M.Sc. (Semester (CBCS) Examination Nov/Dec-2018
Statistics
REAL ANALYSIS
Time: 2½ Hours Max. Marks: 70
Instructions: All Questions are compulsory.
Figures to the right indicate full marks.
Q.1 Choose the most correct alternative. 14
The set has total of limit point/s.
two one
zero infinitely many
If A and B are open sets, then A U B is
always open always closed
may or not be open neither open nor closed M
A set is said to be closed, if
it includes all of its interior points
if every point of set is its limit point
if it includes all of its limit points
None of these
If a sequence is then it is always convergent.
bounded monotonic non-increasing
monotonic non-decreasing none of these
A subset of uncountable set
is always uncountable is always countable
may or may not be countable none of these
The derived set for 0,5 is
5 0,5
0,5 R
If a set is not open, then
it must be closed
it may be neither open nor closed
it may be both open as well as closed
all of these
The set of real numbers is
closed set open set
both a and b neither a nor b
A point c is said to be extremum point of function if
0 0
0 None of these
10) The sequences ∈ is
convergent to 1 oscillatory
convergent to 0 none of these
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11) The function − 3 has
maximum at point 1 minimum at point 1
convergent to 0 none of these
12) A continuous function is
always differentiable always right continuous
always bounded all of these
13) Which of the following is a power series distribution?
Poisson distribution Geometric distribution
Binomial distribution All of these
14) A geometric series with common ratio r converges, if
1 1
1 All of these
Q.2 Answer the following (any four) 08
Define and illustrate supremum of a set.
Define and illustrate bounded set.
Define and illustrate sequence.
State
Bolzano-Weistrauss theorem
ii) Heine-Borel theorem
Define and illustrate concept of limit point.
Write notes on following (any two) 06
Geometric series
Mean value theorem
Cauchy sequence
Q.3 Answer the following (any two) 08
What is meant by convergent sequence? Prove that every monotonic
non-decreasing bounded above sequence is convergent.
Find limit inferior and limit superior of the sequence Sn
Where Sn =−1 ∈
Explain any two tests for convergence of a series.
Answer the following (any one) 06
Explain Riemann integration of a continuous function.
Prove or disprove: Every Cauchy sequence is convergent.
Q.4 Answer the following (any two) 10
State and prove fundamental theorem on calculus.
Explain Lagrange's method for obtaining constrained maxima or minima.
What is meant by Open set? Prove that a set is open, if and only if its
compliment is closed.
Answer the following (any one) 04
Define radius of convergence. Also find it for the following power series.
State Taylor's theorem. Find the power series expansion for the
following functions:
f x log 1 x
ii) f x e−x
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Q.5 Answer the following (any two) 14
Define countable set. Prove that countable union of countable sets is always
countable.
Define limit superior and limit inferior of a sequence. Find the same for
following sequences, hence verify their convergence.
2 +−1 ∈
ii) ∈
Examine the convergence of p-series for various values of p.