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

(D)

A

(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

u

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

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 < 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.

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 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

CORE SECTION

16. (a) Solve the initial value problem d²y

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

dx

(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 = 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 c₂ G (a,b) such that

(9)

(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)

l

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

sequence of functions {f_n }.

(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

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

24. (a) Let X_Y, X₂, ..., 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, X₂, ..., X₉ be an independent random sample from (2,4) and Y_lf 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 X_x, X₂, ..., X_n be a random sample from a distribution having pdf

ax,

for x > x,

. a+1

f(x;x₀,a) =

x

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)

x[tx[7xf

will have a -distribution.

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, a²). Find the best test for testing H₀ : // = 0, a² =1 against H_l : ju = 1, cf² = 4 .    (9)

OPTIONAL SECTION: C

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

| l/tre')! d6 < 4r

o

f^{n)( 0)

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)

OPTIONAL SECTION: D

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)

OPTIONAL SECTION : E

R

1

31. (a) Estimate the error in evaluating the integral J(l + x²)e~^xdx 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 = 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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