# Indian Institute of Technology Guwahati (IIT-G) 2005 JAM Maths - exam paper

JAM 2005 Maths

IMPORTANT NOTE FOR CANDIDATES

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 x^{1} - - 5x--b 9y = 0 is

dx^{2} dx

(B) (cj + c_{2} Inx)x^{3}

(C) (cj + c_{2}x)x^{3}

(D) (cj + c_{2} In x)e^{x}

(Here c_{l} and c_{2} are arbitrary constants.)

4. Let r =xi + 3/j+zk. If <p(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 x^{2} + y^{2} + z^{2} =1 and n be the

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

s

(A) 4 n

(B) -4 n

(C) 8 n

(D)

A

(A) A^{2} + A is non-singular

(B) A^{2} - A is non-singular

(C) A^{2} +3A is non-singular

(D) A^{2} -3A is non-singular

Let T : R^{3}-> R^{3} be a linear transformation and I be the identity transformation of R^{3}.

If there is a scalar c and a non-zero vector x e R^{3} 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

dy

dy

dy

dy

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

^{(d}> r r r**

u

11. Let f : R-> R be continuous and g, h : R^{2}-> 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(*,y)) + f(g(x,y))

+ f(g(x,y))

dh_dh dx dy

dh_dh dx dy

dh_{L}_dh_{L} dx dy

dg dg dx dy

(A) |
f(g(x,y)) |

(B) |
f(h(x,y)) |

(C) |
f(h(x,y)) |

(D) |
f(g(x,y)) |

A

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

Then

(A) 2ttKL <S < 2nA

(B) S < 2nA < 2nKL

(C) 2A <S< 27iKL

(D) 2nA < 2tiKL < S

13. Let f: R-> R be defined by f(t) = t^{2} and let U be any non-empty open subset of JR.

Then

(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 xg (-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

CORE SECTION

16. (a) Solve the initial value problem d^{2}y

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

dx

(b) Solve the differential equation

(2ysinx+ 3y^{4}sinx cosx)dx - (43/^{3}cos^{2}x + cosx)dy = 0

Let G be a finite abelian group of order n with identity e. If for all a e G, a^{3} =e, then, by induction on n, show that n = S^{k} for some nonnegative integer k. (21)

18. (a) Let f \\a,b]> R be a differentiable function. Show that there exist points c_{l5} c_{2} G (a,b) such that

(9)

(b) Let

(x^{2} +y^{2})[\n(x^{2} +y^{2})+l] for (x,y)*(0,0)

f(x,y) = -S

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

a

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 x^{2} + y^{2} + z^{2} = 1, 2 > 0 . Use Stoke's theorem to evaluate

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

where C is the circle x^{2} + y^{2} = 1, z = 0, oriented anticlockwise. (12)

(b) Show that the vector field F = (2xy -y^{4} +3)i + (x^{2} -4xy^{3} )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 : R^{3}->R^{3} be defined by T(x,y,z) = (y + z,z,0). Show that T is a linear

transformation. If v R^{3} is such that T^{2}(v)0, then show that B = {v, T(v), T^{2} (v)} forms a basis of R^{3}. Compute the matrix of T with respect to B. Also find a v e R^{3} such that T^{2}(v) 0. (21)

4 n x

f_{n}W

4n \x

n

0, 2 n

J_ j.

2n n

1

, 1

n

l

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

sequence of functions {f_{n} }.

(12)

(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

y_{e}-y(*-n) _{x}>0,y>0 0 otherwise

f(x, y) =

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

(9)

24. (a) Let X_{Y}, X_{2}, ..., X_{n} be independently identically distributed random variables (rv's)

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

0

_ 1 ^{n}

the moment generating function (mgf) of X = / X_{t}. Also find the mgf of the rv

Y = 2nX/0. (9)

(b) Let X_{x}, X_{2}, ..., X_{9} be an independent random sample from 2V(2,4) and Y_{lf} Y_{2}, Y_{3}, Y_{4} 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 X_{x}, X_{2}, ..., X_{n} be a random sample from a distribution having pdf

ax,

for x > x,

. a+1

f(x;x_{0},a) =

x

0 otherwise

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

(b) Let X_{1}, X_{2}, , X_{5} be a random sample from the standard normal population. Determine the constant c such that the random variable

c(X_{1} +X_{2})

Y

will have a -distribution.

(9)

26. (a) A random sample of size n = 1 is drawn from pdf f_{x} (x,0) = ; * > 0, <9 > 0. It is

decided to test H_{Q} : 6 = 5 against H_{x} : 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 X_{1}} X2 ? ..., X_{n} be a random sample from a normal population n(ju, a^{2}). Find the best test for testing H_{0} : // = 0, a^{2} =1 against H_{l} : ju = 1, cf^{2} = 4 . (9)

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

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

(b) Compute I* (2z + l)e^{2 + }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

o

Find -- for all n > 0. (9)

n!

(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)

R

31. (a) Estimate the error in evaluating the integral J(l + x^{2})e~^{x}dx by Simpson's rd rule

o

with spacing h = 0.25. (9)

(b) Using Newton-Raphson method, compute the point of intersection of the curves y = x^{3} 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 p_{3}(x) = x^{3} + x^{2} - 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)

o

An integrating factor of x + (3x + 1) y = xe~^{2x} is

dx

(A) xe^{31}

(B) 3xe*

(C) xe*

(D) xV

Attachment: |

Earning: Approval pending. |