BSc Sem I 2023 2024 Dec 2024 MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 . (a) Attempt any one from following. 8 marks
   • 1) State and prove binomial theorem for n N where N is set of natural numbers
   • 2) Prove that for given integer a and b > 0, there exist unique integers q
   • b) Attempt any two from following. 12
   • 1) State Euler’s Phi function and calculate and
   • 2) If (a,b) = 1 and c| (a + b) then prove that (a,c) = 1, where (a,b) is g.c.d of
   • 3) Using mathematical induction prove that n(n + 1) is divisible by 2
2. Q2 . (a) Attempt any one from following. 8 marks
   • 1) f:X and g:Y two function such that g f (composition function)is
   • i. f is surjective, then prove that,g is injective
   • ii. g is injective, then prove that, f is surjective
   • 2) define binary operation, Commutativity, associativity, existence of identity element, existence of inverse element. Also check all properties fora + b,a,bER,R Is set of real numbers
   • b) Attempt any two from following. 12
   • 1) Check whether relation R defined as aRb,iffa<b,ER R(set of real numbers) is equivalence relation
   • 3) Show that the function f:R > R, f(x) = 2x + 5 is bijection
3. Q3 . (a) Attempt any one from following. 8 marks
   • 1) State and prove Remainder theorem for R[x], Also show that A polynomial f (x) in R[x] Is divisible by (x — a) iff f(a) = 0, where R[x] is set of real coefficients
   • 2) Prove that every Polynomial of degree n, with n > 2 is reducible in R[x]
   • b) Attempt any two from following. 12
   • 1) If p is a positive prime number, then prove that is an irrational number
   • 2) Find multiplicity of roots of f(x) = —x-1
   • 3) Find fourth roots of unity
4. Q4 . Attempt any three following 15 marks
   • 1) Show that = 1 mod(105)
   • 2) Verify Wilson’s theorem for p = 7
   • 3)a*b = test commutativity and associativity of ‘*’
   • 4) Prove that — R is given by f(x) = 5x — 8 is bijection
   • 5) Find g.c.d of polynomials,
   • 6) Prove that multiplication is binary operation in R[x], Where R[x] is set of real ALL THE BEST

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