1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0

Transcription

1 4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for which the quadratic equation (x a) (x ) = has integral roots B) 8 or 8 D) and 8 4. The system of linear equations 3x + y + z = 3, x + y + z =, 6x + y + 4z = 6 is Inconsistent B) Consistent and has a unique solution Consistent and has infinite number of solutions D) Consistent and has three solutions 5. If and β are the roots of the equation x ax + b =, then the equation whose roots are α and β is x ax + 4b = B) x ax + b = x ax + b = D) x ax + b = 6. If n = mc, then the value of nc is (m + ) C 4 B) 3((m ) C 4 ) (m + ) C 4 D) 3((m + ) C 4 ) 7. If A = {,, 3, 4}, then the number of mappings from A into A whose range set contains two or more elements is 56 B) 5 6 D) 4 8. The domain of y = sin x - 3 log (4 x) is (, 4) B) [, 4) (, 3) D) [3, 4] 9. e e The inverse of the function y x -x e - e is given by log (x -) B) log ( - x) e e x x log e D) log x - e x x -x

2 i. If A is the matrix then A 5 is - i I B) I A D) A. The points (, ) and (5, ) are two vertices of a rectangle. If the other two vertices lie on the straight line y = 3x + c, then the value of c is 5 B) 3 D) 7. The number of common tangents to the circles x + y + x y + = and x + y x y + = is B) 3 D) 3. The area bounded by the curves x =, y = and x + 5y = in square units is B) D) 5 4. If x + y z = 3 and 3x +4y + 5z = 3 are two planes P andp respectively, then x - y - 3 z - 4 the line Is parallel to P but not perpendicular to P B) Is perpendicular to P but not parallel to P Is parallel to P and is perpendicular to P D) Is perpendicular to P and is parallel to P 5. The centre and radius of the sphere x + y + z x + 6y + 4z 35 = are (, 3, ) and 7 B) (, 3, ) and 5 (, 3, ) and 7 D) (, 3, ) and 7 6. If the line y = x + c is a tangent to the parabola y = 4ax, then a + c = B) ac = c = a D) a = c 7 5 B) e D) 8. If xe xy = y + sin dy x, then at (, ) is equal to dx B) D) e 9. The area included between the curves y = e x and y = e x and the lines x = and x = is - e + e (e e ) - (e e ) B) D) e e-

3 4. The value of the integral x 3 - x 7 B) 8 D) 4 dx is. lim n n is n B) e D). The radius of convergence of the power series n B) n! z n! is D) 3. Let the function f be defined on the set of real numbers Rby 3 x,for x f (x) = x, for - x, otherwise Then the set of points at which f is not differentiable is {,, } B) {, } {, } D) {, } 4. The series ( ) n n! n is Absolutely convergent B) Convergent, but not absolutely convergent Divergent and diverges to infinity D) Oscillating 5. Let f be defined on Rby, if x is rational f(x) x, otherwise Then the Lebesgue integral fdµ has the value B) ½ D) 6 Let f and be defined on [, ] by f(x) = x and (x) = x 3. Then the Riemann Stieltjes integral fdα has the value ½ B) 3/5 /3 D) /5 3

4 7. Which of the following is not a property of the Lebesgue measure? Countable sub additivity B) Countable additivity onotonicity D) Non negativity 8. Let E be a subset of R with Lebesgue outer measure zero ie.m*(e) =. Then which of the following statements is not necessarily true? E is measurable B) E is countable If A is any subset of R, then m*(a E) = m*( D) Every subset of E is measurable sinz 9. The residue of at z =, is z B) ½ D) /3 3. The singularity of the function f(z) = /z + sin /z at z =, is A simple pole B) A pole of order An essential singularity D) A removable singularity 3. Suppose f has an isolated singularity at z =. If z = is a simple pole of f, then Is finite and non-zero B) Is zero Is infinity D) Does not exist 3. The value of the integral dz it where γ (t) e, t is z 4z 3 γ πi B) πi πi D) 33. Let γ(t) = 3e it dz, t 4π. Then has the value z γ B) πi πi D) 4πi 34. Let γ be the positively oriented rectangular path with vertices,, +i, i. Then z dz is equal to γ ¼ B) D) /3 35. Which of the following obius transformation maps the open unit disk {z: z < } onto itself z - z - T (z) B) T (z) - z z - 3 3z - z T 3 (z) D) T 4 (z) - z z - 4

5 36. Let f be a non constant analytic function on a region G. Then from the following statements pick the one which is incorrect? f is infinitely differentiable on G B) f(g) is open f attains its maximum on G D) The set of zeros of f in G has no limit point in G. 37. The coefficient of in the Laurent series expansion of f (z) = z z(z -) < z < is B) D) in the region 38. There exists no analytic branch of the logarithm in the region {z : Re z > } B) {z : Im z > } {z : z < } D) {z : z > } 39. Which of the following regions is connected, but not simply connected? {z : z < } B) {z : z > } {z : z + z+ < } D) {z : Re z > } 4. Let f be an analytic function on a region G. Let U = Re f and V = Imf. Then which of the following functions is not necessarily harmonic in G? U B) U U V D) UV 4. Let Z be the ring of integers. Which of the following values of m and n gives mz + nz = Z. m =, n = 6 B) m = 3, n = 6 m = 4, n = 6 D) m = 5, n = 6 4. Let R[x] be the ring of polynomials over the reals. Let f(x) = x + 3 and g(x) = x + 3. Let I be the ideal generated by f(x) and J be the ideal generated by g(x). Then which of the following is true? I J B) J I I + J = R[x] D) I J = ф 43. Let R = Z be the ring of integers mod. Let A be the ideal generated by 4 and B be the ideal generated by 6. Then which of the following holds? R/A is a field B) R/B is a field R/(A+B) is a field D) R/(A B) is a field 44. Which of the following is an irreducible polynomial in Z [x]? x 3 + x + B) x 3 + x + x + x 4 + x + x + D) x 4 + x + 5

6 45. If a, b are divisors of zero in a ring R then which of the following is not necessarily a divisor of zero? a + b B) ab a D) aba 46. Which of the following pairs of fields are isomorphic?, Q Q, Q(i) B) Q 3 Q i, Q - i D) Q i, Q 3 i 47. Let f(x) and g(x) be polynomials of degree 4 over a field F. Then which of the following is true always? deg f(x) + deg. g(x) is 8 B) deg f(x) + deg. g(x) is 4 deg f(x) + deg. g(x) is 3 D) deg f(x) + deg. g(x) is less than or equal to Which of the following is not an algebraic extension? π over Q π Q π over Q 5 over Q D) Q Q B) Q 49. Let E be the splitting field of x 3 over Q. Then [E : Q] = B) 3 4 D) 6 over Q 5. Let {,,, + } be a field of four elements. Then + = B) + D) 5. Let {υ, υ, υ 3, υ 4 } be a linearly independent set in a vector space. Then which of the following is also a linearly independent set? {υ, υ + υ, υ +υ 3 + υ 4,υ +υ 3 +υ 4 } B) {υ, υ + υ, υ +υ 3 + υ 4, υ + υ + υ 4 } {υ + υ, υ +υ 3, υ 3 + υ 4, υ + υ 4 } D) {υ + υ, υ 3 +υ 4, υ + υ 3 + υ 4, υ + υ 3 + υ 4 } 5. Consider the system of equations x + 3y + z + ω = 3x y z + ω = 4x 7y 5z + 3ω =. Then the dimension of the space of solutions is B) D) Which of the following triples of points lie on a straight line? (, ), (, ), (, 3) B) (, ), (, ), (, 4) (, ), (, ), (3, ) D) (, ), (, ), (, ) 6

7 54. Let T : R R be defined by T(x,y) = (x + y, x + y). Then which of the following is a matrix of T with respect to some basis of R? 3 B) D) 55. Which of the following is a diagonalizable matrix? B) - D) 56. Let S 4 be the group of all permutations on four symbols. Let = ( 3) and β = ( ). Then the order of the subgroup generated by and β is B) 3 4 D) The number of homomorphisms from the group Z to the group Z is B) 3 D) Let G be the group of quaternion units. Then the order of the centre of G is B) 4 D) Which of the following is a generator of the group Z 5 x Z? (, 6) B) (, 3) (3, 4) D) (3, 5) 6. The number of mutually non-isomorphic groups of order 49 is B) 3 D) 4 6. Let d denote the sup metric on the set of bounded real valued functions on [, π]. Let f (x) = sin x and g(x) = cos x. Then d (f,g) = B) D) 6. Let A and B be open balls of radius and centre (, ) and (, ) respectively in R x R with usual metric. Then which of the following is a point in A B? 3 3, B),, D), Let ح be the topology on the set R of reals consisting of R, ф and all intervals of the form (a, b) where a < and b >. Then the closure of [, ] in this topology is [,] B) ( -, ] [, ) D) R 7

8 64. Which of the following is a compact subset of the real line?,,,, B) (, ) U (, ) [, ) D) (, ) 65. Let Q be the subspace of rationals of the real line R. Let f : R Q be a continuous function. Then which of the following holds? f () > f () B) f () > f () f () < f () D) f () = f () 66. Which of the following pairs of spaces ishomeomorphic. Here R is the real line and the subsets given are subspaces of R; N is the set of naturals and Z is the set of integers? [, ) and (, ) B) R and R x R R \ {, } and R \ {} D) R \ N and R \ Z 67. Let X = {,, 3, 4, 5} and ح = {X, ф, {,, 3}, {4, 5}}. Let R be the real line. Which of the following is a continuous function f : X R? f () = f () =, f (3) = f (4) = f (5) = B) f () = f () = f (3) =, f (4) = f (5) = f () = f () =, f (3) = f (4) =, f (5) = D) f () =, f () =, f (3) = f (4) = f (5) = 68. Which of the following is not a continuous image of the real line? The open interval (, ) B) The closed interval [, ] The subspace {x: < x <} D) The subspace {x: x > } 69. Let X = {, } be the discrete space and f: R X be defined by if x f (x) if x Let ح be the weak topology on R induced by f. Then which of the following is an?(ح (R, open set in (, ) B) [, ] (, ] D) [, ) 7. Which of the following sequence (x n ) is a cauchy sequence in the metric space R with discrete metric? if n is even if n x n B) x if n is odd n if n x n for all n D) x for all n n n n 8

9 7. Let V be the normed linear space R 3 with Euclidean norm and W= span {(,,), (,,)}. If W and W denote respectively the closure and interior of W, then which one of the following is not true? W W W B) W W ф W W V D) W W W 7. For x = (x(),x())ε R p p, let /p x p x() x(). If E = x : x p is not convex, then the possible value of p from the following is p = B) p = p = D) p = ½ 73. Let X be the normed linear space l and T :X X be defined by x() x(3) T : x (), x(), x(3), x(),,, for x x (), x(), in X. 3 Then which one of the following is not correct? T is continuous at the origin (zero element) B) T is uniformly continuous T is unbounded D) One of the above is not true 74. Let be a closed subspace of a normed linear space X. Then X is a Banach space if X is a Banach space B) is a separable Banach space and X are separable spaces D) and X are Banach spaces 75. Let X be the normed linear space {x ε l : x(j) converges in K} and = span {(,,, ---), (,,, ---), (,,,, ---)}. Let F: X X be the quotient map. If C = {x ε l : x(j) = for all but finitely many j}, then F(C ) is linearly homeomorphic to X B) F(C ) is open in X F(C ) is closed in X D) F() is a nonzero subspace of X 76. Let H be the complex Hilbert space, L int. If x (t) e dt for x ε H and n =, ±, ±, ---, then x = B) x(t) = (t) = for t ε [, π] x(t) = sin t D) x(t) = e int 9

10 77. Let H be the real Hilbert space R, e = (, ) and e = (, ). Let A εbl(h) be - represented by the matrix with respect to the basis { e, e }. If B = {A(e ), A(e )} and B' = {A * (e ), A * (e )} then B is a basis for H while B' is not a basis for H B) B' is an orthonormal basis for H while B is not an orthonormal basis for H Both B and B' are orthonormal basis for H D) Neither B nor B' is an orthonormal basis for H 78. If x, y x y for two vectors x, y in an inner product space X, then x and y are linearly independent B) x and y are linearly dependent x y always D) x y x - y 79. For x, y ina Hilbert space H, if x, x - y 3 and y = 5, then x y is 5 B) 4 7 D) 6 8. If x and y are two vectors in a Hilbert space H such that x y, then which one of the following is not true? x y x y B) x y x y Span {x} Span {y} D) Span {x} Span {y} ***************