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

Questions asked in this paper

1. Q1 . (a) Attempt any one from following. 8 marks
   • 1)Prove that the series for 0 < p S 1is divergent series
   • 2) Prove that the series p ar” is convergent if and only if |r|<1
   • b) Attempt any two from following. 12
   • 1)The telescoping series = is convergent if and only if (t,, ) is convergent
   • 2) Check convergence of the series
   • 3)Prove that Series sin(n) + is divergent series
2. Q2 . (a) Attempt any one from following. 8 marks
   • 1) Let R be bounded function then f is integrable if and onlyifVe> P partition of [a, b] such that U(P, f) — L(P, f) <
   • 2) _Let R be a bounded function an let P and Q be two partitions of [a,b] P Q Then show that L(P, f) < L(Q,f)
   • b) Attempt any two from following. 12
   • 1)Let f: [a,b] > R be a bounded is integrable if and only if there is sequence of partition (P,) such that f) 0
   • 2)Check integrability of function f(x) = 200 in [0,3] and evaluate f(x) = 200 dx
   • 3) Show that R = c constant) is Reimann integrable function,
3. Q3 . (a) Attempt any one from following. 8 marks
   • 1)Let and — R be continuously differentiable function. f:[a,b] >R Is a continuous function such that (a) = a and = band & then
   • 2) Let f, g: [a,b] R be differentiable and g' be Reimann integrable on [a,b] then show that dx = — dx
   • b) Attempt two trom 12
   • 1)Find the surface area of solid of revolution obtained by rotating the curve :
   • 2)show that B(m,n) = B(n,m) = + 1,n)
   • 4). Attempt any three from following 15
   • 1) Apply suitable test to check convergence of
   • 2) By Leibnitz test discuss convergence of + +
   • 3) f: [1,3] R, where f(x) is defined for x [1,3] as f(x)=-2. Show that f(x) is integrable and evaluate it
   • 4)Prove that is Let f: [a,b] R: be a bounded function and let P be partition of [a,b] then
   • 6)Prove that (5) =

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