BSc Mathematics SEM I 2017 18 2017-18 ATKT MATHS Paper 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
   • (c) x=0 (d) None of these
   • (a) x=Oandy=0 (b) x=0ory=0
   • ii. Between any two distinct real numbers there always exists
   • (a) Only rational numbers (b) and
   • (c) Only irrational None of these
   • iv. Every nonempty subset of R which is bounded below has
   • v. The sequence (x,,) where x, = VnEN is
   • (a) convergent (b) divergent unbounded (d) None of these
   • vi. The sequence where x,=n? ,V is
   • (c) Cauchy (d) bounded The value of 3x" for is
   • (a) 1 (b) Does not exist
   • (c) (d) None of these
   • Q.P.Code: 30030
   • ix. Which of the following function has removable discontinuity at
   • (c) (d) None of these
   • x. The function f(x) = sin(2x +3),
   • (a) Continuous everywhere (b) Discontinuous at x =
   • (c) Continuous only for x (d) None of these
2. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. Prove that a nonempty subset of R which is bounded Ifx andy real numbers with x < y then prove that there exists a rational number r such that x <r < y
   • b) Attempt any TWO questions from the following: 12
3. Q1 Let S be anonempty subset of IR which is bounded above Prove that lub S is unique Let S be a nonempty subset of R. For a R, define aS = S}. If S is bounded above then prove that lub aS =
   • ii. Ifx then show that there exists m N such that
   • iv. State and prove Hausdorff property of R Attempt any ONE question from the following: (08)
   • i. Define subsequence of a sequence in R. Prove that every subsequence of a convergent sequence is convergent Prove that every monotonic decreasing sequence of real numbers is convergent if it is bounded below
   • b) Attempt any TWO questions from the following: 12
4. Q1 Prove that the limit of sequence (x,) is using definition
   • Q.P.Code: 30030 il. Prove that every Cauchy sequence of real numbers is
   • ii. Let (x,) and (y,) be two convergent sequences of real numbers converging to p and q respectively. Prove that the sequence (x, — y,) converges to q
   • iv. Show that the sequence (x,) is divergent where
5. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. Let > R be two functions and let a R, if m , then prove tha Let function at p R. Prove that sequence (f(x, )) converges to f(p) for any sequence (x,)
   • b) Attempt any TWO questions from the following: 12
   • l. Prove that 6 (4x + 6) = 30 using — 6 definition of
   • ii. Draw graph of a function e**? for x R ui. Let f: R > R be a function and let | R. Give definition of
   • iv. Discuss the continuity of the following function at x = 6
6. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) State and prove the Arithmetic-Geometric mean inequality for
   • b) Prove that — forallx,yER
   • Q.P.Code: 30030
   • c) Let x, = = EN. Show that (x,) is a Cauchy sequence
   • d) Show that converges to 3 using definition Let Prove that if f(x) = |l|. Is the converse true? Justify your answer
   • f) State Sandwich theorem for limit of function. Use it to find lim f(x) where 1 — for allx R

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