B.Sc. (II YEAR) SEMESTER-IV MATHEMATICS MODEL QUESTION PAPER ADIKAVI NANNAYA UNIVERSITY

MATHEMATICS MODEL PAPER

FOURTH SEMESTER – REAL ANALYSIS

COMMON FOR B.A & B.Sc

(w.e.f. 2015-16 admitted batch)

Time: 3 Hours                                                                              Maximum Marks: 75

SECTION-A

Answer any FIVE questions. Each question carries FIVE marks.                 5 x 5 = 25 Marks

1. Prove that every convergent sequence is bounded.

2. Show that the series

 converges absolutely for all values of x.

3. Show that the series  

converges conditionally.

4. Examine the continuity of the function defined by f(x) = |𝑥| + |𝑥 − 1| at x= 0, 1.

5. Verify Rolle’s theorem in the interval [a, b] for the function f(x) = (𝑥 − 𝑎)(𝑥 – 𝑏)^𝑛 ; m, n being             +ve integers.

6. Prove that f(x) =

  if x≠ 0 and f(0) = 0 is continuous at x = 0 but not derivable at x = 0.

 7. Evaluate  

8.  Prove that  

SECTION-B

Answer the all FIVE questions. Each carries TEN marks.          5 x 10 = 50 Marks

 9 (a). Prove that a monotone sequence is convergent if and only if it is bounded.

Or

   (b). Prove that the sequence

  is not convergent.

10 (a). State and prove Cauchy’s 𝑛^𝑡ℎ Root test

Or

     (b) Test for convergence

 (x > 0, a > 0)

11 (a). If f is continuous on [a, b] then prove that f is bounded and attains its infimum and                  supremum.

Or

     (b). If f is continuous on [a, b] then prove that f is uniformly continuous on [a, b].

 12 (a). State and prove Lagrange’s mean value theorem.

Or

(b) Find c of Cauchy’s mean value theorem for

and

 in [a, b] where 0

13 (a). If f:[a, b] ⟶R is continuous on [a, b] then prove that f is integrable on [a, b].

Or

            (b). State and prove Fundamental theorem of integral calculus.