root of the derivative 'p between two real roots of the polynomial p2. There is exactly one root of the derivative

'p between any two real roots of p3. There is exactly one root of the derivative

'p between any two consecutive roots of p4. There is at least one root of the derivative

'p between any two consecutive roots of p26. Which of the following matrix

4 8 4?3 6 1

2 4 0

1. 1 2 00 0 1 2. 1 1 00 0 13. 0 1 00 0 1 4. 1 1 00 1 0

27. Let k be a positive interger. The radius of

convergence of the series 0!!k n

nn zkn

is1. k 2. kk3. kk 4.

28. The determinant of the n n permutation matrix11

11

1. 1 n 2. 21n

3. 1 4. 129. Let nb and nc sequences of real numbers.

Then a necessary and sufficient condition for thesequence of polynomials 2n n nf b x c xx toconverge uniformly to 0 on the real line is1. lim 0n nb and lim 0n nc 2. 1| |nn b and 1|c |nn 3. There exists a positive integer N such

that 0nb and 0nc for all n N4. lim 0n nc

30. The determinant2

2

2

1 1 11 1 11 1 1

x x xy y yz z z

is equal to

1. z y y xz x 2. x y y zx z 3. 2 2 2x y y z z x 4. 2 2 2 2 2 2x y y z z x

31. Let p be a 2 2 complex matrix such that *P Pis the identity matrix, where *P is the conjugatetranspose of P . Then the eigenvalues of P are1. real2. complex conjugates of each other3. reciprocals of each other4. of modulus 1

32. Which of the following matrices is not diagonaliz-able over ?


1.1 1 00 2 00 0 1

2.

1 1 00 2 10 0 3

3.1 1 00 1 00 0 2

4.

1 0 10 2 00 0 3

UNIT - 233. Find the degree of the field extension

842, 2, 2 over .1. 4 2. 83. 14 4. 12

34. Let 0 1 .. nnp a a z a zz and 21 2 ... nnq b z b z b zz be complex polynomi-

als. .If 0 1,a b are non-zero complex numbers then

the residue of p zq z at 0 is equal to

1. 01

ab 2.

10

ba

3. 11

ab 4.

01

aa

35. In the group of all invertible 4 4 matrices withentries in the field of 3 elements any 3- Sylowsubgroup has cardinality:1. 3 2. 813. 243 4. 729

36. The number of conjugacy classes in the permuta-tion group 6S is:1. 12 2. 113. 10 4. 6

37. The number of surjective maps from a set of 4elements to a set of 3 elements is1. 36 2. 643. 69 4. 81

38. Let {1, 2,...} be the set of posititve integersLet. 1 subspace topology on induced fromthe usual topology of , 2 : order topology on

, . .,i e the topology with base { : 1 } :x x b b ,{ : 0 } : ,x x b a b

3 : discrete topology.Then1. 1 3 and 1 2 2. 1 2 and 1 3 3. 1 3 and 2 3 4. 1 2 3

39. Let G be the Galois group of a filed with nineelements over its subfield with three elements.Then the number of orbits for the action of G onthe field with nine elements is1. 3 2. 53. 6 4. 9

40. Let 0 nna z be a convergent power series such

that 1lim 0.nnn

a Ra

Let p be a polyno-mial of degree d . Then the radius of conver-gence of the power series 0 nnn p a zn equals1. R 2. d3. Rd 4. R + d

UNIT -341. For , consider the boundary value problem

22 2 0, 1,22

01 2

d y dyx x y x Pdxaxy y

Which of the following statemenet is true?1. There exists a 0 such that P

has a nontrivial solution for any 0 .2. { : P has a nontrivial solution}

is a dense subset of .3. For any continuous function : 1, 2f

with 0f x for some ,1, 2x there exists a solution u of p for some such that

2

10fu

4. There exists a such that P has


two linearly independent solutions.42. Let :y be differentiable and satisfy the

ODE:

,00 1

dy f xydxy y

where :f be a Lipschitz continuousfunction.1. 0y x if and only if 0, 1x 2. y is bounded3. y is strictly increasing4. dydx is unbounded

43. Let :P be a polynomial of the form 20 1 2 ,P a a x a xx with 0 1 2, ,a a a and

2 0.a Let 11 0 1 0 12E P dx P Px 12 0 12E P dx px

If | |x is the absolute value of ,x then1. 1 2| | | |E E 2. 2 1| | | |E E3. 2 1| | | |E E 4. 2 1| | 2| |E E

44. The Charpits equation for the PDE2 2 0,up q x y ,u up qx y

are given by

1. 3 2 2 21 1 2 2dx dy dup qp p u q

2 2dp dqpu q

2. 2 22 2 2 2dx dy dupu q p u q 3 21 1

dp dqp qp

3. 2 2 0dx dy du dp dq

x yup q

4. 2 22 2dx dy du dp dqq pu x y p qp

45. Let : 0,y be twice continuouslydifferentiable and satisfy 30 / 6xy y ds xx s sx

Then

1. 3016xy s sin dsx sx

2. 3016xy s cos dsx sx

3. 0xy s sin dsx sx 4. 0xy s cos dsx sx

46. Consider a body of unit mass falling freely fromrest under gravity with velocity v . If the airresistance retards the acceleration by cv where cis a constant, then

1. 1 ctgv ec 2. 1 ctgv ec

3. 1 ctgv ec 3. 1 ctgv ec

47. Consider the functional 212 0 ' ,1J y y dxy x 10y

where 2 ([0, 1]).y C If y extremizes J then1. 211 2y xx

2. 11 2y xx

3. 11 2y xx

4. 211 2y xx

48. Let , i xu e v tx t with 10v be asolution to

23 .

u ut x Then

1. 2, i x tu ex t 2. 2, i x tu ex t 3. 2, i x tu ex t


4. 3, i x tu ex t UNIT - 449. Suppose 1 2,X ,...XnX is a ramdom sample from

, 0.0,U Let ...1 2X X X n be the order statistics. Consider the two unbiasedestimators for 1: 2T X and

21 .nnT Xn Then

21

arlimnV TVar T

1. 0 2. 13. 4. 12

50. Suppose that 1 2,X X and 3X are independent andidentically distributed random variables, each havinga Bernoulli distribution with parameter 1/2.Consider the 2 2matrix 1

2 3

0 .XA X X

Then , P (A is invertible) equals1. 0 2. 13. 1/4 4. 3/4

51. Consider the linear programming problemMinimize 2 5z x y Subject to 3 4 5, 0, 0.x y x y Which of the following is correct?1. Set of feasible solutions is empty.2. Set of feasible solutions is non-empty but

there is no optimal solution

3. Optimal value is attained at 50, 4

4. Optimal value is attained at 5 ,03

52. How many distinct samples of size n can be drawnwith replacement from the population 1 2,u ,..., nu uof n units?

1. nn 2.2 1nn

3.2 1

1nn 4. 1

53. For testing the effectiveness of four teachingtechniques, five teachers of a college were in-volved. A class of 120 students was divided into 5

groups of 24 each at random; one group wasassigned to each of the five teachers. Each teacherfurther divided his/her group into four equalssubgroups at random and used one technique persubgroup. All of them used the same coursematerial .After all the classes were over, a commonexamination was conducted and the marks werenoted Suppose we want to test whether all the fourteaching techniques are equally effective. What isthe degree of freedom associated with the residualsum of squares?1. 60 2. 1003. 119 4. 460

54. Suppose 1 2, ,...X X are random variable on acommon probability space with 2 .,n n nX N Then , nX converges in probability to 2 if and only if1. 0n and 2 2n 2. 2n and 2 0n 3. 0n and 2n converges4. 2 0n and n converges

55. Let 1 2, ,...X X be independent and identicallydistributed random variable with 0iE X and

1iVar X for all .i Let 1 ...n nS X X .Let x denote the cumulative distribution function of

a standard normal random variable . Then for any0x . limn nP nx S nx equals

1. 2 1x 2. x3. 1 4. 1 2x

56. Let 1 nX ,...,X be a random sample of size n froma p-variate Normal distribution with mean andpositive definite covariance matrix . Choose thecorrect statement1. 11 1'X X has chi-square

distribution with 1 d.f .2. XX ' has Wishart distribution with p d.f.3. 1n ii iX 'X has Wishart

distribution with n d.f.4. 1 2X X and 1 2X X are independently

distributed.


57. Suppose 01X N , and the conditional distribu-tion of Y given X x is 1N ,x, for 0 1. When we regress Y on X, the coefficient ofdistribution 2R is1. 2 2.

3. 21 4.

221

58. Let X and Y be integer-valued, bounded randomvariables. Then which of the following statement isFALSE?1. ( | ) ( )yE E X Y y P Y yX 2. ( | ) ( )yV V X Y y P Y yX 3. ( | ) ( )yP P X x Y y P Y yX x 4. ( | ) ( )yE yP X Y y P Y yXY

59. Suppose X Poisson , 0. Let the priordistribution of be .0, 1U If X = 0 is observed,then the posterior probability of the set 10 2 is1. 0.5 2. 1

3. 1e ee 4.

1e

60. Five persons A, B, C, D and E are seated atrandom on eight numbered chairs which arearranged in a circle. What is the probability that Aand B are separated by at least 2 chairs?1. 3/7 2. 1/23. 4/7 4. 1/4

PART CUNIT -1

61. Let f be a continuously differetiable function on . Suppose that lim 'xL f fx x exists . If 0 ,L then which of the following statements is/are correct?1. If lim 'x f x exists, then it is 02. If limx f x exists then it is L3. If lim 'x f x exists then

lim 0x f x

4. If limx f x exists then lim 'x f Lx

62. For a non-empty subset S and a point x in aconnected metric space ,,X dlet inf{ ( , ) : }.,d d x y y Sx S Which of the

following statements is/are correct ?1. If S is closed and 0,d x S then x is

not an accumulation point of S2. If S is open and 0.,d x S then x is not

an accumulation point of S3. If S is closed and 0.,d x S then S does

not contain x4. If S is open and 0,d x S then x S

63. Let ka be an unbounded strictly increasingsequence of positive real numbers and

1 1k kk kx / a .a a Which of the followingstatements is/are correct1. For all 1 mn kk m

n

an m , x a 2. There exists n m such that

11 2n kk m x

3. 1 kk x converges to a finite limit4. 1 kk x diverges to .

64. Suppose that P is a monic polynomial of degree n inone variable with real coefficients and K is a realnuumber. Then which of the following statement

is/are necessarily true?1. If n is even and 0.K then there exists

0x such that 00 xP K ex 2. If n is odd and 0,K then there exists

0x such that 00 xp ex 3. For any natural number n and

0 1,K there exists 0x suchthat 00 xp Kex

4. If n is odd and ,K then there exists


0x such that 00 xp Kex 65. Let E be a subset of . Then the characteristic

function :E is continuous if and only if1. E is closed2. E is open3. E is both open and closed4. E is neither open nor closed

66. Let A be a real n n orthogonal matrix, that is,,t t nA A AA l the n n identity matrix. Which

of the following statemenets are necessarily true ?

1. ,, , nx yAx Ay x y 2. All eigenvalues of A are either + 1 or 13. The rows of A form an orthogonal basis of

n4. A is diagonalizable over

67. Let X be a metric space and :f X be acontinuous function Let {( , ( ) :G x f x x X}be the graph of f . Then1. G is homeomorphic to X2. G is homeomorphic to 3. G is homeomorphic to X 4. G is homeomorphic to X

68. Consider the normed linear spaces11 ( [0, 1], )X C and ( [0, 1], ),X C

where [0, 1]C denotes the vector space of allcontinuous real valued functions on [ 0, 0] and

101 | ( ) | ,f t d tf

sup{| ||, [1, 0]}.f tf t Let 1U and U bethe open unit balls in 1X and X respectively..Then1. U is a subset of 1U2. 1U is a subset of U3. U is equal to 1U4. Neither U is a subset of 1U nor 1U is

subset of U69. Let f be a monotonically increasing function from

[0, 1] onto [0, 1]. Which of the followingstatements is/are true?1. f must be continuous at all but finitely

many points in [0, 1]2. f must be continuous at all but countably

many points in [0, 1]3. f must be Riemann integrable4. f must be Lebesgue integrable

70. Let A be a subset of . Which of the followingproperties imply that A is compact?1. Every continuous function f fromA to

is bounded2. Every sequence { }nx in A has a conver-

gent subsequence convering to a point in A3. There exists a continuous function from A

onto [0, 1]4. There is no one- one and continuous

function from A onto (0, 1)71. Let A be 5 5 matrix and let B be obtained by

changing one element of A. Let r and s be theranks of A and B respectively. Which of the follow-ing statements is/are correct?1. 1s r 2. 1r s 3. 1s r 4. s r

72. The matrix5 9 81 8 29 1 0

A

satisfies:

1. A is invertible and the inverse has allinteger entries.

2. det (A) is odd3. det (A) is divisible by 134. det (A) has at least two prime divisors

73. Let f be a non-zero symmetric bilinear form on3 . Suppose that there exist linear transformations

3: , 1,2iT i such that for all 3 1 2, , .,f T T Then:

1. rank f = 12. dim 3{ : 0,f for all

3} 2 3. f is a positive semi-definite or negative

semi-definite4. { : ( , ) 0}f is a linear subspace

of dimension 274. Let A be a3 4 and b be a 3 1 matrix with

integer entries. Suppose that the system Ax b hasa complex solution. Then


1. Ax b has an integer solution2. Ax b has a rational solution3. The set of real solutions to 0Ax

has a basis consisting of rational solutions4. If 0b then A has positive rank

75. Which of the following matrices have Jordancanonical form equal to

0 1 0?0 0 0

0 0 0

1.0 0 10 0 00 0 0

2.

0 0 10 0 10 0 0

3.0 1 10 0 00 0 0

4.

0 1 10 0 10 0 0

76. Let A be a 4 7 real matrix and B be a 7 4 realmatrix such that 4,AB I where 4I is the 4 4identity matrix. Which of the following is/arealways true?1. rank A = 42. rank B = 73. nullity B = 04. BA = 7,I where 7I is the 7 7 identity

matrix

77. For arbitrary subspaces ,U V and W of a finitedimensional vector sapce, which of the followinghold:1. U U V U WV W 2. U U V U WV W 3. WU V U W V W 4. WU V U W V W

78. Let nM K denote the sapce of all n n matriceswith entries in a field K . Fix a non-singular matrix

ij nA MA K and consider the linearmap :T M MK Kn n given by:

.T AXX

Then1. trace 1nn AT i ii 2. trace 11 nn j ijin AT 3. rank of T is 2n4. T is non - singular

UNIT -279. Which of the following is/are true?

1. Given any positive integer n , there exists afield extension of of degree n .

2 Given a positive integer n , there existsfields F and K such that F K and K isGalois over F with [ : ]K F n .

3. Let K be a Galois extension of with[ : ] 4K . Then there is a field L suchthat . [ : ] 2K L L and L is aGalois extension of .

4. There is an algebra extension K of suchthat [ : ]K is not finite.

80. Let x be the polynomial ring over in onevariable. Let I x be an ideal . Then1. I is a maximal ideal if and only if I is a

nonzero prime ideal2. I is a maximal ideal if and only if the

quotient ring / Ix is isomorphic to 3. I is a maximal ideal if and only if

,I f x where f x is a non-constantirreducible polynomial over

4. I is a maximal ideal if and only if thereexists a nonconstant ploynomial f Ix of degree 2

81. Which of the following are eigenvalues of thematrix0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1 ?1 0 0 0 0 00 0 0 0 0 00 0 1 0 0 0

1. +1 2. -13. +i 4. -i

82. Let G be a nonabelian group. Then its order can be


1. 25 3. 553. 125 4. 35

83. Let 2 2 2{ : 1},X a ba b be the unitcircle inside 2. Let :f X be a continuousfunction. Then1. Image f is connected2. Image f is compact3. The given information is not sufficient to

determine whether Image f is bounded4. f is not injective

84. Let 0 nnnf a zz be an entire function and let rbe positive real number. Then1. 2 2 2| | sup | || |0 na r f zn z rn 2. 2 22sup | | | |0| |

nf a rz nnz r

3. 12 2 22| | | |0 02n ia r f dren n 4. 222| | 01sup | | | |2 iz r f f dz re

85. Let ,x yA y x

where ,x y such that

2 2 1.x y Then we must have

1. For any cos sin1 , sin cosnn A

where cos , sin/ /x yn n

2. 0tr A 3. 1tA A4. A is similar to a diagonal matrix over

86. Let G be a group of order 45. Then1. G has an element of order 92. G has a subgroup of order 93. G has a normal subgroup of order 94. G has a normal subgroup of order 5

87. Let f be an entire function on and let be abounded open subset of Let .Re Im |S f f zz z Which of the following statements is/arenecessarily correct?

1. S is an open set in 2. S is a closed set in 3. S is an open set of 4. S is a discrete set in

88. Let 3 23 2 .u x xy xx iy For which of thefollowing functions ,v is u iv a holomorphicfunction on ?

1. 3 23 2v y x y yx iy 2. 2 33 2v x y y yx iy 3. 3 23 2v x xy xx iy 4. 0v x iy

89. Let x denote the vector sapce of all realpolynomials .Let :D x x denote themap , .dfDf fdx Then1. D is one-one2. D is onto3. There exists :E x x so that

, ,D f fE f 4. There exists :E x x so that

E(D( f )) f , f .90. Let f be an entire function on . Let

.g f zz Which of the following statementsis/are correct1. if f z for all z then f = g2. if f z for all

{ |Im 0} { |Im },z z z z z a for some a > 0, then f fz ia z ia for all z

3. if f z for all{ |Im 0} { |Im },z z z z z a

for some a > 0, then 2f fz ia z for all z .

4. if f z for all


{ |Im 0} { |Im }z z z z z a for some a > 0, then f fz ia zfor all .z

UNIT-391. Let ,P Q be continuous real valued functions

defined on 1, 1 and : , 1, 21, 1ju i besolutions of the ODE:

22 0 , 1, 1d u duP Q u xx xdxdx satisfying 1 2, u 0u and 1 2 00 0u u Let w denote the Wronskian of 1u and 2u then1. 1u and 2u are linearly independent2. 1u and 2u are linearly dependent3. 0w x for all 1, 1x 4. 0w x for some 1, 1x

92. The system of ODE

2

2

,1x,1

,0 0 ,

dx y txdtdy txdtx y a d

has a solution:1. only if (a, b) = (0, 0)2. for any (a, b) 3. such that 2 2 2 2x y a bt t for

all t 4. such that 2 2x yt t as t if

0a and 0b 93. Consider the Cauchy problem of finding

,u u x t such that0u uut x

for ,x t > 0

0, 0u u xxx Which choice (s) of the following functions for 0uyiled a 1C solution ,u x t for all x and t > 0.

1. 0 21

1u x x 2. 0u xx 3. 20 1u xx 4. 0 1 2u xx

94. Let :y be a solution of the ODE

22 ,

00 0

xd y y e xdx dyy dx

then1. y attains its minimum on 2. y is bounded on 3. 1lim 4

xx e y x

4. 1lim 4xx e y x

95. Let y t satisfy the differential equation ' ; 10y y y .Then the backward Euler

method, for 1n and h > 01

0; 1n n ny y y yh yields

1. a first order approximation to nhe2. a polynomial approximation to nhe3. a rational function approximation to nhe4. a Chebyshev polynomial approximation to

nhe96. Let 2 0, 1u c satisfy for some 0 and

0a 10 | |2u x s u ds ax bsx

Then u also satisfies

1.22 0

d u udx

2.22 0

d u udx


3. 102x sdu u ds asdx x s

4. 102x sdu u ds asdx x s

97. Let , tu u x be the solution of the Cauchy

problem2

1u ut x , , 0x t

2, 0u xx x . Then1. ,u x t exists for all x and 0t .2. | |,u x t as *t t for some * 0t

and 0x 3. 0,u x t for all x and for all

1/ 4.t 4. 0,u x t for some x and

0 1/ 4.t 98. Let :f be a smooth function with non-

vanishing derivative. The Newtons method forfinding a root of 0f x is the same as1. fixed point iteration for the map

g x fx x 'f x2. Forward Euler method with unit step length

for the differential equation 0'f ydy

dx f y

3. fixed point iteration for g x fx x 4. fixed point iteration for g x fx x

99. Let ,u x t satisfy for , 0x t 2 22 22 0.u u u

tt x A solution of the form ixu e v t with 00v and ' 10v 1. is necessarily bounded2. satisfies | |, tu ex t 3. is necessarily unbounded4. is oscillatory in x

100. Consider a particle moving with coordinates( ( ), ( ))x t y t on a smooth curve ( , ) 0x y . If the

particle moves from ( (0), (0)).x y to ( ( ), ( )x y for 0 such that its kinetic energy is minimizedthen

1. x yx y

2. 2 2 2 20 0x y x y 3. 0x yx y 4. 2 20x x

101. Which of the following approximations forestimating the derivative of a smooth function f ata point x is of order 2(i.e the error term is 2O h )1. ' f fx h xf x h

2. ' 2f fx h x hf x h

3. 3 4 2' 2f f fx x h x hf x h

4. 3 4 2' 2f f fx x h x hf x h

102. Let 2 0,y C satisfying 00y y and 20 1y dxx . extremize the funtional

20 : ' .' x dyJ dx yy y dx Then1. 2 siny xx

2. 2 siny xx

3. 2 cosy xx

4. 2 cosy xx UNIT -4103. Let 1 2, ,..., , 3,nX X X n be a random sample

from , 1N population where is unknown.Define 11 n iinX Xn . Which of the following are


necessarily true?1. 1 2 3 03 2 , nCov X X X X 2. Cramer-Rao lower bound for unbiased

estimators of is 1n

3. nVar X Var 1 2 32 3 ...

12

nX X X nXn n

4. nX is a function of any sufficient statistic

for 104. Consider a Markov cahin with state sapce

(1,2,...,100). Suppose states 2i and 2jcommunicate with each other and states 2i 1and 2j1 communicate with each other for every i, j = 1, 2,...,50 Further suppose that

323,3 4,40, 0p p and 72,5 0.p Then1. The Markov chain is irreducible2. The Markov chain is aperiodic3. State 8 is recurrent4. State 9 is recurrent.

105. Let 1 2, ,..., nX X X be independent and identicallydistributed obsevations from the distribution withdenisity ,| xf e xx where and

is unknown. Let 111 2 niT X and 2 12T Xwhere 1X is the smallest order statistic . To test

0 : 0H versus 1: 0H at level , where0 1. consider the two tests A and B givenbelow.A : Reject 0H if 1 1T C where 1C is such

that 1 1P Y C with 21 2nY B : Reject 0H if 2 2T C where 2C is such

that 2 2P Y C with 22 2nY Then which of the statements are valid?1. Both A and B are level tests2. A is the uniformly most powerful level

test3. B is the uniformly most powerful level

test4. B is more powerful than A at any 0

106. Let 1 2, ,..., nX X X be independent random variablewith 0kE X and Var .k kX Let

1 .n kknS X Then , as n1. 3/2 0n

Sn in probability

2.3/2 0nSn

in distribution

3. 5/2 0n nS Xn in distribution

4. 5/2 0n nS Xn probability

107. Let t be a characteristic function of somerandom variable . Then which of the following arealso characteristic function?1. 2f t t for all .t 2. 2| |f t t for all t .3. f t t for all t .4. 1tf t t for all t .

108. Suppose 1 2 3, ,X X X and 4X are independent andidentically distributed random variable havingdensity function f Then

1. 4 31 2 1max , 6P X XX X 2. 4 31 2 1max , 8P X XX X 3. 4 3 1 2 1max , 12P X X X X 4. 4 3 1 2 1max , 6P X X X X

109. 1 2, , ,...N A A are independent real valued randomvariables such that , 0,1,2,...1 kP p kpN k where 0 1,p

and : 1,2,....iA i is a sequence of independentand identically distributed bounded random variables

Let

1

0 0, 1,2,...k jj

if N wX w A if N k kw

Which of the following are necessarily correct?1. X is a bounded random variable2. Moment generating function Xm of X is


1 ,1x A

pm tt pm t where Am

is the moment generating function of 1.A3. Characteristic function X of X

1 , ,1X A

p t Rt p t where A

is the characteristic function of 1A4. X is symmetric about 0.

110. Suppose X is random variable such that 20, 2E EX X and 4 4E X . Then

1. 3 0E X 2. 1/ 20P X 3. 0,2X N4. X is bounded with probability 1.

111. Consider a 3-variate population with covariance

matrix

2 2

2 2 2

2 2

0

0

where 2 0, 0. Then which of the followingstatements are true ?

1. 12 2. The proportion of the total population

variance explained by the first principal

component is 1 23

3. The second principal component isuncorrelated with the first and the thirdprincipal component

4. The proportion of the total populationvariance explained by the first two principla

component is 2 23 112. A and B are two methods to determine the levels

of mercury in fish. In a study to compare A and Bamount of mercury was measured using bothmethods on 12n fish. Let 1 1 ,..., ,, n nX YX Ybe those measurements, with 'iX s standing formethod A and 'iX s for method B. It should be

noted that the size of error in measurement candepend on the amount of mercury so theobservations 1 2 ,...,,Y n nX YX many not beidentically distributed to test0 :H There is no difference between method A and B

versus1:H Method B typically given larger reading than

method AWhich of the following test statistic are

appropriate?1. Number of pairs ,i jX Y with

, 1 ; 1j i i n j nY X 2. Sum of the ranks of the Y observations in

the combined sample3. Number of the pairs ,i jX X with

, 1i i i nY X 4. Y X

113. Let Y follow multivariate normal distribution 0,nN I and let A and B be n n symmetric,

idempotent matrices . Then which of the followingstatements are true?1. If 0,AB then 'Y AY and 'Y BY are

independently distributed.2. If 'Y YA B has chi-square distribution

then 'Y AY and 'Y BY are independentlydistributed

3. 'Y YA B has chi-square distribution4. 'Y AY and 'Y BY have chi-square

distribution114. For a Markov chain with finite state space, the

number of stationary distributions can be1. 0 2. 13. 2 4.

115. Let X t = number of customers at time t in thesystem in an M/M/I queueing model with arrivalrate 0 and service rate 0 . Let

lim ( ), 0,1,2...k t P X k kt whenever itexists. Which of the following are true?1. { }X t is a birth and death process with

birth rates , 0, 1, 2,...k k and deathrates , 1, 2,...k k

2. { }X t is a birth adn death process with


birth rates 1, 0, 1, 2,...k k and death

rates 1, 1, 2,..k k

3. Limiting distribution k exists if and onlyif , and is the geometric distribution

with parameter .

4. If an arriving customer finds exactly onecustomer then his total waiting time in thesystem has an exponential distribution withparameter 2

116. Consider the linear model1 12 23 3

22

yyy

where , , are unknown parameters and1 2 3, , are uncorrelated random errors with

meas 0 and constant variance. Then which of thefollowing statemenets are true?1. , and are estimable2. as estimable3. 2 2 is estimable4. is estimable

117. Suppose a sample of size n is drawn using simplerandom sampling without replacement from a finitepopulation of n unit where N > n and denote thesample mesn of the study variables correspondingto the selected units by y . Now suppose we knowone variate value 1y corresponding to one unit anddrawn a simple random sample of size n withoutreplacement from the remaining (N - 1) units anddenote the sample mean of the study variablescorresponding to the selected units bu 0y .Define

11 2 0 1 1, ,1t Ny t y y V Var tN and 22V Var t . Which of the follwoing are

necessarily true?1. 1t is unbiased for population total2. 2t is unbiased for population total

3.2

21 1

N nV N n N where 2 population

variance4. 2 1V V for all ,n N

118. Consider the linear model 2,E X Cov IY Y where X is a matrix of

size n p having rank r p . Then which of thefollowing statements are necessarily true?1. The set of estimable linear functions forms

a vector space of dimension r2. If 0'E c Y for some nonzero vector c,

then there is a function 'l which is notestimable

3. If all linear function 'l are estimable,then r p .

4. The set of functions cY wth 0'E c Y form a vector sapce of dimension r .

119. Consider a 32 factorial experiment with threefactors A, B and C. Suppose eight treatment areassigned in two blocks of each of the four repli-cates in the following way.

bcacab

(1)abcabc

(1)bacabc

acbcab

(1)cababc

abacbc

(1)abbcac

bacabc

Replicate1 Replicate 2 Replicate 3 Replicate 4

Which of the following are necessarily treu?1. This is an example of complete

confounding2. AB is confounded in Replicate 13. AC is confounded in Replicate 24. ABC is confounded in Replicate 4

120. Let 1 2 nX ,X ,...,X be independent and identicallydistributed Bermoulli , where 0 1 andn 1. let the prior density of be proportional to

1 , 0 1.1

Definen ii 1S X . Then

valid statements among the following are:1. The posterior mean of does not exist2. The posterior meas of exists;3. The posterior mea of exists and it is

larger than the maximum likelihoodestimator for all values of S;

4. The posterior mean of exists and it islarger than the maximum likelihoodestimator for some values of S.


1. 3As we have 2 901 2 ... 9 So, st91 to th110 term is 10

2. 11022121 = 1000000 + 121 + 2(11) (1000)

2 21000 11 1011 3. 1

23 1 13m k m M 2 33 1 23m n M

23 33 1 13m k m n M So, L.H.S. is never 0, hence it has no integralsolution.

4. 3According to graph option 3 is correct.

5. 3We can set 4 3 4 3 3 3 = 12 + 12 + 9 = 33 laddoos.

6. 41 ky yx x

xy k (hyperbolic shape)7. 3

They are prime numbers in continuation8. 2

( )3, 0- ( )3, 0 y 6=

y 2x=y 2x= -

Base = 6 & Height = 6

So, Area = 1 6 6 182 9.- 410. 3

Fraction removed =

3

3

43 21

a

a

11 6 2

11. 3

Number of steps will be Bb andAa

12. 4Option 4 gives suitable match for the professions ofthe person.

13. 151024 6408

5640 4008

5400 2508 5250 156.258

14. 2 7 5 6 4860000002 3 5f f

= 4 + 8 + 6 + 0 + 0 + 0 + 0 + 0 + 0 = 1815. 1

Here we have 5 50 140x , 18.x 16. 2

Total number of paths = 2 4 3 2 48 17. 3

1 42 9, K 32n n K n

18. 1If d is the width after corrugation then we have

222.2 2.22 14d d 1.4 .d m

19. 32013 3 671N E T . If ,N E and T are

distinct then they will be 1, 3, 671 to givemaximum sum 1 3 671 675

20. 2By equality of areas, we get

2 2 22 1 2r r r 2 21 22r r 1 22r r

PART BUNIT-121. 2

0Df x for all x emplies that f x will be constant in connected components

of , not in whole .

CSIR 2014 DECEMBER SOLUTION

22. 2na is convergent but na is divergent1/ 1nna as n . Hence radius of conver-

gence of 1n

nna x

is R = 1

23. 2On the arc of circle , we will have tangent vectorperpendicular to the radius vector.

24. 22 2BA B I BA 2 2BA B BA I

2B IA B A As R.H.S. is non-singular, matrix so both thematrices on L.H.S. is non-singularB is non-singular

25. 4By Rolles theorem , there is at least one root of thederivative p between any two consecutive roots of p

26. 14 8 43 6 12 4 0

R 0 0 40 0 12 4 0

By 1 1 3R R 2R and 2 2 33R R R2 Now,

by 1 1 2R R 4R we get0 0 00 0 12 4 0

So, we have Row space as same as that of

1 2 00 0 127. 3

!!k

nna kn , Radius of convervence 1

lim nn naR a

!! 1lim! !1k

knkn n

kn n

...1 2lim ...1 1 1nkn kn kn kn n n

kK28. 2

The given matrix is involutary, so its eigen values is

1 or - 1 . Also if n is even then trace is 0 ; so 2n

eigen values is 1 and 2n eigen values is - 1

But if n is odd trace is 1 So, 12n eigen values is 1

and 12n eigen values is -1 So, 2| | 1

nA

29. 3

It is definition of uniform convergence, so forcertain finite value onward 0na and 0,nb then only it will becomes identity .x R

30 12

2

2

1 1 11 1 11 1 1

x x xy y yz z z

By 3 3 2C C C and followed by 2 2 1C C C we get

2

2

2

111

x xy yz z

x y y z z x z y y xz x

31. 4P*P = I P is unitary matrix, whose all eigenvalues are of modulus 1.

32. 3For

1 1 00 2 00 0 1

A

0 1 0A 1 I 0 1 0

0 0 0

Rank 1A 1I

G.M. of 1 = 31 = 2 = A.M. of 1, so it isdiagonalisable A is diagonalisable in (4) & (2)because its eigen values are distinctFor


1 1 0A 0 1 0

0 0 2

,

0 1 00 0 0A 1I0 0 1

Rank ( A- 1 I) = 2A.M. of 1 is 2 but its G.M. is 3-2 =1 which are notequals so A is not diagonalisable in (3)

33. 2 1/2 1/4 1/82 , 2 , 2Q over Q has degree 8 2 2 2

34. 1 p zq z has simple pole at z = 0 , so residue of p zq z at 0 i.e

0

Re lim0 zs p zpz zq qz z z

0 110 1 2

...lim ...n

nnz n

a a z a zb b z b z

01

ab

35.Number of elements in the given group = O(G) = 4 4 4 2 4 33 1 3 3 3 3 3 3

6 4 3 23 3 13 1 3 1 3 1 As 63 divides 0(G) but not 73 , so number ofelements in 3 sylow group 63 729

36. 26 = 1+5 = 2+4 = 3+3 = 1+1+4=1+2+3 = 2+2+2 = 1+1+1+3=1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1So, number of conjugacy classes is 11.

37. 14 = 1+1+24 can be partitioned as 1, 1, 2 in 4! 62!2!

ways

So, number of onto functions = 6 3! 36 .38. 4

1 2 3 39. 3

Number of orbits for action of G on the field with23 elements over subfield with 3 elements will be 6.

40. No Option is correct (By default 1 is better)Radius of convergence

11

11lim nn n

R ap nap n

= 1 .R

41. 3It is cauchy Euler differential equation with y(1) =y(2) = 0. So, for any continuous function f from[1, 2] to R with f x not vanishing identically in[1, 2], there exists a solution u of p for some

R such that 21 0.fu 42. 2

As f is Lipschitz continuous function, so under thegiven condition y is bounded.

43. 1 11 0 1 0 12E P dx P Px

1 2 0 1 20 1 22 3 2a aa a a a 26

a

12 0 12E P dx Px

1 20 0 1 2

1 12 3 2 4a aa a a a

212a

2 21 2;6 12

a aE E

1 2E E 44. 2

Charpits Equation isdx dy duf f f fp qp q p q

dp dqf f f fp qx u y u

2 2 32 2 2 1dx dy du dppu q p q p

21dqqp


45. 3

0 6x xy y dsx s sx ..........(1)

Differentiating equation-(1) w.r.t.x we get

2

0' 2x xy y dsx x s s ..........(2)

Comparing (1) & (2) we get after solution 0xy S Sin dsx sx

46. 3 1 ctgV ec

47. 2 11 2y xx will extremize J.

48. 1

, i xu e vx t t ,33

u ut x

3i x i xdve v eidt 3dv i vu

3dv i dtv 3lnv i t c

(0) 1 0v c 3i tv e

2, i x tu ex t 49. 1

21

lim 0nVar TVar T

50. 3

P(A is invertible ) 1 1 12 2 4 as 1X and 3Xboth should be non-zero but 2X can be any value

51. 2

(0, 5/4)

Y

X

3x+4y = 5

(5/3, 0)

z= - (2x + 5y) will approach to in the feasibleregion when feasible region is unbounded andhence non-empty.

52. 2It will be number of ways of distributing n identi-cal items among n places or persons such that anyone can get any number of items.It is 1 11 2 1C Cn nn n n 2 1Cnn

53. 2Degree of freedom = 100

54. 22 2pn nX and 2 0n

55. 0nE S , nV nS 0lim 1nn

S xn for any x > 0

lim 1nn P nx S nx 56. 3

i iX X has Wishart distribution withn d.f.

57. 4

C.O.D.221

58. 2Except option (2) all are true

59. 3posterior probability of the 10 2 is 1

e ee

60. 1

Required probability 26! 3 37! 7

C PART C


61. 1,2As, lim 'xL f fx x is finite , so if lim 'x f xexist it cannot be non-zero else f x will becomeunbounded So, if lim 'x f x and lim 'x f x exist ,they will be 0 and L respectively

62. 1,2,3Option 4 is wrong because it 0,1S and x = 1d (x , s) = 0 but .x S Rest follows from definitionof closed set .

63. 1,2,41

1 11k k kk

k k

a a ax a a

1 1...1 1n m nkk m m n

a ax a a

11 1m m

m n

a aa a

as ka is strictly` increasing ..........(option 1)

As inn

kk mx

each term is positive so there exists

n m such that 12n

kk mx

. ......... (option 2)

As each term in the series is positive so, 1 kk x

diverges to (option 4)64. 1,2

If K > 0 then xKe P x has at least one root if p x is of even degree

and if K < 0 then xKe p x has at least oneroot if p x is pf odd degree.

65. 3Characteristic function

1:0E

if x ER Rif x E

E will be continuous iff E or R i.e E is both

open and closed.66. 1,3

, ,AAx Ay x AyttA AA y xA y x

y x x y . Rows of orthogonal matrix fromsan orthonormal basis of nR

67. 1Function from X R is continuous and

,G x f x be the graph of f then G ishomeomorphic to X.

68. 1Open unit ball in X is contained in open unit ballin 1X , so we have U as a subset of 1U

69. 2,3,4As f x is monotonically increasing, so set ofdiscontinuities of f x should be countable , henceit will be Riemann Integrable and hence Lebesgueintegrable.

70. 1,2If every continuous function from A to R is boundedthen A must be closed and bounded subset of R,hence it is compact in this case.Also for closed and bounded subset of R only wehave statement in option (2) as true.

71. 1,2By manipulating either one row or one column ofa matrix its rank is affected by at most 1.So, 1 1 1r s rs r

72. 3,45 9 8 1 23 01 8 2 1 8 29 1 0 9 1 0

A

2 4161 207 532 13 2 13

73, 1,2,3,4 1 2,f T T Rank f = 1 because

f is nozero and each iT will have its Rank 1Also Nullity of each iT is equals to 3-1 = 2 As except 1 eigenvalue all will be zero , so f iseither positive semi definite or negative semidefiniteAlso, Null sapce of ,f is of dimension 2.


74. 2,3,4As ratio of integers is rational number, so Ax = bhas a rational solutionThe set of real solutions to Ax = 0 has a basisconsisting of rational solutions .

For solutionRank(A) = Rank(A:b) if b 0, then Rank (A:b) is

1, so Rank (A) 1, .75. 1,2,3

Jordan canonical form is0 1 00 0 00 0 0

if minimal

polynomial is m (x) = 2x .Except matrix in option(4) each of them have 2( )m x x .

76. 1,34AB I Rank (AB) = 4 Rank (A) 4

& Rank (B) 4. But A is 4 7 and B is 7 4matrix so Rank (A) 4 and Rank (B) 4 Rank (A) = Rank (B) = 4 Nullity (B) =N o. ofcolumn in B- Rank (B) = 4 - 4 = 0

77. None are correctIf U, V and W are taken as zero - spaces(subspace) then all four options becomes equal tozero - space . So, proper superset or proper subsetis not possible

78. 1,3,4

T AXX nij ik kjk 1T x A x trace (T) =

n niii 1 i 1

A n iii 1n A

Rank of T is 2n and T is non-singular79. 1,2,3,4

All four statements are correct80. 1,3

1 & 3 are true but 2, & 4 are false81. 1,2

The given matrix is real symmetric and orthogonalmatrix so its eigenvalues will be + 1 and -1 and itstrace is 0, so it has 3 eigenvalues + 1 and 3 eigenvalues - 1

82. 2,325 25 If 0(G) = 25 G is abelian.35 5 7 and 1+5k divides 35 iff 1+5k = 1 So, if 35,O G G is cyclic so it is abelian

83. 1,2,4

2 2 2: 1,X R a ba b is closedbounded subset of 2R which is also

connected, so Image (f) is also connected boundedand closed and hence compact

2 2 1a b+ =

Also f cannot be injective because for , 1, 0a b & , 1, 0a b satisfy2 2 1.a b

84. 1,3

0

nf a zz nn

By maximum modulus priciple, we have2 2 sup

| |0na rn z rn

2| |f z

And also by Gauss mean value theorem we get

2 2 2 201 | |20n ia r f durenn

85, 1,3,4

2 2& 1x yA x yy x cos & sinx y cos sinsin cosA

n cosn sinA sin cosn

cos sinsin cos

Also A is orthogonal matrix, so we have 1tA AAlso we have A as diagonalisable

86. 2,3,4


20 45 3 5G 23 45| but 33 45does not divide , so have

subgroup of order 9Also, 5 | 45 but 25 45does not divide , so G has asubgroup of order 5

1 5 45k | iff 1 + 5k = 1 & also 1+3k | 45 1+ 3k= 1 , So, G has normal subgroup of order 9 & 5.

87. 1 S Re f Im f |zz z

|x y x iy . As is bounded opensubset of C, so we have S is an open

subset of R.88. 2

3 23 2u x xy xx,y 2 23 3 2x yu x y V 6y xu xy V

2 33 2V x y y y c 89. 2,3

D : R Rx x where dfDf , fdx As 0D a Ra So, D is notone-one. If 20 1 2a a x a x ... R x

Then 2 30 1 22 3x xD a x a a ...

20 1 2a a x a x ... So, D is onto D fE f if 20 1 2E a a x a x ...

2 30 1 22 3

x xa x a a ... Once differentiated , constant term is lost, so wecannot find E such that E f ; fD f

90. 1,2,3 f R f Rz z f Rz . So, f f gzz z If

f Rz 0z z|I m z az|Im z

For some a > 0 , then f f , z cz ia z ia & 2f f , z cz ia z but not f fz ia z for all z c .

91. 2,3 1 20 0u u & 0 is lying inside [ -1, 1] i.e 0 is

interior point of [-1,1], so we have 1u and 2u aslinearly dependent and also Wronskian w (x) = 0 forall 11x ,

92. 2,3

0dx dyy x from the given equation

2 22 2 0xdx ydyx y C

0 0x ,y a,b 2 2 2 2x y a bt t t R .It has solution

for any (a, b) R R. 93. 2,4

1 0dt dx du

u u c & x ct 00u or ux, x can be taken as 0u xx or 0 1 2u xx

94. 1,3 2 1 xy eD . C.F is x xy Ae Be P.I is

11 1

xy eD D 2

xxe

2x

x x xey Ae Be

2x x

x xdy e xeAe Bedx

0 00y A B 100 2

dy A Bdx 1 14 4A &B


14 2x

x x xey e e

1xxlim e y x x

& y attains its minimum on R.

95. 1,3 1 0; 1nn ny y h yy

11

1n ny yn It is first order appoximation to nhe and also it isrational function approximation

96. 1,4 10| |2u x s u dssx

ax b By differentiating both sides w.r.t. x, we get

10 |2x sdu u ds asx sdx

( By Leibnitz rule) Again differentiating both sides

w.r.t. x, we get22 0

d u udx 97. 2,498. 1,2

1 '

nn n

n

xfx x xf , so, Newtons method is sameas fixed point iteration for the map '

f xg xx f x Also,

0'

f ydydt f y

'

f ydydt f y

1 '

nn n

n

yfy y yf So, it is also same99. 2,3,4100. 1,3101. 2,3,4

2

f fx h x hh

2 3' " "'2! 3!

h hf hf f fx x x x

2... " ...2!

2

hf hf fx x xh

2' "' ...6

hf fx x So, order of error is 2 similary 3 & 4 are correct

102. 1,2

2 siny xx Extremizes the given functional.

103. 1,2,3,4, 104. 4 105. 1,3,4106. 1,2,3,4, 107. 1,2,3, 108. 1,3109. 3 110. 1,2,4 111. 1,3,4112. 3 113. 1,2,4 114. 2,4115 1,3 116. 2,3 117. 1,2,3,4118. 1,2,3 119. 3,4 120. 2,4>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>


Documents

Filename: December 2014 question & solution.pdf