RGU Question Paper Mathematics Semester II 2022: Real Analysis

 

2022

(May/June)

BA/B.Sc (II Semester) Examination (CBCS)

MATHEMATICS

Paper: MAT-CC-121

(Real Analysis)

Full Marks: 80

Pass Marks: 35%

Time: Three Hours

Note: 1. Questions are divided into three Sections-A, B and C.

2. Answer the questions as per instructions given in each Section.

3. The figures in the margin indicate full marks for the questions.

Section-A

1. Answer any four parts of the following: 5×4=20

(a) Define least upper bound and greatest lower bound. When will the given set be bounded? Give an example of a bounded set.

(b) Define Riemann Integral and show that the set

is not integrable in Riemann sense.

(c) State and prove the general criterion of Cauchy's convergence for series.

(d) State D' Alembert's ratio test. Show that the series

$n\ge 1$, converges for x≤1 and diverges for x>1.

(e) Prove that every bounded sequence with a unique limit point is convergent.

(f) Define closure, interior and boundary of a set. Differentiate between boundary point and frontier point with the help of suitable example.

Section-B

2. Answer any three parts of the following: 10×3=30

(a) (i) Define ordered set. What do you mean by order completeness? Is the set of rational number complete?

(ii) Define open set. Whenever will a given set be closed? Further show that the set

is closed. 5+5=10

(b) (i) Define Riemann upper sums and Riemann lower sums. State and prove the necessary and sufficient conditions of Riemann integrability.

(ii) State and prove the fundamental theorem of calculus. 5+5=10

(c) (i) Define continuity. Show that the function

is discontinuous at x=0.

(ii) Find limit inferior and limit superior of the sequence

5+5=10

(d) (i) State comparison test, condensation test and Cauchy's root test.

(ii) Show the series

is absolutely convergent. 5+5=10

(e) (i) If $\underset{x\to a}{\lim }f\left(x\right)$ exists, then show that it must be unique.

(ii) Evaluate:

5+5=10

Section-C

3. Answer any two parts of the following: 15×2=30

(a) (i) What do you mean by integral as a limit of a sum?

(ii) Evaluate the definite integral

using the definite integral as the limit of a sum.

(iii) Show that

for $a\le c\le b$. 5+5+5=15

(b) (i) Define a countable set. Prove that a countable union of countable sets is countable.

(ii) State and prove Bolzano-Weierstrass theorem.

(iii) State and prove Heine-Borel theorem. 5+5+5=15

(c) (i) If a function f is continuous on [a, b] and $f\left(a\right)\ne f\left(b\right)$, then prove that it assumes every value between f(a) and f(b).

(ii) Examine the validity of the hypothesis and conclusion of Rolle's theorem for the function $f\left(x\right)=x^3-4x$ on the interval [-2, 2].

(iii) Evaluate:

5+5+5=15

(d) (i) Define subsequence of a sequence with suitable example. What can you say about its convergence?

(ii) If $<a_n>$ is a bounded sequence such that $a_n>0$ for all $n\in \mathbb{N}$, then prove that

if $\overline{lim}\left(a_n\right)>0$ where $\overline{lim}$ and $\underset{\_}{lim}$ denote limit inferior and limit superior, respectively.

(iii) Show that

5+5+5=15