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
matha1
 converges absolutely for all values of x.

3. Show that the series  
MATHA1
converges conditionally.


4. Examine the continuity of the function defined by f(x) = |๐‘ฅ| + |๐‘ฅ − 1| at x= 0, 1.

5. Verify Rolle’s theorem in the interval [a, b] for the function f(x) = (๐‘ฅ๐‘Ž)(๐‘ฅ๐‘)๐‘› ; m, n being             +ve integers.

6. Prove that f(x) =
MATHA1
  if x≠ 0 and f(0) = 0 is continuous at x = 0 but not derivable at x = 0.


 7. Evaluate  
MATHA1


8.  Prove that  
MATHA1



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
MATHA1
  is not convergent.


10 (a). State and prove Cauchy’s ๐‘›๐‘กโ„Ž Root test
Or
     (b) Test for convergence
MATHA1
 (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
MATHA1
 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.

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