BSc Mathematics SEM I 2015 16 2015-16 ATKT MATHS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 For Q.1, Q.2, Q.3 Attempt any one subquestion (each 8 marks) from Part (a) and any three subquestion (each from Part (b)
2. Q3 For Q.4, attempt any three (each 5 marks)
3. Q4 Attempt any one (Each 8 marks)
4. Q1 Fora, b, ceIR, Prove following
5. Q2 State and Prove Archimedian Property of IR
   • B) Attempt any three (Each 4 marks) Show that if V a, beIR, (ab)
6. Q2 Define bounded sequence of IR and prove that if a, is convergent then it is bounded
7. Q3 State and Prove AM GM inequality of IR
8. Q4 Define Least Upper Bound (LUB) and Greatest Lower Bound (GLB) of a non empty Set A and find LUB and GLB of
   • Q. Attempt any one (Each 8 marks) +) State all algebraic properties of a sequences of
9. Q2 Define the cauchy sequence in IR and show that following sequence are not cauchy in TR three (Each 4 marks) Define Monotonic sequence in IR. Examine whether the following sequences are monotonic
   • i) a, ii) a, = sin n that if (a,) and convergent sequences in IR such that a, > respectively then prove that (a, +b
10. Q3 Prove that ifa, > a, the a-a, >
11. Q4 State and-Prove Cauchy Completeness of IR a A) Attempt any one (Each 8 marks)
12. Q1 Let f, g be real valued function defined on subset J of IR and lim = lim g (x) =m, then prove that lim 1 2) Let R be a function where I is an open interval in IR Let ae! then prove that
    • B) Attempt any three (Each 4 marks)
13. Q1 Define right hand limits and left hand limits of f Evaluate right hand, left hand limits of fa lim (x) in following case
14. Q2 Give definition of limit of fat P and show that lim (27 + 3)=11
15. Q3 Prove that = [x] is discontinuous at every integer
16. Q4 Let f, be continuous at PeJ where J is an open interval in IR Then Prove that (f+ g) is also continuous at P
17. Q4 Attempt any three (Each 5 marks)
18. Q1 Let f be real valued function defined on subset J of IR Let PeJ lim f (x) exists then prove that it is unique Define neighbourhood in IR and prove that intersection of any two neighbourhood js
19. Q3 Discuss the boundedness of S = {x/3< x < 7) 4). Define defination of continuity of a function at a point P and explain graphical of continuity of function
20. Q5 Show that every monotonic decreasing sequence is bounded above
21. Q6 Define the following function with example

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