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Pune University - FY BSc MATHEMATICS - I (Old Course)

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Total No. of Questions : 5] P219 [Total No. of Pages : 3 [3717] - 51 F.Y. B.Sc. MATHEMATICS Algebra (Paper - I) (Old Course) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Candidates are advised to see the relevent question paper and solve the same. 2) All questions are compulsory. 3) Figures to the right indicate full marks. Q1) Attempt all the subquestions: a) [16] Consider the set N x N of all ordered pairs of natural numbers. For (a, b), (c, d) in N x N define an equivalence relation. (a, b) ~ (c, d) if ad = bc then find the equivalence class of (3, 5). b) Let A = {a, b, c}, B = {1, 2}. Find a possible function f from A to B which is onto but not one c) Let a, b, c, d be integers. If a | b and c | d then prove that ac | bd. d) Find (7 ) + (12) , where e) 2 +i . Find real and imaginary part of z, where z = 3 2i f) 2 2 If z = 3 cos + i sin 3 3 is Euler s s - function. , then by using De-Moivre s theorem 1 3 z4 = 3 4 + i prove that 2 2 g) If the equation x3 x2 8 x + 12 = 0 has one root 3 then find the other roots. h) If f(x) = 2x3 + 4x2 + 3x + 2 and g(x) and g(x) = 3x4 + 2x + 4 then find the deg [f(x) + g(x)] and deg [f(x) g(x)]. P.T.O. Q2) Attempt any four of the following: [16] a) Let ~ be an equivalence relation on a set S. For a, b S prove that b a if and only if a = b . b) Find the greatest common divisor of the integers 847 and 203 and express it in the form 847 m + 203 n where m and n are integers. c) If a is any odd integer, then prove that a2 1 (mod 4). d) Show that for i n , there exists j n with i j = 1 if and only if (i, n) = 1. e) Compute f) Show that (a, b) = (a, b + ax), for any integer x. 1 , where = (1 , 4 , 5 ( 2,1) and ) = (1 , 3 , 6 , 2 in S6. ) Q3) Attempt any two of the following: [16] a) Prove that any two non-zero integers a and b have a unique positive g.c.d. d = ( a, b) and d can be expressed in the form d = ma + nb for some m, n Z. b) Prepare the composition tables for addition and multiplication in Z8. Also find the elements which have multiplicative inverses and evaluate 8 + ( 7 + 1 3 ) in Z9. 1 c) Define congruence relation of integers modulo n. Prove that (i) If ax bx (mod n) and (x, n) = 1 then a b (mod n), where a, b , x Z, n N. (ii) If (x, n) = d and ax bx (mod n) then a b (mod w), where n = dw, n N and a, b, x, d, w are integers. d) i) Let P(X) be the power set of X = {a, b, c}. Then show that the relation on P(X) defined by A B means A is a subset of B is a partial order relation. Also draw the Hasse diagram for it. ii) Let f : R R be defined by f(x) = 2x 5 . Then show that f i s 3 bijective. Also find the formula for f 1. Q4) Attempt any four of the following: a) [16] If z1 , z 2 0 , then prove that if z1 + z2 = z1 + z 2 , then arg g (z 2 ) 2n [3717] - 51 = arg(z 1 ) , where n = 0, 1, 2, . . . . . . . 2 b) Solve the equation x3 x2 + x 1 = 0, by using De-Moivre s theorem. c) If z1 and z2 are any two complex numbers, then prove that, z1 z2 z1 z 2 . d) If , , are the roots of the equation x3 5x2 + 7x 5 = 0, then find the value of e) 2 . Find the equation of the lowest degree with rational coefficients, having 2 + 3 and 2 + 7 as two of its roots. f) If = u + iv is a root of a real polynomial equation f(x) = 0 then show that = u iv is also a root of f(x) = 0. Q5) Attempt any two of the following: [16] a) Solve the equation x4 3x2 4x 3 = 0 by Ferrari s method. b) Explain Cardan s method for solving a cubic equation. c) State and prove De-Moivre s theorem. d) i) Prove using De-Moivre s theorem that cos4 ii) = cos4 6 cos2 sin2 If z = 1 and arg z = , then show that 1+ z = i cot . 2 1 z Y [3717] - 51 + sin4 . 3

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Additional Info : F.Y.B.Sc. MATHEMATICS Algebra (Paper - I) (Old Course), Pune University
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