Indian Institute of Technology Guwahati (IIT-G) 2007 JAM (M.ScEntrance) Mathematics - Question Paper
JAM (M.Sc. Entrance) Mathematics
Full ques. Paper in attachment
Mathematics Paper 2007
IMPORTANT NOTE FOR CANDIDATES
Attempt ALL the 29 questions.
Questions 1-15 (objective questions) carry six marks each and questions 16-29 (subjective questions) carry fifteen marks each.
Write the answers to the objective questions in the Answer Table for Objective Questions provided on page 7 only.
Which of the following sets is a basis for the subspace x y
W
0 t
of the vector space of all real 2x2 matrices?
|
1 o |
'0 0" | ||
|
9 |
1 o 0 1 |
> |
0 1 |
1 0 0 0 2 1 0 -1 -1 1 2 -1 1 -1" 0 1
(B)
(C)
(D)
(A)
|
'l -l | |
|
> |
\ r-i o |
J
Let G be an Abelian group of order 10. Let S ={geG : g'1 =g}. Then the number of non-identity elements in S is
(A) 5
(B) 2
(C) 1
(D) 0
Let R be the ring of polynomials over Z2 and let / be the ideal of R generated by the polynomial x3 +x + l. Then the number of elements in the quotient ring RlI is
(A) 2
(B) 4
(C) 8
(D) 16
Ji.
4. Let f: R > R be a continuous function. If jf(2t)dt=sin(x) for all x e R, then f(2) is
o 71
equal to
(A) -1
(B) 0
(C) 1
(D) 2
5. Suppose (c ) is a sequence of real numbers such that lim |c|1/n exists and is non-zero.
n <*>
If the radius of convergence of the power series xn is equal to r , then the radius of
oo
convergence of the power series X"2 c is
n = 1
(A) less than r
(B) greater than r
(C) equal to r
(D) equal to 0
|
'1 |
4 |
8 ' | ||
|
6. The rank of the matrix |
2 |
10 |
22 |
is |
|
0 |
4 |
!2 |
(A) 3
(B) 2
(D) 0
7. If k is a constant such that xy+k=e(x~l> 12 satisfies the differential equation
x -{x2 -x -1 )y + (x 1), then k is equal to dx
(A) 1
(B) 0
(C) -1
(D) -2
8. Which of the following functions is uniformly continuous on the domain as stated?
(A) f(x) = x2 , x G R
(B) f{x) = ~, X6[l,oo)
X
(C) fix) = tanx , xe {-nl2,nl2)
([x] is the greatest integer less than or equal to x )
9. Let Ait) denote the area bounded by the curve y =e" , the x-axis and the straight lines x--t and x= t. Then lim Ait) is equal to
(A) 2
(B) 1
(C) 1/2
(D) 0
10. Let C denote the boundary of the semi-circular disk and
let <p(x,y)=x2 +y for (x,y)eD. If ii is the outward unit normal to C , then the integral (p)-fids , evaluated counter-clockwise over C , is equal to
(A) 0
(B) n-2
(C) *
(D) n + 2
11. Let a = (aex siny-4x )i+(2y+e* cosy )j+ azk, where a is a constant. If the line integral
u-dr over every closed curve C is zero, then a is equal to
c
(A) -2
(B) -1
(C) 0
(D) 1
12. One of the integrating factors of the differential equation (y2 -3 xy)dx + (x2-xy )dy-0 is
0 if (x,y) = (0,0).
Which of the following statements holds regarding the continuity and the existen partial derivatives of f at (0,0)? '
:e of
0,0) :k is
(A) Both partial derivatives of f exist at (0,0) and / is continuous at (0,0)
(B) Both partial derivatives of f exist at (0,0) and / is NOT continuous at (0,0)
(C) One partial derivative of f does NOT exist at (0,0) and f is continuous at (0,0)
(D) One partial derivative of f does NOT exist at (0,0) and / is NOT continuous at
14. Let (a ) be an increasing sequence of positive real numbers such that the series
k = i
n n a
divergent. Let sn=2_,ak for n = 1,2,... and tn=2_,~ fr n~ 2,3,.... Then lim
*-i k = 2Sk-\sh n
equal to
(A) l/ox
(B) 0
(C) l/( al+a2)
(D) a1 + a2
15. For every function f: [0,1] > R which is twice differentiable and satisfies f'(x)> 1 f( x [0,1], we must have
r all
(A) f"(x)>0 for all [0,1]
(B) f(x)>x for all xe [0,1]
(C) f(x2)-x2 <f(x1)~x1 for all xx,x2 [0,1] with x2 >
(D) f(x2)-x2 >f(x1)-x1 for all xltx2 [0,1] with x2 > xx
2 i i + 3 l-i 3t 0 i
1 +i 0 0
Determine the eigenvalues of the matrix
16. (a) Let M =
B = M2 -2M +/ .
(9)
(b) Let N be a square matrix of order2. If the determinant of N is equal to 9 and the sum of the diagonal entries of N is equal to 10, then determine the eigenvalues of N. (6)
2 d2y dy 2 x +x--y=x, dx2 dx
given that x and are two solutions of the corresponding homogeneous equation: x
(9
(b) Find the real number a such that the differential equation
4- + 2(a-l)(a-S)+(a-2)y=0 dx dx
has a solution y (x)=a cos(/3x)+b sin(/?x ) for some non-zero real numbers a,b, ft. (6 equation a (jc + a/2 f \ + frix + V2 + cy = 0. (9)
dxz dx
(b) Solve the differential equation
dx + (e-ysm;y - x )(y cosy + siny )c?y = 0 . (6)
Let fix, y)=xix-2y2) for ix, y)e R2. Show that f has a local minimum at (0,0) on every straight line through (0, 0). Is (0, 0) a critical point of f? Find the discriminant of f at (0, 0). Does f have a local minimum at (0, 0)? Justify your answers. (15)
20. (a) Find the finite volume enclosed by the paraboloids z-2-x2-y2 and z =
3
x2+y2. (9)
(6)
(b) Let f: [0, 3] R be a continuous function with J f(x)dx=3 . Evaluate
o
3 jc
|[jc/'(rc)+J f(t)dt] dx .
21. (a) Let S be the surface {(x,y,z)eR3: x2 +y2 +2z=2, z>0), and let n be the outward
unit normal to S . If F =y i+ xzj+(x2 +y2) k , then evaluate the integral JjF/zdS. (9)
s
A A A t *
(b) Let i+y j + 2 k and r=|r|. If a scalar field (p and a vector field u satisfy V = Vx u+f(r) r , where f is an arbitrary differentiable function, then show that V2<p=r f'(r)+3f(r). (6)
22. (a) Let D be the region bounded by the concentric spheres Sx : x2 + y2 + z2 =a2 and S2 : x2 +y2 +z2 =b2, where a <6. Let n be the unit normal to directed away from the origin. If V2 (p =0 in D and <p=0 on S2, then show that
JJJ|Vp|2dV+JjV(Vp)-ndS=0. (9)
D Sj
(b) Let C be the curve in R3 given by x2 +y2 =a2 , z= 0 traced counter-clockwise, and let
. . a a a r~*_
F = x y i + j + z k . Using Stokes theorem, evaluate vF dr . (6)
c
Let V be the subspace of R4 spanned by the vectors (1,0,1,2), (2,1,3,4) and (3,1,4,6). Let T:V > R2 be a linear transformation given by T(x,y,z,t)= (x-y, z-t) for all (x,y,z,t) e V . Find a basis for the null space of T and also a basis for the range space of T .
(15)
bounded by the straight lines ;y=* + 3, y=x-3, y = -2x + 4 and y = -2x-2.
(9)
nil
Evaluate J o
sinx
(b)
dx
dy
(6)
n! 2
]idx dy+ j
.n! 2
25. (a) Does the series - converge uniformly for x e [ -1,1]? Justify. (9)
A = 1
(b) Suppose (fn ) is a sequence of real-valued functions defined on R and f is a real-valued function defined on R such that \ fn(x) - /'(jc)| <\an | for all neN and an >0 as n . Must the sequence (fn ) be uniformly convergent on R ? Justify. (6)
Suppose f is a real-valued thrice differentiable function defined on R such that j f"'(x)>0 for all xeR. Using Taylors formula, show that il
f(x2)~ f(x1)>{x2 ~ )/''l2 j fQr ancj x jn w-tj1 x >x
Let (a ) and (bn) be sequences of real numbers such that an < an+1 < bn+1 < bn for all I n N. Must there exist a real number x such that an < x < bn for all n e N? Justify | your answer. (6)
(b)
Let G be the group of all 2x2 matrices with real entries with respect to matrix multiplication. Let Gx be the smallest subgroup of G containing J and , and
G2 be the smallest subgroup of G containing and . Determine all elements
of Gx and find their orders. Determine all elements of G2 and find their orders. Does there exist a one-to-one homomorphism from Gj onto G2? Justify. (15)
(b) Consider the ring R = { a + ib : a,beZ} with usual addition and multiplication. Find all invertible elements of R . (6)
29. (a) Suppose E is a non-empty subset of R which is bounded above, and let a - supE.
If E is closed, then show that a eE . If E is open, then show that a&E . (9)
(b) Find all limit points of the set E - In : n, m e N }. (6)
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