A) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
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1 M.A./M.Sc. (Mathematics) Etrace Examiatio Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks will be give. For wrog aswer, 1 mark will be deducted. Scietific calculators are allowed. I the followig R, N, Q, Z ad C deote the set of all real umbers, atural umbers, ratioal umbers, itegers ad complex umbers respectively. (1) Let X be a coutably ifiite subset of R ad A be a coutably ifiite subset of X. The the set X \ A = {x X x / A} A) is empty. B) is a fiite set. C) ca be a coutably ifiite set. D) ca be a ucoutable set. () The subset A = {x Q : x < 4} of R is A) bouded above but ot bouded below. B) bouded above ad sup A =. C) bouded above but does ot have a supremum. D) ot bouded above. (3) Let f be a fuctio defied o [0, ) by f(x) = [x], the greatest iteger less tha or equal to x. The A) f is cotiuous at each poit of N. B) f is cotiuous o [0, ). C) f is discotiuous at x = 1,, 3,.... D) f is cotiuous o [0, 7]. (4) The series x + x + 33 x 3 + is coverget if x belogs to the iterval! 3! A) (0, 1/e). B) (1/e, ). C) (/e, 3/e). D) (3/e, 4/e). (5) The subset A = {x Q : 1 < x < 0} N of R is A) bouded, ifiite set ad has a limit poit i R. B) ubouded, ifiite set ad has a limit poit i R. C) ubouded, ifiite set ad does ot have a limit poit i R. D) bouded, ifiite set ad does ot have a limit poit i R. (6) Let f be a real-valued mootoe o-decreasig fuctio o R. The A) for a R, lim f(x) exists. x a B) f is a ubouded fuctio. 1
2 M.A./M.Sc. Admissio Etrace Test C) h(x) = e f(x) is a bouded fuctio. D) if a < b, the lim f(x) lim f(x). x a + x b (7) Let X = C[0, 1] be the space of all real-valued cotiuous fuctios o [0, 1]. The (X, d) is ot a complete metric space if 1 A) d(f, g) = f(x) g(x) dx. B) d(f, g) = max f(x) g(x). 0 x [0,1] { 0, if f = g C) d(f, g) = sup f(x) g(x). D) d(f, g) = x [0,1] 1, if f g. (8) The series k=0 k + 3k + 1 (k + )! coverges to A) 1. B) 1/. C). D) 3. (9) We kow that xe x = =1 x ( 1)!. The series =1! coverges to A) e. B) e. C) 4 e. D) 6 e. (10) Let X = R with metric defied by d(x, y) = 1 if x y ad d(x, x) = 0. The A) every subset of X is dese i (X, d). B) (X, d) is separable. C) (X, d) is compact but ot coected. D) every subspace of (X, d) is complete. (11) Let d 1 ad d be metrics o a o-empty set X. Which of the followig is ot a metric o X? A) max(d 1, d ). B) d 1 + d. C) 1 + d 1 + d. D) 1 4 d d. (1) Let f : R R be give by f(x, y) = xy. The at origi A) f is cotiuous ad f exists. x B) f is discotiuous ad f exists. x C) f is cotiuous but f does ot exist. x D) f is discotiuous but f exists. x (13) The sequece of real-valued fuctios f (x) = x, x [0, 1] {}, is A) poitwise coverget but ot uiformly coverget. B) uiformly coverget.
3 M.A./M.Sc. Admissio Etrace Test C) bouded but ot poitwise coverget. D) ot bouded. (14) The itegral 0 si x dx A) exists ad equals 0. B) exists ad equals 1. C) exists ad equals 1. D) does ot exist. (15) If {a } is a bouded sequece of real umbers, the A) if B) lim if C) lim if D) if (16) The series a lim if a if a if a lim if =1 a ad sup a. ()! (!) a ad sup a ad lim sup a lim sup a. a lim sup a. a sup a. A) diverges. B) coverges to 1. C) coverges to 1. D) coverges to 1. 7 (17) Cosider the fuctio f : R R defied by xy f(x, y) = x + y, x y 0, x = y. The A) directioal derivative does ot exist at (0, 0). B) f is cotiuous at (0, 0). C) f is differetiable at (0, 0). D) each directioal derivative exists at (0, 0) but f is ot cotiuous. (18) Let f : [ 1, 1] R be defied by x f(x) = x, x 0 0, x = 0 ad F be its idefiite itegral. Which of the followig is ot true? A) F (0) does ot exist. B) F is a ati-derivative of f o [ 1, 1]. C) F is Riema itegrable o [ 1, 1].
4 4 M.A./M.Sc. Admissio Etrace Test D) F is cotiuous o [ 1, 1]. (19) Let f(x) = x, x [0, 1]. For each N, let P be the partitio of [0, 1] give by P = {0, 1,,..., }. If α = U(f, P ) (upper sum) ad β = L(f, P ) (lower sum) the A) α = ( + )( + 1)/(6 ). B) β = ( )( + 1)/(6 ). C) β = ( 1)( 1)/(6 ). D) lim α lim β. (0) Let I = π/ 0 log si x dx. The A) I diverges at x = 0. B) I coverges ad is equal to π log. C) I coverges ad is equal to π log. D) I diverges at x = π. 4 (1) Which of the followig polyomials is ot irreducible over Z? A) x x + 5x + 5. B) x 3 + 6x + 1. C) x 3 + x + 1. D) x 4 + x 3 + x + x + 1. () A complex umber α is said to be algebraic iteger if it satisfies a moic polyomial equatio with iteger coefficiets. Which of the followig is ot a algebraic iteger? A). B) 1. C) 1 5. D) α, α is a algebraic iteger (3) If A = 0 1, the the value of A 4 A 3 4A + 4I is A) B) C) D)
5 M.A./M.Sc. Admissio Etrace Test (4) Let T : R R 3 be a liear trasformatio defied by T (x, y) = (x + y, x y, y). If {(1, 1), (1, 0)} ad {(1, 1, 1), (1, 0, 1), (0, 0, 1)} are ordered bases of R ad R 3 respectively, the the matrix represetatio of T with respect to the ordered bases is A) C) B). D) (5) Let P 4 be real vector space of real polyomials of degree 4. Defie a ier product o P 4 by a i x i, b i x i = a i b i. i=0 The the set {1, x, x, x 3, x 4 } is i=0 A) orthogoal but ot orthoormal. B) orthoormal. C) ot orthogoal. D) oe of these. (6) If {a + ib, c + id} is a basis of C over R, the i=0 A) ac bd = 0. B) ac bd 0. C) ad bc = 0. D) ad bc 0. ( ) 1 0 (7) Cosider M 1 =, M 3 = ( ) 0 0 of M 0 0 (R). The ( ) ( ) 5 6, M 3 = ad M 3 4 = A) {M, M 3, M 4 } is liearly idepedet. B) {M 1, M, M 4 } is liearly idepedet. C) {M 1, M 3, M 4 } is liearly idepedet. D) {M 1, M, M 3 } is liearly depedet. ( ) 1 1 (8) If A = = αm βm + γm 3, where M 1 = I, M = ( ) 1 1 M 3 =, the 1 1 ( ) ad
6 6 M.A./M.Sc. Admissio Etrace Test A) α = β = 1, γ =. B) α = β = 1, γ =. C) α = 1, β = 1, γ =. D) α = 1, β = 1, γ =. (9) Let W be the subset of the vector space V = M (R) cosistig of symmetric matrices. The A) W is ot a subspace of V. B) W is a subspace of V of dimesio ( 1). C) W is a subspace of V of dimesio (+1). D) W is a subspace of V of dimesio. (30) Let T : R 3 R 3 be a liear trasformatio ad B be a basis of R 3 give by B = {(1, 1, 1) t, (1,, 3) t, (1, 1, ) t }. If T ( (1, 1, 1) t) = (1, 1, 1) t, T ( (1,, 3) t) = ( 1,, 3) t ad T ( (1, 1, ) t) = (,, 4) t (A t beig the traspose of A), the T ( (, 3, 6) t) is A) (, 1, 4) t. B) (1,, 4) t. C) (3,, 1) t. D) (, 3, 4) t. (31) Let T : R 3 R 3 be a liear trasformatio ad B = {v 1, v, v 3 } be a basis for R 3. Suppose that T (v 1 ) = (1, 1, 0) t, T (v ) = (1, 0, 1) t ad T (v 3 ) = (, 1, 1) t the A) w = (1,, 1) t / Rage of T. B) dim (Rage of T ) = 1. C) dim (Null space of T ) =. D) Rage of T is a plae i R 3. (3) The last two digits of the umber 9 (99) is A) 9. B) 89. C) 49. D) 69. ( ) a b (33) Let G be the group of all matrices uder matrix multiplicatio, c d where ad bc 0 ad a, b, c, d are itegers modulo 3. The order of G is A) 4. B) 16. C) 48. D) 81. (34) For the ideal I = x + 1 of Z[x], which of the followig is true? A) I is a maximal ideal but ot a prime ideal. B) I is a prime ideal but ot a maximal ideal. C) I is either a prime ideal or a maximal ideal. D) I is both prime ad maximal ideal. (35) Cosider the followig statemets:
7 M.A./M.Sc. Admissio Etrace Test Every Euclidea domai is a pricipal ideal domai;. Every pricipal ideal domai is a uique factorizatio domai; 3. Every uique factorizatio domai is a Euclidea domai. The A) statemets 1 ad are true. B) statemets ad 3 are true. C) statemets 1 ad 3 are true. D) statemets 1, ad 3 are true. (36) The ordiary differetial equatio: dy dx = y x with the iitial coditio y(0) = 0, has A) ifiitely may solutios. B) o solutio. C) more tha oe but oly fiitely may solutios. D) uique solutio. (37) Cosider the partial differetial equatio: 4 u x + 1 u x y + u 9 9u = 9. y Which of the followig is ot correct? A) It is a secod order parabolic equatio. B) The characteristic curves are give by ζ = y 3x ad η = y. C) The caoical form is give by u u = 1, where η is a characteristic variable. η D) The caoical form is u + u = 1, where η is a characteristic variable. η (38) Cosider the oe dimesioal wave equatio: u t = u 4 x, x > 0, t > 0 subject to the iitial coditios: ad the boudary coditio: ( The u π, π ) is equal to 4 u(x, 0) = si x, x 0 u t (x, 0) = 0, x 0 u(0, t) = 0, t 0.
8 8 M.A./M.Sc. Admissio Etrace Test A) 1. B) 0. C) 1. D) 1. (39) The iitial value problem has x dy dx = y + x, x > 0, y(0) = 0 A) ifiitely may solutios. B) exactly two solutios. C) a uique solutio. D) o solutio. (40) I a tak there is 10 litres of brie (salted water) cotaiig a total of 50 kg of dissolved salt. Pure water is allowed to ru ito the tak at the rate of 3 litres per miute. Brie rus out of the tak at the rate of litres per miute. The istataeous cocetratio i the tak is kept uiform by stirrig. How much salt is i the tak at the ed of oe hour? A) kg. B) kg. C) kg. D) kg. (41) If the differetial equatio t d y dt + tdy dt 3y = 0 is associated with the boudary coditios y(1) = 5, y(4) = 9, the y(9) = A) B) 13.. C) 19. D) (4) The third degree hermite polyomial approximatio for the fuctio y = y(x) such that y(0) = 1, y (0) = 0, y(1) = 3 ad y (1) = 5 is give by A) 1 + x + x 3. B) 1 + x 3 + x. C) x + x 3. D) oe of the above. (43) Let y be the solutio of the iitial value problem dy = y x, y(0) =. dx Usig Ruge-Kutta secod order method with step size h = 0.1, the approximate value of y(0.1) correct to four decimal places is give by A) B).714. C) D).7716.
9 M.A./M.Sc. Admissio Etrace Test (44) Cosider the system of liear equatios 1 0 x x 0 1 x 3 = With the iitial approximatio [x (0) 1, x (0), x (0) 3 ] T = [0, 0, 0] T, the approximate value of the solutio [x (1) 1, x (1), x (1) 3 ] T after oe iteratio by Gauss Seidel method is A) [3.,.5, 1.5] T. B) [3.5,.5, 1.65] T. C) [.5, 3.5, 1.65] T. D) [.5, 3.5, 1.6] T. (45) For the wave equatio the characteristic coordiates are u tt = 16 u xx, A) ξ = x + 16 t, η = x 16 t. B) ξ = x + 4 t, η = x 4 t. C) ξ = x + 56 t, η = x 56 t. D) ξ = x + t, η = x t. (46) Let f 1 ad f be two solutios of a 0 (x) d y dx + a 1(x) dy dx + a (x)y = 0, where a 0, a 1 ad ( a) are cotiuous ( ) o [0, 1] ad a 0 (x) 0 for all x [0, 1]. 1 1 Moreover, let f 1 = f = 0. The A) oe of f 1 ad f must be idetically zero. B) f 1 (x) = f (x) for all x [0, 1]. C) f 1 (x) = c f (x) for some costat c. D) oe of the above. (47) The Laplace trasform of e 4 t is A) 1/(s + ). B) 1/(s ). C) 1/(s + 4). D) 1/(s 4). (48) Let f(t) = 4 si t ad let =0 a cos t be the Fourier cosie series of f(t). Which oe is true? A) a 0 = 0, a = 1, a 4 =. B) a 0 =, a = 0, a 4 =. C) a 0 =, a =, a 4 = 0. D) a 0 = 0, a =, a 4 =
10 10 M.A./M.Sc. Admissio Etrace Test (49) For a, b, c R, if the differetial equatio is exact, the (ax + bxy + y )dx + (x + cxy + y )dy = 0 A) b =, c = a. B) b = 4, c =. C) b =, c = 4. D) b =, a = c. (50) Let u(x, t) be the solutio of the wave equatio The the value of u(1, 1) is u xx = u tt, u(x, 0) = cos (5πx), u t (x, 0) = 0. A) 1. B) 0. C). D) 1.
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