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Indian Institute of Technology Guwahati (IIT-G) 2005 JAM MATHS - Question Paper

Wednesday, 23 January 2013 06:25Web



2005 - MA Test Paper Code : MA Time : 3 Hours    Max. Marks : 300 INSTRUCTIONS


Test Centre :

2005 - MA


1.    The question-cum-answer book has 44 pages and has 32 questions. Please ensure that the copy of the question-cum-answer book you have received contains all the questions.

2.    Write your Roll Number, Name and the name of the Test Centre in the appropriate space provided on the right side.

3.    Write the answers to the objective questions against each Question No. in the Answer Table for Objective Questions, provided on page No. 7. Do not write anything else on this page.

4.    Each objective question has 4 choices for its answer: (A), (B), (C) and (D). Only ONE of them is the correct answer. There will be negative marking for wrong answers to objective questions. The following marking scheme for objective questions shall be used :

(a)    For each objective question, you will be awarded 6 (six) marks if you have written only the correct answer.

Do not write your Roll Number or Name anywhere else in this question-cum-answer book.

(b)    In case you have not written any answer for a question you will be awarded 0 (zero) mark for that question.

(c)    In all other cases, you will be awarded -2 (minus two) marks for the question.

5.    Answer the subjective question only in the space provided after each question.

I have read all the instructions and shall abide by them.

6.    Do not write more than one answer for the same question. In case you attempt a subjective question more than once, please cancel the answer(s) you consider wrong. Otherwise, the answer appearing later only will be evaluated.

7.    All answers must be written in blue/ black/blue-black ink only. Sketch pen, pencil or ink of any other colour should not be used.

Signature of the Candidate

8.    All rough work should be done in the space provided and scored out finally.

9.    No supplementary sheets will be provided to the candidates.

I have verified the information filled by the Candidate above.

10.Logarithmic    Tables / Calculator of any kind / cellular phone / pager / electronic gadgets are not allowed.

11.The    question-cum-answer book must be returned in its entirety to the Invigilator before leaving the examination hall. Do not remove any page from this book.

Signature of the Invigilator

12.Refer    to special instruction/useful data on the reverse.


Name :


Objective Part :

Attempt ALL the objective questions (Questions 1 - 15). Each of these questions carries six marks. Write the answers to the objective questions ONLY in the Answer Table for Objective Questions provided on page 7.

Subjective Part :

Attempt ALL questions in the Core Section (Questions 16 - 22). Questions 16 - 21 carry twenty one marks each and Question 22 carries twelve marks. There are Five Optional Sections (A, B, C, D and E). Each Optional Section has two questions, each of the questions carries eighteen marks. Attempt both questions from any two Optional Sections. Thus in the Subjective Part, attempt a total of

11 questions.

cL v cJi/v

3. The general solution of x2 - - 5x--h 9y = 0 is

dx2 dx

(A)    (cj + c2x)e3x

(B)    (cj + c2 Inx)x3

(C)    (cj + c2x)x3

(D)    (cj + c2 In x)ex

(Here cx and c2 are arbitrary constants.)

4. Let r = xi + ;yj+zk. If 0{x,y,z) is a solution of the Laplace equation then the vector field

(A) neither solenoidal nor irrotational

(B) solenoidal but not irrotational

(C) both solenoidal and irrotational

(D) irrotational but not solenoidal

5. Let F = x i+ 2y j+ 3z k, S be the surface of the sphere x2 + y2 + z2 =1 and n be the

inward unit normal vector to 5 . Then J*J* ndS is equal to


(A) 4 n

(B) -4 n

(C) 8 n



(A)    A2 + A is non-singular

(B)    A2 - A is non-singular

(C)    A2 +3A is non-singular

(D)    A2 -3A is non-singular

Let T : R3-> R3 be a linear transformation and I be the identity transformation of R3.

If there is a scalar c and a non-zero vector x e R3 such that T(x) = cx, then rank(T - cl)

(A)    cannot be 0

(B)    cannot be 1

(C)    cannot be 2

(D)    cannot be 3

In the group {l, 2, ..., 16} under the operation of multiplication modulo 17, the order of the element 3 is

(A)    4

(B)    8

(C)    12

(D)    16

A ring R has maximal ideals

(A)    if R is infinite

(B)    if R is finite

(C)    if R is finite with at least 2 elements

(D)    only if R is finite





jo |_ jo V.J0 2 r f2 / rl-y

(d> r r r**


11. Let f : R-> R be continuous and g, h : R2-> R be differentiable. Let F(u,v) = j* f(t)dt,

d F dF

where u = g (x,y) and v = h (x,y). Then -+-

d x d y

dg , dg dx d y

dg [ dg dx dy

dg , dg dx dy

dh_ dh_ dx dy

- f (h(x,y)) + f(g(x,y))

+ f(g(x,y))


dx dy

dh_dh dx dy

dhL_dhL dx dy

dg dg dx dy









12.    Let y = f (x) be a smooth curve such that 0 < f(x) < K for all x e \a,b\. Let L = length of the curve between x = a and x = b

A = area bounded by the curve, x -axis, and the lines x = a and x = b

S = area of the surface generated by revolving the curve about x -axis between x = a and x = b


(A)    2ttKL <S < A

(B)    S < 2nA < 2ttKL

(C)    2A <S< 27iKL

(D)    2A < 2ttKL < S

13.    Let f: R-> R be defined by f(t) = t2 and let U be any non-empty open subset of JR.


(A)    f(U) is open

(B)    f~\U) is open

(C)    f(U) is closed

(D)    f~\U) is closed

14.    Let f : (-1,1) -> R be such that f'l\x) exists and | /(x)| < 1 for every n > 1 and for

every xg (-1,1). Then f has a convergent power series expansion in a neighbourhood of

(A)    every ig (-1,1)

(B)    every xe    only

(C)    no x (-1,1)

(D)    every xe 0, only

15.    Let a > 1 and f,g,h: \-a, a] -> R be twice differentiable functions such that for some c

with 0 < c < 1 < a ,

f (x) = 0 only for x = -a, 0, a ; f\x) = 0 = g (x) only x = -1, 0,1 ; g'(x) = 0 = h (x) only for x = -c, c .

The possible relations between f, g, h are

(A)    f - g and h = f'

(B)    f' = g and g = h

(C)    f = -g' and h' = g

(D)    f = -g' and h' = f


16. (a) Solve the initial value problem d2y

. .,~y =*(sinx + e*), 3,(0) = /(0) = l    (12)


(b) Solve the differential equation

(2ysinx+ 3y4sinx cosx)dx - (43/3cos2x + cosx)dy = 0

Let G be a finite abelian group of order n with identity e. If for all a e G, a3 =e, then, by induction on n, show that n = Sk for some nonnegative integer k.    (21)

18. (a) Let f \\a,b]> R be a differentiable function. Show that there exist points cl5 c2 G (a,b) such that


(b) Let

(x2 +y2)[\n(x2 +y2)+l] for (x,y)*(0,0)

f(x,y) = -S

for (x,y) =(0,0)


Find a suitable value for a such that f is continuous. For this value of a, is f differentiable at (0,0) ? Justify your claim.    (12)

19. (a) Let S be the surface x2 + y2 + z2 = 1, 2 > 0 . Use Stoke's theorem to evaluate

j* \(2x - y)dx - ydy - zdz] c

where C is the circle x2 + y2 = 1, z = 0, oriented anticlockwise.    (12)

(b) Show that the vector field F = (2xy -y4 +3)i + (x2 -4xy3 )j is conservative. Find its potential and also the work done in moving a particle from (1,0) to (2,1) along some curve.    (9)

Let T : R3->R3 be defined by T(x,y,z) = (y + z,z,0). Show that T is a linear

transformation. If v R3 is such that T2(v)0, then show that B = {v, T(v), T2 (v)} forms a basis of R3. Compute the matrix of T with respect to B. Also find a v e R3 such that T2(v) 0.    (21)

4 n x


4n \x


0, 2 n

J_ j.

2n n


, 1



Compute ftl{x)dx for each n. Analyse pointwise and uniform convergence of the

sequence of functions {fn }.


(b) Let f: R > R be a continuous function with |/'()-/'(3;)||x-j| for every x,y G R . Is f one-one? Show that there cannot exist three points a, b, c g R with a <b < c such that f(a) < fic) < fib).    (9)

Find the volume of the cylinder with base as the disk of unit radius in the xy -plane centred at (1, 1, 0) and the top being the surface z = [(x-l)2 + (y-l)2] .    (12)

23. (a) Bag A contains 3 white and 4 red balls, and bag B contains 6 white and 3 red balls.

A biased coin, twice as likely to come up heads as tails, is tossed once. If it shows head, a ball is drawn from bag A, otherwise, from bag B. Given that a white ball was drawn, what is the probability that the coin came up tail?    (9)

(b) Let the random variables X and Y have the joint probability density function f(x,y) given by

ye-y(*-n) x>0,y>0 0    otherwise

f(x, y) =

Are the random variables X and Y independent? Justify your answer.


24. (a) Let XY, X2, ..., Xn be independently identically distributed random variables (rv's)

with common probability density function (pdf) fx (x,6) = e~x/0; x > 0, 6 > 0. Obtain


_ 1 n

the moment generating function (mgf) of X = / Xt. Also find the mgf of the rv

Y = 2nX/0.    (9)

(b) Let Xx, X2, ..., X9 be an independent random sample from (2,4) and Ylf Y2, Y3, Y4 be an independent random sample from N (1,1). Find P(x >Y), where X and Y are sample means.

[Given P(Z > 1.2) = 0.1151, where Z ~ N(0,1)]    (9)

25. (a) Let Xx, X2, ..., Xn be a random sample from a distribution having pdf


for x > x,

. a+1

f(x;x0,a) =


0    otherwise

where x0 > 0, a > 0. Find the maximum likelihood estimator of a if x0 is known. (9)

(b) Let X1, X2, ..., X5 be a random sample from the standard normal population. Determine the constant c such that the random variable

c(X1 +X2)



will have a -distribution.


26. (a) A random sample of size n = 1 is drawn from pdf fx (x,0) =     > 0, (9 > 0 . It is

decided to test HQ : 6 = 5 against Hx : 0 = 7 based on the criterion: reject 7/0 if the observed value is greater than 10. Obtain the probabilities of type I and type II errors.    (9)

(b) Let X1} X2 ? ..., Xn be a random sample from a normal population n(ju, a2). Find the best test for testing H0 : // = 0, a2 =1 against Hl : ju = 1, cf2 = 4 .    (9)


27. (a) Let f,g:R-> R be such that for x,y& R,

<p(x+iy) = ex\f(y) + ig(y)\ is an analytic function. Find a differential equation of order 2 satisfied by f.    (9)

(b) Compute I* (2z + l)e2 + l'z dz.    (9)

JI z-fl1=2

28. (a) Let f(z) be analytic in the whole complex plane such that for all r > 0,

2 n

| l/tre')! d64r


f{n)( 0)

Find -- for all n > 0.    (9)


(b) Find all values of aeC such that f(z) = (z +z)2 +2a\z |2 + a(z)2 is analytic at some point 2 having non-zero real part.    (9)


A hemispherical bowl of radius 12 cm is fixed such that its rim is horizontal. A light rod of length 20 cm with weights w and W attached to its two ends is placed inside the bowl. In equilibrium, the weight w is just touching the rim of the bowl. Find the ratio w : W. (18)

A uniform ladder of length 2a and mass m lies in a vertical plane with one end against a smooth wall, the other end being supported on a horizontal floor. The ladder is released from rest when inclined at an angle a to the horizontal. Find the inclination of the ladder to the horizontal when it ceases to touch the wall.    (18)




31. (a) Estimate the error in evaluating the integral J(l + x2)e~xdx by Simpson's rd rule


with spacing h = 0.25.    (9)

(b) Using Newton-Raphson method, compute the point of intersection of the curves y = x3 and y = 8x + 4 near the point x = 3, correct up to 2 decimal places.

[Round-off the first iteration up to 2 decimal places for further computation]    (9)

The polynomial p3(x) = x3 + x2 - 2 interpolates the function fix) at the points

x = -1, 0, 1 and 2. If the data /(3) = -14 is added, find the new interpolating polynomial

by using Newton's forward difference formula. Also find f{2.5) by using Newton's

backward difference formula with pivot value 3. Justify whether the value obtained will be the same if pivot value 2 is taken.    (18)



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