BSc Mathematics SEM I 2019 20 Nov 2019-20 MATHS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to right indicate full marks
3. Q1 Choose correct alternative in each of the following (2 marks each) <yz d)none of these
4. Q2 Infimum of the set iS = d) none of these
   • a) convergent oscillates infinitely none of these
   • a) bounded and b) bounded but not convergent
   • c) unbounded but converngent d) none of these
5. Q5 A monotonic sequence
   • a) must be ~ b) may be bounded not be none of these 6)The sequence (1 + =)" i$
   • a) monotonic and bounded b) neither monotonic nor bounded
   • c) monotonic but not none of these 7)A convergent sequence of rational numbers
   • a)has a rational limit may have irrational limit cannot have irrational limit d) none of these 8)Which of the following is true?
   • a) A sequence is Cauchy if it is bounded a sequence is convergent if it is bounded
   • c) a sequence is convergent if it is cauchy d) none of these
6. Q9 is
   • b)1 c) d)does not exist
   • a) fis continuous at all points in R c) fis discontinuous at all points in R
   • b) fis continuous at all points other than zero in none of these
7. Q2 a) Attempt any ONE question from the following.(8marks each) ) State and prove Arithmetic Mean and geometric mean inequality for two nonnegative real
8. Q2 State and prove Cauchy Schwarz inequality of R and using it prove that if a,b are real numbers then prove that 3(a* + +c?)
   • b) Attempt any TWO question from the following.(6marks each) State and prove Hausdorff Property of R and by applying it find disjoint neighbourhood of
9. Q2 Define the following term Upper bound , Lower bound, least upper bound lower bound and prove that For a nonempty subset S of R , Supremum of S whenever exists is unique
10. Q3 Prove that a nonempty subset of is bounded below has the infimum in R Also find infimum and supremum of the following set
11. Q4 Let S be a nonempty subset of R. Prove that a real number m is the infimum of S iff
12. Q3 a) Attempt any ONE question from the following.(8marks each)
13. Q1 i) If ~ x and @ R, then prove that > ax
    • ii) Ifx, #0 O then prove that (x,) >
14. Q2 State and prove Sandwich theorem for sequence. Hence discuss the convergence of i) ay = ii) by
    • b) Attempt any TWO question from the following.(6marks each)
15. Q1 i) Examine whether the following sequences are monotonic? i) a, = = siin
    • ii) Prove that every monotonic increasing sequences converges to its lub if bounded above Define a subsequence of a sequence x,, of R and prove that If is a subsequences of Converging to p in R then prove that (Xn,,) to p
    • ii) Find the two convergent subsequences of (x,) and discuss the convergence of
16. Q3 A sequence is given by a, = V2 and = then show that (a,) converges 4)i) Show that the following sequences are convergent and find where the limit of each lies
    • ii) prove that every convergent sequence inR is a Cauchy sequence
17. Q4 a) Attempt any ONE question from the following.(8marks each)
18. Q1 1) If f(x) exists then prove that it is unique
    • ii) Let f(x) exists at a R then prove for k non zero real number,
19. Q2 State and prove the sequential continuity theorem for real valued continuous function
    • b) Attempt any TWO question from the following.(6marks each) 1)) Let f(x) & g(x) exist ata ER then prove that
20. Q2 State and prove Sandwich theorem for real valued functions
21. Q3 Using — 6 definition show that i) 2x +5 it) 4 = 4 marks
22. Q4 Discuss the continuity of following function
23. Q5 Attempt any FOUR question from the following.(Smarks each) ) Define the convergence of a sequence of (x,)of R and Prove that the sequence (=) converges to zero
24. Q2 Prove that every convergent sequence is bounded
25. Q3 Discuss a"for the given cases
26. Q4 Using — No definition show that =5
    • S) Find the domain and the range of each function Also draw the graph of given function {
27. Q6 lf 2x < g(x) R find :

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