BSc Mathematics SEM I 2022 2023 Dec 2023 MATHEMATICS II Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate marks

1. Q1 a) Attempt any ONE question from the following: 8 marks
   • i. State and prove Binomial theorem for neN
   • ii. Prove, for every integer n > 1 can be expressed as a product of positive primes and this expression is unique for order in which prime factors occur
   • b) Attempt any TWO questions from the following: 12
   • i. Prove that by using finite method of induction — 8n — 9 is divisible
   • ii. Find the smallest positive integer to which 10515 is congruent modulo 7
   • iii. Prove that ((a,b),c)) = (a, (b,c))
2. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. Leta,b Define a relation R in Z as, aRb iff a = b (mod n) then prove that R is an equivalence relation
   • ii. Define: Binary operation, commutativity, associativity, existence of identity element and existence of inverse element. Check all the properties for
   • b) Attempt any TWO questions from the following: 12
   • i. Write the distinct residue classes modulo 6 and the addition table modulo 6
   • ii. Define partition of a set and list any 5 partitions of set {a, b, c, d, e} ili. Check whether f: \ \ {=} given by f(x) = bijective
3. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Define Divisibility in IR [x]. State Division Algorithm in IR [x]. Also state and prove Remainder Theorem
   • ii. State and prove Unique Factorization Theorem in IR [x]
   • b) Attempt any TWO questions from the following: 12
   • i. Find the G.C.D. of polynomials f(x) = - 1 and g(x) = x!- 1 over Q[x]
   • ii. Define irreducible polynomials. And if p(x) is irreducible polynomial in IR[x]. Prove that, If p(x) does not divide a(x) in IR [x], then (p(x), a(x))
   • iii.Find the multiplicity of each root of f(x) =4x? + 4x? - x - 1
4. Q4 Attempt any THREE questions from the following: 15 marks
   • a) Show that 3927 and -377 are co-prime. ‘
   • b) Define Euler function and hence find
   • c) Check whether following relation is reflexive, symmetric, transitive or
   • d) Check whether the function f: Q IR given by f(x) = 2x + 3 is bijective or not
   • e) r2, are the roots of polynomial x? 4x2 + 5x + 1, without actually calculating the values of r2, write polynomial with roots 3r2, and 3r3
   • f) Find the cube roots of unity

Read from the scan above, so a character or two may differ. The scan is the original.