#1
19th August 2014, 03:08 PM
| |||
| |||
Previous Papers for Integrated Ph.D Mathematics entrance exam
Please provide me question paper for IISC Integrated Ph.D Mathematics entrance examination ?
|
#2
20th August 2014, 08:30 AM
| |||
| |||
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 |