Indian Institute of Technology Guwahati (IIT-G) 2005 JAM MATHS - Question 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 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
![](images/stories/user/1920/pd/2118-29156-ma_files/2118-29156-ma-2.png)
(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
s
(A) 4 n
(B) -4 n
(C) 8 n
(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
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 : 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))
dh_rk
dx dy
dh_dh dx dy
dhL_dhL 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)) |
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
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 = 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
(9)
(b) Let
(x2 +y2)[\n(x2 +y2)+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 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
fnW
4n \x
n
0, 2 n
J_ j.
2n n
1
, 1
n
l
Compute ftl{x)dx for each n. Analyse pointwise and uniform convergence of the
sequence of functions {fn }.
(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
f(x, y) =
Are the random variables X and Y independent? Justify your answer.
(9)
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
0
_ 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
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)
Y
will have a -distribution.
(9)
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')! d6 < 4r
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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 + x2)e~xdx by Simpson's rd rule
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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)
![](images/stories/user/1920/pd/2118-29156-ma_files/2118-29156-ma-3.png)
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