Master of Science I (Statistics)Examination: Oct Nov 2016
Semester I (Old CBCS)
SLR No. Day
Date Time Subject Name Paper
No. Seat No.
SLR SB
681
Friday
18/11/2016
10.30 AM
to
01.00 PM
Real Analysis
C
II
Instructions: Answer any five questions.
Q. No. and Q. No. are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Total Marks:70
Q.1 A. Select the correct alternative: 05
The compliment of an open set is
Always open Always closed
May be closed or may be
open
Neither open nor closed
Every point of an open set is its
Interior point Limit point
Boundary point None of these
The set of rationals is set.
A finite A countable
An uncountable None of these
Subset of uncountable set is
Always countable Always countable
May or may not be
countable
None of these
Every Cauchy sequence is a sequence.
Convergent Divergent
Monotonic Oscillatory
B. Fill in the blanks: 05
Countable union of countable sets is
A set of all limit points of a set is called
Greatest lower bound of a set is also called as
A set is open, iff its compliment is
The infimum of the set of all rational numbers in is
Page 1 of 2
C. State whether following statements are true or false: 04
Every point of a set is its limit point.
If exists, infimum is always unique.
An open set includes all of its limit points.
A set can be both open as well as closed.
Q.2 A. State the following: 06
Least upper bound property of R
Heine Borel theorem
Labnitz rule
B. Write short note on the following 08
Countability and uncountability of sets
Limit superior of a sequence
Q.3 A. Define open set. Is arbitrary union of open sets always open? Justify. 05
B. Define countable set. Prove that countable union of countable sets is
countable.
05
C. Is the set countable? Justify. 04
Q.4 A. Prove that a sequence is convergent, iff it is a Cauchy sequence. 08
B. Describe the Lagrange's method of undetermined multipliers. 06
Q.5 A. Describe root test and ratio test of convergence of a series. 07
B. Prove that limn ∞ αn 0 is the necessary but not sufficient condition for
convergence of series Σαn .
07
Q.6 A. Define lower and upper Riemann integral of a function Also state the
condition under which function is said to be Riemann integrable.
07
B. Find Riemann integral of the following functions over

ii) x2
07
Q.7 A. Find minimum value of the function x2+y2+z2 when 08
B. Write a short note on Taylor's theorem. 03
C. State the implicit function theorem. Also give its one application. 03
Page 2 of 2