. n x dx 3 n n is equal to : 0 4. Let f n (x) = x n n, for each n. Then the series (A) 3 n n (B) n 3 n (n ) (C) n n (n ) (D) n 3 n (n ) f n is n (A) Uniformly convergent on [0, ] (B) Po intwise convergent but is not uniformly convergent on [0, ] (C) Point wise convergent on [0, ] (D) Not point wise convergent on [0, ]. Let (X, B) be a measurable space. Let f be an extended real-valued measurable function on X. Then (A) f + is measurable f is measurable (B) f is measurable f is measurable (C) f is measurable f is measurable (D) f is measurable f is measurable 3. Which of the following series is convergent? 5. Define f on [0, ] by, when x is irrational f(x) = 0, when x is rational Then (A) n n n (A) f is Riemann integrable but is not Lebesgue integrals (B) n sin n n (B) f is Lebesgue integrable but is not Reimann integrable (C) (D) n n n ( ) log n log n n (C) f is both Riemann integrable and Lebesgue integrable (D) f is neither Riemann integrable nor Lebesgue integrable 3
6. Let x and x be two non-negative real number 8. The field of rational numbers is such that x < x. Define x n+ = (x n x n ) for n. Let S = x, x 3, x 5,... and S = x, x 4, x 6,.... Then (A) Both subsequences S and S are increasing (B) Both subsequences S and S are decreasing (C) Subsequence S is incresing and subsequence S is decreasing (A) Co mplete o rdered field with Archimedean property (B) Complet e ordered field without Archimedean property (C) Incomplete ordered field with Archimedean property (D) Incomplete ordered field witho ut Archimedean property (D) Subsequence S is decreasing and subsequence S is increasing 9. For x, y R define d (x, y) = (x y), d (x, y) = x y, d 3 (x, y) = x y and d 4 (x, y) = x y. Then 7. Suppose f : [, ] R is continuous such that f( ) = and f() =. Then (A) None of d, d, d 3 is a metric but d 4 is a metric (A) f(x) = 0 for at least one value of x in [, ] (B) f(x) = 0 for atmost one value of x in [, ] (C) f(x) = 0 for at least one value of x in [, 0] (B) None of d, d 3, d 4 is a metric but d is a metric (C) None of d, d 3, d 4 is a metric but d is a metric (D) f(x) = 0 for atleast one value of x in [0, ] (D) None of d, d, d 3, d 4 is a metric 4
0. If A and B are two square matrices of order n such that : AB + A = I AB, then (A) B is non-singular (B) A is non-singular (C) Both A and B are non-singular (D) Neither A nor B is non-singular 3. If T is a positive linear transformation on a finite dimensional complex inner product space such that trace (T) = 0 then (A) T = 0 (B) Trace (T ) > 0 (C) T n = I for some natural number n (D) T n 0 for any natural number n 4. Let f(x) be the minimal polynomial of the following matrix. Which of the following is not a linear transformation from (A) T(x, x ) = (x, x ) (B) T(x, x ) = ( + x, x ) R to R? (C) T(x, x ) = (x + x, x x ) (D) T(x, x ) = (x, 0) 0 0 0 0 0 0 A 0 0 0 0 0 0 Then rank of the matrix f(a) is (A) 0 (B) (C) (D) 4. Let V = then 3 R and S = {(, 0, 0), (0,, 0), (0, 0, )}, (A) S is a basis for V (B) S is linearly independent but is not a basis (C) S is basis but not linearly independent (D) S is linearly dependent 5. Let W and W be two subspaces of a vector space such that dim (W ) = 3, dim (W ) = 4 and 4 < dim (W + W ) < 7. Then dim (W W ) must be (A) 0 or 7 (B) 3 or 4 (C) or (D) or 3 5
6. The bilinear transformationwhich maps z = 0, i, into, i, 0 respectively, is (A) W = z 9. The number of primitive 9th roots of is (A) 8 (B) 5 (B) W = z (C) W = z (C) 6 (D) 7 (D) W = z 0. Let f(z) = e z. Then 7. Let S = Then S = (A) 64 (B) 646 3 0 q q (3p ) sin cos p q 4p (A) f(z) has a removable singularity at 0 (B) f(z) has a pole of order at 0 (C) f(z) has essential singularity at 0 (D) None of the above (C) 648 (D) 658 8. The function f(z) = z Re (z) is (A) Differentiable nowhere (B) Differentiable everywhere (C) Differentiable only at z = 0 (D) Differentiable everywhere except at z = 0. The number of reflexive relations on a set with 8 elements is equal to (A) 64 (B) 56 (C) 48 (D) 40 6
. Let Q and R respectively denote field of rational numbers and field of real numbers. Then the field extension R over Q is (A) Algebraic and finite (B) Algebraic but not finite (C) Finite but not algebraic (D) Neither finite nor algebraic 5. Let R and S be two equivalence relations on a set X. Then (A) R S is an equivalence relation (B) R O S is an equivalence relation (C) R S is an equivalence relation (D) None of the above 3. Which of the following is correct? (A) There is a field with 35 elements 6. Let X = {a, b, c}. Which of the followings is not a topology on X? (A) {, X, {a}, {b, c}} (B) There is a field with 48 elements (C) There is a field with 64 elements (D) There is a field with 80 elements (B) {, X, {a, b}, {b, c}} (C) {, X, {a, b}, {b, c}, {b}} (D) {, X, {a}, {a, b}} 4. Let F be a field extention of E of degree and K be a field extension of F of degree 0. Then K is a field extension of E of degree (A) (B) (C) 60 7. Let X be discrete topological space and Y indiscrete space. Then (A) Every function on X into Y is continuous (B) Every function on Y into X is continuous (C) Every function on X into Y is open (D) 0 (D) Every function on Y into X is closed 7
8. Consider the initial value problem (IVP) dy y, y(0), (x, y) IR IR dx Then there exists a unique solution of the IVP on (A) (, ) (B) (, ) (C) (, ) (D) (, ) 9. The differential equation of the family of circles touching the y-axis at the origin is 30. A homogeneous linear differential equation with real constant coefficients, which has y = xe 3x cos x + e 3x sin x as one of its solutions, is given by (A) (D + 6D + 3)y = 0 (B) (D 6D + 3)y = 0 (C) (D 6D + 3) y = 0 (D) (D + 6D + 3) y = 0 3. The partial differential equation representing the set of all spheres of unit radius with center in the plane xoy is (A) + p + q = 0 (B) yp xq = 0 (C) z ( + p + q ) = (D) None of these (A) dy y x dx xy Where p = z z and q x y (B) dy y x dx xy (C) dy xy dx y x (D) dy xy dx y x 3. The singular solut ion of t he partial differential equation z = px + qy + pq is (A) z = 0 (B) z = ax + by + ab (C) z xy = 0 (D) z + xy= 0 z z Where p = and q and a, b are x y arbitrary constants. 8
33. The partial differential equation : u u u sin x sin x cos x x xy y u u sin x cos y 4u 0 x y (A) Circular (B) Parabolic (C) Elliptic (D) Hyperbolic is 36. The solutio n of initial value problem y' = dy dx = f(x, y), where y(x 0 ) = y 0 by Runge- Kuttu fourth order method is y y (K K K K ) 6 (A) n n 3 4 y y (K K K K ) 6 (B) n n 3 4 y y (K K K K ) 6 (C) n n 3 4 34. The rate of convergence of Newton-Raphson method is of (A) First - order (B) Second - order (C) Third - order (D) None of these 35. Set, and E represent forward difference, backward difference and shift operators respectively. Then which of the following is incorrect? (A) E = + (B) = ( ) (C) = E (D) = ( + ) 9 y y (K K K 3K ) 6 (D) n n 3 4 In all these expressions : h K = hf(x 0, y 0 ), K = hf x 0, y0 K, K 3 = hf x0 h, y0 K, K 4 = hf(x 0 + h, y 0 + k 3 ) 37. The shape which has maximum area for a given perimeter is (A) Circle (B) Triangle (C) Square (D) Rectangle
38. The value of for which the homogeneous Fredholm integral equat ion u(x) = x t 0 e.e u(t) dt has a non trivial solution is (A) e (B) e 40. The Hamiltonian for the simple harmonic oscillator, which consists of a mass m moving in a quadratic pot ent ial field with characteristic coefficient k, is (A) H = (B) H = (C) p kx m p kx m p H kx m (C) e (D) p H kx m (D) e 39. Let T be kinetic energy and V a potential function, then T V is called 4. To compare the variability between two series having different unit of measurements, the measure usually used is (A) Coefficient of variation (B) Standard deviation (C) Mean deviation (D) Inter quartile range (A) Euler function (B) Hermite function (C) Lagrange s function (D) Hamilton function 4. Which one of the following distribution is said have memoryless property in a certain sense? (A) Binomial (B) Poisson (C) Exponential (D) Cauchy 0
43. Which one of the following inequalities is a very st rong too l fo r showing weak convergence (A) Markov s inequality (B) Boole s inequality (C) Schwartz s inequality (D) Chebychev s inequality 44. A non-parametric tests in which ranks are not used (A) Mann-Whitney U-test (B) Kruskal-Wallis test (C) Kolmogorov-Smirnov test (D) Friedman test 45. Convergence with probability one does not mean (A) Strong convergence (B) Almost sure convergence (C) Almost everywhere convergence (D) Convergence in mean square 46. From a moment generating function (mgf), one can obtain moments of various order, by using the coefficients of (A) Taylor series expansion (B) Log series expansion (C) Maclaurin series expansion (D) None of the above 47. The graph of a Negative Binomial distribution is (A) Negatively skewed and Mesokurtic (B) Negatively skewed and Platykurtic (C) Positivelyskewed and Leptokurtic (D) Positively skewed and Mesokurtic 48. Let (t) be the characteristic function of a random variable, then which statement is true (A) (t) is continues everywhere (B) (t) is defined in every finite t interval (C) ( t ) ( t) (D) All the statements are true 49. The maximum likelihood estimates (MLE) are generally (A) Consistent and invariant (B) Unbiased and consistent (C) Unbiased and inconsistent (D) Unbiased and invariant 50. Which statement(s) about the standard error (S.E.) is true (A) S.E. helps in obtaining the probabilistic limits (B) The index for the precision of the estimate of a parameter is given by the magnitude of S.E. (C) The reciprocal of SE is the meausre of reliability of the sample (D) All the above are true
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