BSc Mathematics SEM III 2022 2023 Oct 2023 MATHEMATICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 (a) Attempt any one. [each 8Mks]
   • 1) Define convergent and divergent.series and Prove that for n=1 to 00 the
   • 2) State and prove Leibniz test for series
   • (b) Attempt any two. [each 6Mks]
   • 1) Investigate the convergence of
   • 2)Prove that if is a convergent series of R then lim a, as n tends tu is equal to 0
   • 3)Prove that if Lan are two convergent series then prove that , L(an-bn)is convergent series
2. Q2 (a) Attempt any one. [each
   • 1) Let be bounded function and P be partition of [a, b] then prove that
   • 11) m(b-a)<U(P, f) <M(b-a)where m, M are Infimum and supreme of f(x) over [a, b]
   • 2)Let f:[a, be bounded function then prove that f is integrable iff for every e>0 there exist a partition P of [a, b] such that U(P, f) -L(P, f) <e
   • (b) Attempt any two. [each
   • 1) Prove that If fis integrable on [a, b] then f? is also integrable on [a, b]
   • 2) Define L(P, f) , U(P, f) where P is partition of [a, b] for bounded function f:[a, and prove that for bounded function f:[a, , Partition P of [a, b] , VCD/ SYBSC- SEM III MATHEMATICS- I 75MARKS
   • 3) Prove that a consiant function is Riemann integrable
3. Q3 Attempt any one. [each 8Mks]
   • 1) Let a>0, then prove that converges iff
   • 2) Define Beta function B(m, n) and prove that B(m, n)=B(n, m) .Define Gamma function and
   • (b) Attempt any two. [each 6 Mks]
   • 1) i) Define Indefinite of fon [a, b] Find indefinite integral of f for li) Define Primitive if function Give an example to show that primitive of f if exist,need not
   • 2)Prove that Let f:[a, be a continuous function then there exist cefa, such that af
   • 3)State Leibnitz rule and Use it to find the derivative of the following f(x) =cosxf ‘mn, dt
4. Q4 Attempt any three. [each 5 Mks]
   • 1) Discuss the convergence of following series
   • 2) Find the sum of the series
   • 3) Let as =3x Prove that f is integrable and find of '3x dx
   • 4) Find the norm of following partition of intervals | =[-2,5], P={-2,-1.5,
   • 5) Find the value of c in [0,1] such that for , xe[0, 1]
   • 6) Find the area between the curves y=2x? and y=x between and x=2

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