Part II FINAL PART -B

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1 Part II FINAL Branch 1(A) - Mathematics Paper I - FINITE MATHEMATICS AND GALOIS THEORY PART -A 1. Find the coefficient of xk, k ~ 18 in the expansion of + X r. 2. Find the number of primes between 2 and 48 inclusive. 3. Prove that every connected graph contains a spanning tree. 4. If G is a Hamiltonian then prove that for every nonempty proper subset S of V, w(g-s)!~isj. 5. State and prove Gauss lemma. 6. Prove that the degree of the extension of the splitting field of x 3 2 QE xj is Let. F be a field of characteristic * 2. Let x 2 - a F [ x ] be an 'irreducible polynomial over F. Then prove that its Galois group is of order Prove that the Galois group of x4 +X2 +1 is the same as that of x 6 1 and is of order 2. PART -B Answer ONE question from each Unit. Each question carries 15 marks.. (Marks :4x 15= 60) UNIT I 9. (a) Show that S(rn)= S(r - 1,n - i)+ ns(r - 1,n) where r,n E N with r 2! n. (b) Solve the system of recurrence relations a + 2a_1-4b,, = 0 b,, + 5a..1-7b,, 1 = 0 Given that a1 =4 and b 1 =1. UNIT IV 15. State and prove fundamental theorem of Galois theory. 16. (a) Let F be a field and let U be a finite subgroup of the multiplicative group F* F M. Then prove that U is cyclic. (b) Let E be the splitting field of x" a e F[x]. Then prove that G(E I F) is a solvable group.

2 Part II Final Branch I (A) Mathematics Paper II TOPOLOGY AND FUNCTIONAL ANALYSIS P A R T A 1. Show that any intersection of closed sets in a topological space Xis closed. 2. Show that every separable metric space is second countable. 3. Show that every sequentially compact metric space is totally bounded. 4. Show that any continuous image of a connected space is connected. 5. Let N be a non-zero normed linear space and prove that N is' a 13anach space iff E I = 1 } is complete. 6. If P is a projection on a Banach space B and if M and N are its range and null space, then show that'm and N are closed linear subspaces of B such that B = M N. 7. Show that a closed convex subset C of a Hilbert space H contains a unique vector of smallest norm. 8. If A/ and A, are self-adjoint operators on H, then their product A.1.42 is self-adjoint iff A1A2 = A2A1. PART B Answer ONE question from each Unit. Each question carries 15 marks. (Marks : 4 x 15 = 60) UNIT I 9. (a) If f and g are continuous real or complex functions on a topological space X then show that f v g and f A g are continuous. (b) State and prove Liondelofs theorem. 10. (a) State and prove Heine-Borel theorem. (b) Show that every sequentially compact metric space is compact. UNIT II 11. State and prove Uryshon's lemma. 12. State and prove Ascolis theorem.

3 13. State and prove Hahn-Banach theorem State and prove closed graph theorem. UNIT IIT (Jr UNIT IV 15. (a) If N and N' are normed linear spaces, then the set BV\ r, N') of all continuous linear transformation of N into N' is itself a normed linear space with. respect to the pointwise linear operation and the norm defined by TI,SudiTx11/11x1k1}. If N' is a Banach space then show that B(N, N') is also a Banach space. (b) Prove that the adjoint operation T T' or B(1I) has the following properties. ) (T, +T 2 )' = T: (ii) (dr = (Ti = T2-7. 1, (iv) T =T (v) 11T 1=111 (vi) ; IT-TI 16. (a) Define a normal operator. If NI and N 2 are normal operators on II with the property that either commutes with the adjoint of the other then show that N 1 4- N, and N, N. are normal. (b) If P and Q are the projections on closed linear subspaces Al and.1\i' of H, then show that M I N PQ = 0 = 0.

4 Part II Final Branch I (A) - Mathematics Paper III - OPERATIONS RESEARCH P A R T A 1. Explain the linear programming problem giving two examples. 2. Explain the use of artificial in L.P. 3. What is a balanced transportation problem? What are its applications? 4. Explain the difference between a transportation problem and an assignment problem. 5. Explain the advantages and disadvantages of dynamic programming. 6. What is a non-linear programming problem and give the general form of NLPP? 7. What is queuing theory? What are the limitations of queuing theory? 8. Enumerate the various types of inventory models. PART B Answer ONE question from each Unit. Each question carries 15 marks. (Marks: 4 x 15 = 60) UNIT - I 9. (a) Explain the Big M method to solve L.P.P (b) Use penalty method to Maximize z =x1 +2X2+ 3x3 -x4 Subject to the constraints Xi+ 2x2 + 3x3 = 15 2x x3 = 20 x1+2x2+x3+x4 =10 x11x21x1x4 >O.

5 10. (a) Explain two phase simplex method to solve L.P.P. (b) Use two-phase simplex method to Maximize z = 3x 1 - x2 Subject to the constraints 2x 1 + x2?, 2 x +3x,, < 2.<1.-4,xpx, O.. '". UNIT -II 11. (a) Explain 'Vogel's approximation method to solve L.P.P. (b) By using above method obtain an initial basic feasible solution of the transportation problem D E F G Available A ` C l _, 400 D e m an d (a) Explain the North-West cornermethod to solve L.P.P. (b) Obtain an initial basic feasible solutions to the following transportation problem using the North-West corner rule. D E F G Available A I I Requirement U N I T -I l l 13. (a) What are the basic features and applications of dynamic programming? (b) Use dynamic programming to solve the L.P.P. Maximize z = 3x x 2 Subject to the constraint x I + 4 x 2 X.5_ 2 and x > 0 > 0 ".-2

6 14. (a) Explain the significance of lagrange multipliers. (b) Solve by using lagrangian multipliers Maximize z = 4x1 +6x9 2x1,x2 2x92 Subject to the constraints + 2x, =2 andx1,x2 O. UNIT IV 15. Determine the steady state solution of the queuing models. (a) (M/M/1X./FCFS) (b) (M/M/I) (N/FCFS) and obtain the average queue length in each tsable. 16. (a) Derive the E0Q formula for the manufacturing model without shortages. (b) An item is produced at the rate of 50 items per day. The demand occurs at the rate of 25 items per day. If the set up cost is Rs. 100 and holiday cost is Re unit of item per day. Find the economic lot size for one run, assuming that the shortages are not permitted. Also find the time of cycle and minimum total cost for one run.

7 Part II - Final Branch I (A) - Mathematics Paper IV - NUMBER THEORY PART A 1. If f and g are multiplicative, then prove that their Dirichiet product f * g is also multiplicative. 2. Show that 1 (mod in), if (a, in) = If x > 1 prove that I= logx + C +0 ~1). n:5x 4. Show that lirn1 - H ( x ) = 0. X xlogx1 5. If A is the conjugate transpose of the matrix A, then prove that AA = ni, where I is the ii x n identity matrix. 6. Show that there are infinitely many primes of the form If S ( n ) = f(d)gij where f and g are multiplicative, then prove that d(n,k) S,(ab) = S..., WS,, (b), whenever (a, k) = (b, m) = Let x be an odd integer. If a ~: 3, prove that x 22 1 (mod 2). PART - B Answer ONE question from each Unit. Each question carries 15 marks. (Marks: 4 x 15 = 60) 9. (a) If 1, prove thatç o ( d ) = n. UNIT I rifr (b) State and prove the Selberg identity (a) State and prove Chinese remainder theorem. (b) Show that the lattice points in the plane visible from the origin contains arbitrarily large square gaps.

8 UNIT H 11. State and prove Eu]er's summation formula. 12. For every integer n 2. prove that n <11 <. G log72 tog /I UNIT 1.3. Show that a finite abelian group G of order n has exactly 77. distinct characters. 14. (a. ) Prove that there are infinitely many primes of the form e l. n. -,-- 1. (b) For x >1, prove that. lotg p 1, 1. e.,,,i, N-- -- = -- iof,, - x -2., ( p) log p - I - V -- + L.--,. / 7.(h)I. ' - I./ ( L ). pfi. x P (p(h) - q ) ( k ),,ff-, 2 p-_,:x P p... h (modk) UNIT IV 15. (a) if p is an odd prime, then for all 11 show that (nip) n IP (mod p). (b) Ifp and q are distinct odd primes, then prove that (p 1 q)(q I p)=( 16. (a) State and prove Lagrange's interpolation theorem. (b) If p is an odd prime and a 1, prove that there exist odd primitive roots olmodulo p" and each such g is also a primitive root modulo 2 p'.

9 Part II- Final Branch I(A) - Mathematics Paper V MATHEMATICAL STATISTICS PART - A Answer any FOUR questions. Each question carries 5 marks. (Marks : 4 x5 = 20) 1. Let X be a random variable with p.d.f. f(x), 0 <x < g and zero otherwise. Find g 2, the pdf of Y = sin x. 2. Let X and Y have the p.d.f f(x,y)=x+y; 0 <x< 1, 0 <y< 1 and zero elsewhere, find E(XY2). 3. Let X be n(5,10). Find Pr[0.04 < (X - 5) 2 <38.4j. 4. Let T have a t distribution with 14 degrees of freedom. Determine b so that Pr(-b <T <b)= Let P = 0.95 be the probability that a man, in a certain age group lives atleast 5 years. If we are to observe 60 such men and if we assume independence, find the probability that atleast 56 of them live 4 or more years. 6. Let the observed value of the mean X of a random sample of size 20 from a distribution that is n 80) be Find a 95 percent confidence interval for,u 7. Let X 1,X 9,... X n be a random sample of size n from a geometric distribution that has p.d.f f(x;0)= - 6 0, x 0, 1, < 0 <1 and zero elsewhere. Show that 1- :X is a sufficient statistic for Show that the mean X of a random sample of size ti from a distribution which is b(1,0), 0 <0 <1 is an efficient estimator of 0.

10 PART B Answer ONE question from each Unit. Each question carries 15 marks. (Marks :4x15=60) UNIT (a) Let X and Y have the p.d.f. fx,y) = 1, 0 <x <1, 0 < <1 and zero elsewhere. Find the p.d.f of the product Z XY (b) Let 1(x) =, x = 1,2,3... and zero elsewhere, be the p.d..f of the random variable X. Find the moment generating, the mean and the variance of K. 10. (a) Show that the random variables K3 and K2 with joint p.d.f. fx1,x2) = 12x1x2(1 x2), 0 <x1 <1, 0 <x2 <1 and zero elsewhere, are Stochastically independent. (b) Find Pr <K1 <, 0< K2 <J if the random variables K1 and K2 havethe, joint p.d. ffx1,x9) = 4 x 1(i'- x 2),0<x3 <1, 0<x., <1 and zero elsewhere. UNIT Il 11. (a) Find the moment-generating function of a ganmia distribution and deduce mean and variance. (b) If X has a gamma distribution with a = 3 and fi = 4 find Pr(3.28 <X< 25.2). 12. (a). Find, the mean and variance of a normal distribution. (b) if K is the mean of a random sample of size n from a normal distribution with mean je and variance 100, 'find ii SO that, Pr - 5 <X <u -' 5) = UNIT - flj 13. (a) Let X3,X2...X, denote the items of a random sample from a distribution that has mean ft and positive variance a 2. Then prove that the random variable = I x1. -'- has a limiting distribution that is normal with. mean zero and variance 1. (b) Let X denote the mean of a raridorn sample of size n from a distribution that is 17.U,a 2 ). Find the limiting distribution of X 2

11 14. (a) If 8.6, 7.9, 8.3, 6.4, 8.4, 9.8, 7.2, 7.8, 7.5 are the observed values of a random sample of size. 9 from a distribution that is n(8, c 2 ), construct a 90 percent confidence interval for a2. (b) Let Xi,X2... X represent a random sample for the probability density function f (x; 0) = 0 x -1, 0 <x <1, 0 < and zero elsewhere. Find the maximum likely hood estimator 8 of 8. UNIT -IV 15. State and prove Neymann-Pearson theorem. 16. State and prove Rao-Blackwell theorem.

ASSIGNMENT - 1, DEC M.Sc. (FINAL) SECOND YEAR DEGREE MATHEMATICS. Maximum : 20 MARKS Answer ALL questions. is also a topology on X.

ASSIGNMENT - 1, DEC M.Sc. (FINAL) SECOND YEAR DEGREE MATHEMATICS. Maximum : 20 MARKS Answer ALL questions. is also a topology on X. (DM 21) ASSIGNMENT - 1, DEC-2013. PAPER - I : TOPOLOGY AND FUNCTIONAL ANALYSIS Maimum : 20 MARKS 1. (a) Prove that every separable metric space is second countable. Define a topological space. If T 1 and