BSc Mathematics SEM I 2018 19 Nov 2018-19 MATHEMATICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q3 Use of Calculator is not allowed
2. Q1 Choose correct alternative in each of the following: 20 marks
   • a) only if x# 0 b) cannot say
   • c) always d) none. of the above
   • ii. If S={xER: |x—7|< 1 } then
   • a) S is bounded b) S is only bounded above S bounded below d) cannot say Which of the following sets is aneighbourhood of -1 with radius 1, N(-1,1) in R?
   • c) [0,3] d) none of these
   • iv. The sequence (n*) in R is
   • a) divergent b) convergent Cc) bounded d) none of these The sequence (x,) xy N is Cannot say d) none of these Vi. Every constant sequence in
   • c) bounded d) none of these Vii. value of lim for 0<x<1 is Cc) d) none of these
   • c) 1 d) none of these
   • a) does not exist b) -1
   • c) 1 d) none of these The function f(x)= x , R is
   • a) continuous everywhere b) continuous if x >0
   • c) discontinuous everywhere d) none of these
3. Q2 a) Attempt any ONE question from the following: 8 marks
4. Q1 Prove that any two distinct real numbers can be separated by disjoint neighborhoods in R. Hence find disjoint neighborhoods of 2.33 and 2.333 li. Define infimum of a non-empty set Prove that a lower bound m non-empty set S is the infimum of S iff for
   • b) Attempt any TWO questions from the following: 12
5. Q1 Show that for any two real numbers a and ll. Prove that for positive real numbers x and y, iii are bounded subsets of real numbers then prove that
   • iv. Prove that for positive real numbers a,b and c,
6. Q3 Attempt any ONE question from the following: 8 marks
7. Q1 Prove that a real sequence is convergent if and only if it is a Cauchy sequence li. Prove that every monotone, bounded sequence of real numbers is convergent
   • b) Attempt any TWO questions from the following: If lim a, =a, lim b, = b and > 0 is arbitrary, then prove that there exists 12
   • ii. Let a, and Prove that sequence (a,) is increasing and bounded above by 2 Let and be two convergent sequences such that lim (3a, + 4b,) = 10 and lim (a, — = 11. Then find lim a,, and Iv. If (a,) is a sequence of non-negative real numbers and lim a, = a. Then prove that a > 0 and Van = Ja
8. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. Let R f(x) and =m then prove that lim[ f(x) — li. Let f: Rbea function and pe R. If converges to f(p) for any sequence (xn) converging to p then prove that f is continuous
   • b) Attempt any TWO questions from the following: 12
9. Q1 Draw the graph of the function f where f(x) x1+3 for-3 <x <3 li. Show that lim (14 — 2x)= 8 using — 6. definition State and prove Sandwich theorem for limit of functions in R Iv. Let f: R— R be a function which is continuous at R Then prove that there exists 6>0 and M that! M,
10. Q5 Attempt any FOUR questions from the following: 20 marks
    • a) Show that x += > 2 for x>0
    • b) Let A and B be nonempty bounded subsets of IR such that A B. Prove that sup A <
    • c) Let x, = cos (=) N. Show that is not convergent by exhibiting two convergent subsequences of (x,,) converging to two different limits
    • d) Show that = 0 using Sandwich theorem
    • e) Let f, g: R R be functions and p R. pf = land g(x) = mand f(x) = x R then show that 1 > m
    • f) Find the value of b so that f becomes continuous at — where (x) = 2

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