Master of Science I (Statistics) Examination: Oct Nov 2016
Semester I (New CBCS)
SLR No. Day
Date Time Subject Name Paper No. Seat No.
SLR SB
674
Wednesday
16/11/2016
10.30 AM
to
01.00 PM
Real Analysis
C
HCT 1.1
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
B. Fill in the blanks: 05
Superset of uncountable set is
A set is compact if and only if it is closed and
Every bounded set has at least one limit point.
A set of all limit points of set is called as set.
Least upper bound is also called
C. State whether following statements are true or false: 04
A countable set is always closed.
An infinite set is always uncountable.
A series of positive terms is always convergent.
The set is bounded above, but not bounded below.
Q.2 A. Define open set. Prove or disprove: Finite union of open sets is open. 06
Explain with illustration the concept of bounded sets.
B. Write short note on the following 08
Countable and Uncountable sets
Supremun of a set
Q.3 A. Prove or disprove: Countable union of countable sets is always countable 07
B. Prove: A set is open, iff its compliment is closed. 07
Q.4 A. Show that every monotonic bounded sequence converges. 07
B. Discuss the convergence of geometric series for various values of r. 07
Q.5 A. Discuss root and ratio test of convergence for series. 07
B. Explain Riemann upper and lower integral. 07
Q.6 A. Use the Taylor's series formula to expand:
ii) cos x
07
B. Find the stationary value of x2+y2+Z2 subject to the condition x3+y3+Z3=3a3. 07
Q.7 A. Find limit superior and limit inferior of the following sequences.
Hence, verify their convergence.
07
B. 07