Mathematics -Exam Paper 2004 - Entrance examination for Ph. D. (PHD), Exams of Mathematics

Download Mathematics -Exam Paper 2004 - Entrance examination for Ph. D. (PHD) and more Exams Mathematics in PDF only on Docsity! C4 University of Hyderabad, Entrance Examination, 2004 Ph.D. (Mathematics/ Applied Mathematics) [ Hall Ticket No. Answer Part A by circling the Time: 2 hours Max. Marks: % correct letter in the array below: oe Part B: 50 tialblelale Instructions |? ajblela e| Sla/bjcl|dle 1. Calculators are not allowed. 4jalbic}dle 2. Part A carries 25 marks. Each cor- 5 aibjeidje rect answer carries 1 mark and each G6 Jal blefd]e wrong answer carries — Z mark. [7 |aybjc|dje} So do not gamble. Tf you want to 8 falbjcidle change any answer, cross out the old 9 jalbic]dje one and circle sthe new one. Over INialb]cldle wrilten answers will be ignored. lailalolelale 3. Part B carries 60 marks. Instructions 12] albje a & for answering Part B are given at the I3ja;bicldje beginning of Part B. ldjajbicidje 15 albje]d]e 4, Use a separate booklet for Part B. 16 albjejd e| Wialbleldicl lalbic|die 1i9talbicldle a}albfeldle 2ljajble|dje 2talbje}dle @Bialbjc|dje Wa@iaibje|dle 2lalbjcidie  C- 47 3 (13) Consider the quotient ring Z/95Z, where Z is the ring of integers. The numbor of clements x € Z/95Z such that. x? = T is (a) 1. (b) 2. (c) 3. (d) 4. {e) none of these. (14) The number of proper subfields of the field Fas (finite field with 2% elements) is {a} 1. (b) 2. {c) 3. (d) 4. {e) none of these. (15) Denote by Zp, the field of integers modulo p, where p is a prime. In the vector space Z, the number of distinct one dimensional subspaces is (a) p+2. (b)pri. (c)p—1. (d) p—2. (e) none of these. (16) If A is a 4 x 4 matrix with characteristic polynomial (2? — 1)? and minimal polynomial (x? ~ 1), then its Jordan canonical form is 11 0 #90 -l1 100 ol 60 0 0 -1 00 @ oo 11 |) o o 14 00 0 -l 0 0 01 1 io 0 10 0 06 01 0 0 01 0 0 } og -1 0 |Mloo0-1 0 04 0 -1 00 0 -1 (e) none of the above. (17) Let X and Y be two topological spaces and f : X + Y bea continuous surjection. Then (a) V is an open set in X implies f(V) is open in ¥. (b) F is a closed set in X implies f() is closed in Y. (c) D is dense in X implies f(D) is dense in Y. (d) A is an infinite subset of X implies f(A) is an infinite subset of Y, (e) none of the above. (18) Let m be the Lebesgue measure on R and E be a Lebesgue measurable subset of A. For any real a, if we denote af = {ax in € FB}, then {a) m(aB) = m(£). (b) m(aE) = am(E). {c) m(aF) = |alm(E). (d) m(aF) = a+m(£). (ce) none of the above. C-47 (19) Ou the vector space C0, 1] of all real-valued continuously differen- tiable functions defined on [0,1], consider the following norms IIflleo = SuPocecs Lf (2)!, [Fh — fo LF) Idx, lle = [lflloo + HY'Foos and || ff}s = maxf{ll/ lle: IIF'llec}- Then {a) Hl -[leo and || - 2 are equivalent. (b) {+h and |[--[]2 ave equivalent {c) i --[h and || --[]s are equivalent. (d) jf] --I}.o and || --i}s are equivalent. {e) {| --llp and || --[fs are equivalent. (20) Among the following statements, which one does not characterize a finite-dimensional normed linear space X? (a) every linear subspace of X is closed. (b) the closed unit ball B = {x € X : {la|| < 1} is compact. (c) every bounded sequence in X has a convergent subsequence. (d) every continuous functional on a linear subspace of X has a unique Ifahn-Banach extension. (e) every functional on X is continuous. (21) Consider the linear subspace Y = {(z,y,z) € R?: 2 = y.z = 0} of R® with the norm ||(#,y, 2)|| — [2] + |y| + {z| and fo : Y > RB be a linear functional given fo(x,y,z) — 22, where (a,y,2) © Y. Then (a) fi : R® - R defined by fi(a.y,z) = 2c + 2 is a Hahn-Banach extension of fo. (b) fe : IR? —+ R defined by fa(x,y,z) = 2y | 2 is a Hahn-Banach extension of fo. (c) fg : RS > R defined by fa(e,y,2) = 2 — y is a Hahn-Banach extension of fo. (d) fy: B® > R defined by fa(z,y,2) = 2+ y — 2 is a Hahn-Banach extension of fy. (e) none of the above is a Hahn-Banach extension of fy (22) The general solution of the fourth order differential equation yf) By” + By” + dy’ — 8y = 0 is (a) y— (er | coe + cr" )e®* + eye *. (b) y= ce?™ | age. (c) y= ey tepte” | cgaPe™® -- cye™*. (a) y = (er + eae + eye? )e~™ + ce”. (c} none of the above. C4? (23) The critical point (0,0) of the following system dx 2 he = (y +1)? —cosz, “y = sin(e + y), is {a) a saddle point. (b) an unstable node. (c) a stable node. (d) an unstable spiral , (e) a stable spiral. (24) The extremum of the functional J[y| — Kee + 2yy dz, satisfying y(Q) = 1 and y(1) =0 {a} is a straight line (b) is a parabola. (c) is an ellipse (d) does not exist (e) noue of the above. (25) The equation tise + Uz, + YUyy — TY%, = 0 is hyperbolic for 2 x a > (a) y 4 ae b)y<—. (b) y z x (chy= TT (d) yao. {e) none of the above. Part B (50 Marks) "There are 15 questions in this part. Each question carries 5 marks. Attempt any 10 questions. The answers should be written in the separate booklet given. (1) Consider the sequence {f,} of real valued functions, where fal) = tna’ az € R. Answer the following questions with proper justification. (a) Is {fa} pointwise convergent on (0, 1]? (b) Is {/,} uniformly convergent. on (0, 1]? (c) Is {fn} uniformly bounded on [0,1]? (d) Is {f,} bounded in L1(0, 1]? {e) Is {f.} convergent on L*[0, 1]?