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20th August 2014, 08:30 AM
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Join Date: Apr 2013
Re: Previous Papers for Integrated Ph.D Mathematics entrance exam

Here I am giving you question paper for IISC Integrated Ph.D Mathematics entrance examination in a file attached with it so you can get it easily.

1. Consider the following system of linear equations.
x + y + z + w = b1:
x _ y + 2z + 3w = b2:
x _ 3y + 3z + 5w = b3:
x + 3y _ w = b4:
For which of the following choices of b1; b2; b3; b4 does the above system have a
solution?
(A) b1 = 1; b2 = 0; b3 = _1; b4 = 2:
(B) b1 = 2; b2 = 3; b3 = 5; b4 = _1:
(A) b1 = 2; b2 = 2; b3 = 3; b4 = 0:
(A) b1 = 2; b2 = _1; b3 = _3; b4 = 3:
2. Let y : [0; 1] ! R be a twice continuously di_erentiable function such that,
d2y
dx2 (x) _ y(x) < 0; for all x 2 (0; 1); and y(0) = y(1) = 0:
Then,
(A) y has at least two zeros in (0; 1):
(B) y has at least on zero in (0; 1):
(C) y(x) > 0 for all x 2 (0; 1):
(D) y(x) < 0 for all x 2 (0; 1):
3. Which one of the following boundary value problems has more than one solution?
(A) y00 + y = 1; y(0) = 1; y(_=2) = 0:
(B) y00 + y = 1; y(0) = 0; y(2_) = 0:
(C) y00 _ y = 1; y(0) = 0; y(_=2) = 0:
(D) y00 _ y = 1; y(0) = 0; y(_) = 0:
4. Let A be an n _ n nonsingular matrix such that the elements of A and A_1 are
all integers. Then,
3
(A) detA must be a positive integer.
(B) detA must be a negative integer.
(C) detA can be +1 or _1:
(D) detA must be +1:
5. Let Q be a polynomial of degree 23 such that Q(x) = _Q(_x) for all x 2 R with
jxj _ 10: If R1
_1 (Q(x) + c) dx = 4 then c equals
(A) 0:
(B) 1:
(C) 2:
(D) 4:
6. Let b > 0 and x1 > 0 be real numbers. Then the sequence fxng1n=1 de_ned by
xn+1 =
1
2 _xn +
b
xn_
(A) diverges.
(B) converges to px1:
(C) converges to p(b + x1):
(D) converges to pb:
7. Let f(x) = (3x
4 if x 2 Q:
sin x if x 2 R n Q:
Then the number of points where f is continuous equals
(A) 1:
(B) 2:
(C) 3:
(D) 1:
8. Let f : R ! R be a continuous function satisfying, f(x) = 5 Rx
0 f(t) dt+1; 8x 2 R: Then f(1) equals
(A) e5:
(B) 5:
(C) 5e:
4
(D) 1:
9. Let f : R ! R be a continuous function and let g(x) = Rx2+3x+2
0 f(t) dt: Then,
g0(0) equals
(A) 3f(2):
(B) f(2):
(C) 3f(0):
(D) f(0):
10. Let xn > 0 be such that P1n=1 xn diverges and P1n=1 x2
n converges. Then xn
cannot be
(A) n
n2+1:
(B) log n
n :
(C) 1
nplog n:
(D) 1
n(log n)2 :
11. If B is a subset of R3 and u 2 R3; de_ne B _ u = fw _ u : w 2 Bg: Let A _ R3;
be such that tu + (1 _ t)v 2 A whenever u; v 2 A and t 2 R: Then,
(A) A must be a straight line.
(B) A must be a line segment.
(C) A _ u0 is a subspace for a unique u0 2 A:
(D) A _ u is a subspace for all u 2 A:
12. Minimum value of jz + 1j + jz _ 1j + jz _ ij for z 2 C is
(A) 2:
(B) 2p2:
(C) 1 + p3:
(D) p5:
13. The minimum value of jz_wj where z;w 2 C such that jzj = 11; and jw+4+3ij =
5 is
(A) 1:
(B) 2:
(C) 5:
5
(D) 6:
14. Let P be the vector space of polynomials with real coe_cients. Let T and S be
two linear maps from P to itself such that T _ S is the identity map. Then,
(A) S _ T may not be the identity map.
(B) S _ T must be the identity map, but T and S need not be the identity maps.
(C) T and S must both be the identity map.
(D) There is a scalar _ such that T(p) = _p for all p 2 P:
15. Let `1 and `2 be two perpendicular lines in R2: Let P be a point such that the
sum of the distances of P from `1 and `2 equals 1: Then the locus of P is
(A) a square.
(B) a circle.
(C) a straight line.
(D) a set of four points.
16. Let 0 < b < a: A line segment AB of length b moves on the plane such that A
lies on the circle x2 + y2 = a2: Then the locus of B is
(A) a circle.
(B) union of two circles.
(C) a region bounded by two concentric circles.
(D) an ellipse, but not a circle.
17. Let u; v and w be three vectors in R3: It is given that u_u = 4; v _v = 9; w_w = 1;
u _ v = 6; u _ w = 0 and v _ w = 0: Then the dimension of the subspace spanned
by fu; v;wg is
(A) 1:
(B) 2:
(C) 3:
(D) cannot be determined.
18. Let an be the number of ways of arranging n identical black balls and 2n identical
white balls in a line so that no two black balls are next to each other. Then an
equals
(A) 3n:
6
(B) _2n+1
n _.
(C) _2n
n _.
(D) _ 2n_1
n(2n+1)_.
20. Let ak = 1
22k _2k
k _ ; k = 1; 2; 3; _ _ _ : Then
(A) ak is increasing.
(B) ak is decreasing.
(C) ak decreases for _rst few terms and then increases.
(D) none of the above.
21. What is the limit of (2n + 3n + 4n)
1
n as n ! 1 ?
(A) 0:
(B) 1:
(C) 3:
(D) 4:
22. What is the limit of e_2n Pn
k=0
(2n)k
k! as n ! 1?
(A) 0:
(B) 1:
(C) 1=e:
(D) e:
23. Let f; g : [_1; 1] ! R be odd functions whose derivatives are continuous. You
are given that jg(x)j < 1 for all x 2 [_1; 1], f(_1) = _1, f(1) = 1 and that
f0(0) < g0(0). Then the minimum possible number of solutions to the equation
f(x) = g(x) in the interval [_1; 1] is
7
(A) 1:
(B) 3:
(C) 5:
(D) 7:
24. Let f : S3 ! Z6 be a group homomorphism. Then the number of elements in
f(S3) is
(A) 1:
(B) 1 or 2:
(C) 1 or 3:
(D) 1 or 2 or 3:
25. Consider the multiplicative group S = fz : jzj = 1g _ C: Let G and H be
subgroups of order 8 and 10 respectively. If n is the order of G \ H then
(A) n = 1:
(B) n = 2:
(C) 3 _ n _ 5:
(D) n _ 6:
26. Let G be a _nite abelian group. Let H1 and H2 be two distinct subgroups of G
of index 3 each. Then the index of H1 \ H2 in G is
(A) 3:
(B) 6:
(C) 9:
(D) Cannot be computed from the given data.
27. A particle follows the path c : [__
2 ; _
2 ] ! R3; c(t) = (cos t; 0; j sin tj): Then the
distance travelled by the particle is
(A) 3_
2 :
(B) _:
(C) 2_:
(D) 1:
28. Let T : R3 ! R4 be the map given by
T(x1; x2; x3) = (x1 _ 2x2; x2 _ 2x3; x3 _ 2x1; x1 _ 2x3):
Then the dimension of T(R3) equals
(A) 1:
(B) 2:
(C) 3:
(D) 4:
29. The tangent plane to the surface z2 _ x2 + sin(y2) = 0 at (1; 0;_1) is
(A) x _ y + z = 0:
(B) x + 2y + z = 0:
(C) x + y _ 1 = 0:
(D) x + z = 0:
30. Let A and B be two 3_3 matrices with real entries such that rank(A) = rank(B) =
1: Let N(A) and R(A) stand for the null space and range space of A: De_ne N(B)
and R(B) similarly. Then which of the following is necessarily true ?
(A) dim(N(A) \ N(B)) _ 1:
(B) dim(N(A) \ R(A)) _ 1:
(C) dim(R(A) \ R(B)) _ 1:
(D) dim(N(A) \ R(A)) _ 1:
31. For a permutation _ of f1; 2; _ _ _ ; ng; we say that k is a _xed point if _(k) = k:
Number of permutations in S5 having exactly one _xed point is
(A) 24:
(B) 45:
(C) 60:
(D) 96:
32. Let A = f1; 2; _ _ _ ; 10g: If S is a subset of A, let jSj denote the number of elements
in S: Then X S_A;S6=_
(_1)jSj
equals
9
(A) _1:
(B) 0:
(C) 1:
(D) 10:
33. Let Pm be the vector space of polynomials with real coe_cients of degree less
than or equal to m: De_ne T : Pm ! Pm by T(f) = f0 + f: Then the dimension
of range(T) equals
(A) 1
(B) (m _ 1):
(C) m:
(D) (m + 1):
34. Let A and B be two _nite sets of cardinality 5 and 3 respectively. Let G be the
collection of all mappings f from A into B such that the cardinality of f(A) is 2:
Then, cardinality of G equals
(A) 3 _ 25 _ 6:
(B) 3 _ 25:
(C) 3 _ 52:
(D) 1
2(35 _ 3):
35. Let G be the group Z2 _Z2 and let H be the collection of all isomorphisms from
G onto itself. Then the cardinality of H is
(A) 2:
(B) 4:
(C) 6:
(D) 8:
36. A line L in the XY -plane has intercepts a and b on X-axis and Y -axis respectively.
When the axes are rotated through an angle _ (keeping the origin _xed), L makes
equal intercepts with the axes. Then tan _ equals
(A) a_b
a+b :
(B) a_b
2(a+b) :
(C) a+b
a_b :
10
(D) a2_b2
a2+b2 :
37. Let B1;B2 and B3 be three distinct points on the parabola y2 = 4x: The tangents
at B1;B2 and B3 to the parabola (taken in pairs) intersect at C1;C2 and C3: If a
and A are the areas of the triangles B1B2B3 and C1C2C3 respectively, then
(A) a = A:
(B) a = 2A:
(C) 2a = A:
(D) a = p2A:
38. Let P be a 3 _ 2 matrix, Q be a 2 _ 2 matrix and R be a 2 _ 3 matrix such that
PQR is equal to the identity matrix. Then,
(A) rank of P = 2:
(B) Q is nonsingular.
(C) Both (A) and (B) are true.
(D) There are no such matrices P;Q and R:
39. The number of elements of order 3 in the group Z15 _ Z15 is
(A) 3:
(B) 8:
(C) 9:
(D) 15:
40. The number of surjective group homomorphisms from Z to Z3 equals
(A) 1:
(B) 2:
(C) 3:
(D) 1:
11


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