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As you want here I am giving model question paper of B. Sc (Mathematics) 3rd semester: I. Answer any fifteen of the following 15x2=30 1. Show that O(a)=O(xax1) in any group G. 2. Define center of a group. 3. Prove that a cyclic group is abelian. 4. How many elements of the cyclic group of order 6 can be used as generator of the group? 5. Let H be a subgroup of group G.Define K={xєG:xH=Hx}Prove that K is a subgroup of G. 2. Find the index of H={0,4} in G= (z8, +8). 3. Define convergence of sequence. 4. Find the limit of the sequence √2, √2 √2, √2√2√2.............. 5. Verify Cauchy’s criterion for the sequence {n/n+1} 6. Show that a series of positive terms either converges or diverges. 7. Show that 1/1.2 + 1/2.3 + 1/3.4+.......... is convergent. 8. State Raabe’s test for convergence. 9. discuss the absolute convergence of 1x2 + ∟2 x4 4!  x6 6!+......When x2=4 10. If a and b belongs to positive reals show that ab a+b +1/3 [ab a+b]3 +1/5 [ab/a+b]5+..... 11. Name the type of discontinuity of (x)={3x+1, x>1 2s1, x≤1} 12. State Rolle’s theorem. 13. Verify Cauchy’s Mean Value theorem for f(x)=log x and g(x)=1/x in {1, e] 14. Evaluate lim (cotx)sin 2x x→ 0 15. Find the Fourier coefficient a0 in the function foo = x, o ≤x <π 2π x, π≤x≤2 π 16. Find the half range sine series of (x)=x over the interval (0, π). II. Answer any three of the following 3x5=15 1. Prove that in a cyclic group (‹a) or order d, a (k 0 1], 0 1], 1 0], 1 1], 0 0]} 3. Prove that if H is a subgroup of g then there exist a onetoone correspondence between any two right cosets of H in G. 4. Find all the distinct cosets of the subgroup H= {1, 3,9} of a grouop G= {1,2,.....12} w.r.t multiplication mod 13. 5. If a is any integer p is a prime number then prove that ap=(amod p). III. Answer any two of the following 2x5=10 1. Discuss the behavior of the sequence {(1+ 1/n)}n 2. If an=3n4/4n+3 and an  ¾/< 1/100, n> m find m using the definition of the limit. 3. Discuss the convergence of the following sequences whose nth term are (i) (n21)1/8  (n+1)¼ (ii) [log(n+1)log n]/tan (1/n) IV. Answer any fifteen of the following 2x5=10 1. State and prove Pseries test for convergence. 2. Discuss the convergence of the series 1.2/456 + 3.4/6.7.8 + 5.6/8.9.10+.......... 3. Discuss the convergence of the series x2 2√1+x3 √3√2 + x4/4√3+........ 4. Show that ∑ (1)n is absolutely convergent if p›1 and conditionally convergent if p z 1 (n+1)p 5. Sum the series ∑(n+1)3/n!] xn V. Answer any two of the following 2x5=10 1. If limx→a f(x)=L, lim x→a g(x)=m prove that lim x→a [ (x)+g (x)=1+m 2. State and prove Lagrange’s Mean Value theorem. 3. Obtain Maclaurin’s expansion for log (1+sin x) 4. Find the yalues of a, b, c such that lim x→a x(2+a cosx)b sin x+c xo5 = 1/15 x5 VI. Answer any two of the following 2x5=10 1. Expand f(x)=x2 as a Fourier series in the interval (─)(ﻼπ,π) and hence Show that 1/12+1/22+1/32+..... π2/6 2. Find the cosine series of the function (x)= πx in 0< x<_ π. 3. Find the halfrange sine series for the function (x)=2x1 in the interval (0.1) Contact details: Bangalore University Mysore Rd, Jnana Bharathi, Bengaluru, Karnataka, 560056 080 2321 3172 Map: Last edited by Aakashd; May 23rd, 2019 at 01:18 PM. 