BSc Sem I 2017 2018 2018 Maths I Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Choose correct alternative in each of the following: 20 marks
2. Q1 Multiplicative inverse of a real number
   • (a) Exists and is unique
   • (b) Does not exist
   • (c) If exists then is unique
   • (d) None of these un. IfA =(2,5] then
   • (c) SupAEA (d) None of these
   • ii. If0
   • (c) x*<x None of these
   • iv. The sequence (x,) is
   • (a) Convergent (b) Bounded
   • (c) Divergent (d) None of these
   • v. Every constant sequence in R is
   • (a) Convergent (b) Bounded but not convergent
   • (c) Never Cauchy None of these
   • (c) 2 (d) None of these
   • (c) 0 (d) None of these
   • viii. If (x,) of real numbers satisfies, <x,< = , then
   • (a) (b) Diverges
   • (c) (d) None of these
   • Q.P.Code : 12086
   • ix. The inequality Ris
   • (c) Triangle inequality
   • (d) None of these
   • x. The function f(x) = is continuous
   • (c) these
3. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. State any four properties of R under addition. Further prove that additive inverse of a real number is unique Ifx,y Rsuch that x < y, then prove that there exists
   • b) Attempt any TWO questions from the following: 12
   • i. Prove the following: For x 0, Let A be any non-empty, bounded above subset of R. Let k > 0. Prove that sup(kA) =k sup A
   • ii. Show x R then there exists n N such that
   • iv. State and prove Hausdorff property of R Attempt any ONE question from the following: (08)
4. Q1 Let and be two sequences converging to p and q respectively. Prove that (x,+y,) converges to p and (cx,) converges to cp where c R
   • i. Prove that every Cauchy sequence of real numbers is
   • b) Attempt any TWO questions from the following: 12
5. Q1 =b", Vn N where 0 < b < 1. Show that =3- EN. Show that is monotonic increasing and bounded above. Is convergent?
   • Q.P.Code : 12086 ili. Prove that every convergent sequence of real numbers is Show that the sequence (cos is divergent
6. Q4 a) Attempt any ONE question from the following: 8 marks
7. Q1 State and prove Sandwich theorem for limit of a Let R be two functions and let a R. If and th that an m , then prove tha
   • b) Attempt any TWO questions from the following: 12
8. Q1 Prove that f(x) = is continuous = 2, using Draw graph of a function f(x) =log.x for x (0,0) ui. Let f:R Rbea function and let R. Give definition of and also find
   • iv. Let a function and a R. Prove that = 0 if and only if
9. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) If A,B are non-empty, bounded subsets of R, then show that the set AN B is bounded
   • b). State and prove the Arithmetic-Geometric Mean inequality
   • c) Give an example of two divergent sequences and such that their product is convergent
   • d) State and prove Sandwich theorem for sequences of real
   • e) Discuss the continuity of the following function at x = 4,8
   • f) Prove that f(x) = if x R\Q is discontinuous at x = 2 by using sequential definition of continuity

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