M.Sc. (Statistics) (CBCS) Examination, 2017
REAL ANALYSIS
Day Date: Tuesday, 18-04-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
N.B. Attempt any five questions.
Q. No. 1 and 2 are compulsory.
Attempt any three from Q. No. 3 to 7
Figures to the right indicate full marks.
Q.1 A. choose the correct alternative:
05
The compliment of open set is
Always closed Always open
May or may not be closed None of the above
The limit point of the set
n is
1 2
3 0
The set of irrationals is
Countably finite Countably infinite
Uncountably infinite None of these
The sequence
converges to
1 0
1.15 Infinity
If Σ
converges, then
Zero 1
infinity
B. Fill in the blanks:
05
Subset of uncountable set is
If A and B are countable sets, then A B is
Every bounded closed set is
Derival set is set of all points of the set.
Greatest lower bound is also called as
C. State whether following statements are true or false:
04
Every closed set is countable.
If a set is open, it must be closed.
A series of positive terms is always convergent.
Sequences can have maximum of two distinct convergence limits.
Q.2 A. Define closed set. Prove or disprove: Arbitrary intersection of closed
sets is closed .
06
Show that arbitrary union of open sets is always open.
B. Write short note on the following
08
Both open and closed sets
Cauchy criterion of convergence of a sequence
Q.3 A. Prove or disprove: Countable union of countable sets is always
countable
07
B. Define convergence of a sequence of real numbers. Show that
convergence limit is unique, whenever exists.
07
Q.4 A. Show that every monotonic non-decreasing bounded sequence
convergences.
07
B. Discuss the convergence of the series Σ
07
Q.5 A. Discuss comparison and ratio test of convergence for series.
07
B. Explain the concept of Reimann integral. 07
Q.6 A. Explain Lagrange's method of undetermined multipliers to maximize a
function.
07
B. Find the stationary value of x2+y2+Z2 subject to the condition
x3+y3+Z3=3a3.
07
Q.7 A. Find the radius of convergence of the following series.
07
B. Show that every continuous function is Reimann integrable.
07