No.

of Printed Pages : 12 BMTC-133

BACHELOR OF SCIENCE (GENERAL)

(BSCG/BAG)
Term-End Examination
June, 2022

BMTC-133 : REAL ANALYSIS

Time : 3 hours Maximum Marks : 100

Note : The question paper has three sections – A, B and C.
All questions in Section A and Section B are
compulsory. In Section C, do any five questions
out of seven questions. Use of calculators is not
allowed.

SECTION A

1. State whether the following statements are True

or False. Give a short proof or a counter-example
in support of your answer. 102=20

(i) There is no bijection from N to Q  [0, 1].

(ii) Every finite set has empty interior.

(iii) Every real valued function f has a point x in

its domain such that f(x) = x.

(iv) If a sequence is bounded, then it has at least

two convergent subsequences.

BMTC-133 1 P.T.O.
 1 1 1 1
(v) The series – + – + ... is a
3 7 11 15
convergent series.

(vi) The function f defined on R by

f(x) = |x + 15| has a local minimum at
x = – 15.

(vii) For  = 1, there does not exist any  > 0 such

that
x2
0 < |x| <   – 2 < .
x 1

(viii) The function f(x) = [x] is integrable on [0, 4].

x 2  nx
(ix) For any x  R, lim = x2.
n n

(x) The sequence (Sn)nN , where

1 1 1
Sn = 1 + + + .... + , is convergent.
2 3 n

BMTC-133 2
 SECTION B

2. (a) Find the supremum of the set


 n 

 n  N . 3
n  1
 


(b) Check whether the sequence

 n2 – n  is monotone or not. 2
 
  nN

3. Find the nth partial sum of the following series,

and hence check whether they are convergent or
not. 5


(i)
 n 1


1
ln  1  
n


(ii)
 n 1
1
n(n  1)

4. Let f be a differentiable function whose

derivative never vanishes on [a, b]. Show that f is
either strictly decreasing or strictly increasing. 5


5. Show that the series

n 1
sin (n 3 x)
n3
is uniformly

convergent on [0, [ . 5

BMTC-133 3 P.T.O.
6. Let f and g be two real-valued functions defined
on [a, b] such that f is Riemann integrable and g
is differentiable with g(x) = f(x)  x  [a, b].
b

Then show that


a
f (x) dx = g(b) – g(a). 5

7. (a) Using the definition show that the

 1 
sequence   is Cauchy. 3
 n  nN

(b) Check whether the set of integers is

countable or not. 2

BMTC-133 4
 SECTION C
8. (a) Show that if f is a real-valued continuous
function defined on a closed interval [, ],
then f is Riemann integrable over [, ]. 6
x –1
(b) Let f be a function defined by f(x) = ,
x2  3
x  R. Using the  –  definition, show that
1
f(x)  whenever x  2. 4
7

9. (a) Let f : [0, 2]  R be defined by

– 1, 0  x  1
f(x) =  .
 1, 1  x  2
Show that there is no real-valued function
defined on [0, 2] whose derivative is f. 5

(b) Check whether the sequence

 –5 
  1  1   is convergent or not. 5
 n 
  nN

10. (a) Find the sequence of partial sums of the


series
 x
n(n  1)
n 1
, where x  [0, [ . Does

the sequence converge pointwise ? Does it

converge uniformly ? Justify your answers. 5

(b) Test the absolute and conditional


convergence of the series
n 1
(– 1) n
(3n  5)
. 5

BMTC-133 5 P.T.O.
11. (a) Prove or disprove that
n
3
3n – 2n >    n  2. 5
2

x2
(b) Prove that x – < ln (1 + x),  x > 0. 5
2

12. (a) Show that the function f(x) = x3 is

uniformly continuous on R. 5

(b) Check whether the function f defined by

3x – 4
f(x) = , x  {0, 1}, has a local
x2 – x

extrema. 5

13. (a) For any two real numbers x and y show

that |x – y|  ||x| – |y||. 4

(b) Show that every convergent sequence is

bounded. Is the converse true ? Justify your
answer. 6

14. (a) Find the primitive of tan–1 x and evaluate

1

the integral

0
tan – 1 x . 5

(b) State and prove Bolzano-Weierstrass

Theorem. 5

BMTC-133 6
 ~r.E_.Q>r.gr.-133

{dkmZ ñZmVH$ (gm_mÝ`)

(~r.Eg.gr.Or. / ~r.E.Or.)
gÌm§V narjm
OyZ, 2022

~r.E_.Q>r.gr.-133 : dmñV{dH$ {díbofU

g_` : 3 KÊQ>o A{YH$V_ A§H$ : 100
ZmoQ> : Bg àíZ-nÌ _| VrZ ^mJ h¢ — H$, I Am¡a J & ^mJ H$ Am¡a
^mJ I Ho$ g^r àíZ A{Zdm`© h¢ & ^mJ J _| gmV àíZm| _|
go H$moB© nm±M àíZ H$s{OE & H¡$ëHw$boQ>am| Ho$ à`moJ H$aZo H$s
AZw_{V Zht h¡ &

^mJ> H$
1. ~VmBE {H$ {ZåZ{b{IV H$WZ gË` h¢ `m AgË` & AnZo
CÎma Ho$ nj _| bKw Cnn{Îm `m à{V-CXmhaU Xr{OE & 102=20
(i) N go Q  [0, 1] na H$moB© EH¡$H$s-AmÀN>mXZ Zht h¡ &
(ii) àË`oH$ n[a{_V g_wƒ` H$m Aä`§Va [aº$ hmoVm h¡ &
(iii) àË`oH$ dmñV{dH$ _mZ \$bZ f Ho$ àm§V _| EH$ q~Xþ x
Eogm hmoVm h¡ {H$ f(x) = x hmo &
(iv) `{X EH$ AZwH«$_ n[a~Õ h¡, Vmo BgHo$ H$_-go-H$_ Xmo
A{^gmar CnAZwH«$_ hmoVo h¢ &
BMTC-133 7 P.T.O.
 1 1 1 1
(v) loUr – + – + ... EH$ A{^gmar loUr
3 7 11 15
h¡ &

(vi) f(x) = |x + 15| Ûmam R na n[a^m{fV \$bZ f H$m

x = – 15 na ñWmZr` {ZpåZîR> h¡ &

(vii) =1 Ho$ {bE Eogm H$moB© ^r >0 H$m ApñVËd Zht
h¡ {OgHo$ {bE
x2
0 < |x| <   –2 < hmoVm hmo &
x 1

(viii) \$bZ f(x) = [x], [0, 4] na g_mH$bZr` h¡ &

x 2  nx
(ix) {H$gr ^r xR Ho$ {bE, lim = x2
n n

h¡ &

(x) AZwH«$_ (Sn)nN , Ohm±

1 1 1
Sn = 1 + + + .... + , A{^gmar h¡ &
2 3 n

BMTC-133 8
 ^mJ> I


 n 

2. (H$) g_wƒ`  n  N H$m Ý`yZV_ Cn[a
n  1
 

n[a~§Y kmV H$s{OE & 3

(I) Om±M H$s{OE {H$ AZwH«$_  n2 – n 


  nN
EH${XîQ> h¡ `m Zht & 2

3. {ZåZ{b{IV lo{U`m| Ho$ nd| Am§{eH$ `moJ\$b kmV H$s{OE,

Am¡a Bg àH$ma Om±M H$s{OE {H$ `o A{^gmar h¢ `m Zht & 5


(i)

n 1


1
ln  1  
n


(ii)

n 1
1
n(n  1)

4. _mZ br{OE f EH$ AdH$bZr` \$bZ h¡ {OgH$m AdH$bO

[a, b] na H$ht ^r eyÝ` Zht h¡ & {XImBE {H$ f `m Vmo {Za§Va
õmg_mZ h¡ `m {Za§Va dY©_mZ h¡ & 5


5. {XImBE {H$ loUr 
n 1
sin (n 3 x)
n3
, [0, [ na EH$g_mZV:

A{^gmar h¡ & 5

BMTC-133 9 P.T.O.
6. _mZ br{OE f Am¡a g, [a, b] na n[a^m{fV Xmo dmñV{dH$-_mZ
\$bZ Bg àH$ma h¢ {H$ f ar_mZ g_mH$bZr` h¡ Am¡a g
AdH$bZr` h¡ Am¡a g^r x  [a, b] Ho$ {bE g(x) = f(x)
b

h¡ & V~ {XImBE {H$


a
f (x) dx = g(b) – g(a). 5

7. (H$) n[a^mfm H$m à`moJ H$aHo$ {XImBE {H$ AZwH«$_

 1 
  H$m°er h¡ & 3
 n  nN

(I) Om±M H$s{OE {H$ nyUmªH$m| H$m g_wƒ` JUZr` h¡ `m

Zht & 2

BMTC-133 10
 ^mJ> J
8. (H$) {XImBE {H$ `{X f EH$ g§d¥V A§Vamb [, ] na
n[a^m{fV EH$ g§VV dmñV{dH$-_mZ \$bZ h¡, Vmo f,
[, ] na ar_mZ g_mH$bZr` h¡ & 6
x –1
(I) _mZ br{OE f EH$ \$bZ h¡ Omo f(x) = ,
x2  3
xR Ûmam n[a^m{fV h¡ & – n[a^mfm H$m à`moJ
1
H$aHo$, {XImBE {H$ f(x)  O~ ^r x  2 h¡ & 4
7
9. (H$) _mZ br{OE f : [0, 2]  R {ZåZ àH$ma go
n[a^m{fV h¡ :
– 1, 0  x  1
f(x) = 
 1, 1  x  2
{XImBE {H$ [0, 2] na n[a^m{fV Eogm H$moB© ^r
dmñV{dH$-_mZ \$bZ Zht h¡ {OgH$m AdH$bO f h¡ & 5
 –5 
(I) Om±M H$s{OE {H$ AZwH«$_   1  1  
 n 
  nN
A{^gmar h¡ `m Zht & 5

10. (H$) loUr 
n 1
x
n(n  1)
Ho$ Am§{eH$ `moJ\$bm| H$m

AZwH«$_ kmV H$s{OE, Ohm± x  [0, [ h¡ & Š`m `h

AZwH«$_ q~Xþe: A{^gaU H$aVm h¡ ? Š`m `h
EH$g_mZV: A{^gaU H$aVm h¡ ? AnZo CÎmam| H$s
nw{îQ> H$s{OE & 5

(I) loUr n 1
(– 1) n
(3n  5)
Ho$ {Zanoj A{^gaU d

gà{V~§Y A{^gaU H$s Om±M H$s{OE & 5

BMTC-133 11 P.T.O.
11. (H$) {gÕ `m A{gÕ H$s{OE {H$ : 5
n
3
3n – 2n >    n  2
2

(I) {gÕ H$s{OE {H$ g^r x > 0 Ho$ {bE,

2
x
x– < ln (1 + x) h¡ & 5
2

12. (H$) {XImBE {H$ \$bZ f(x) = x3, R na EH$g_mZV:

g§VV h¡ & 5

3x – 4
(I) Om±M H$s{OE {H$ f(x) = , x  {0, 1}, Ûmam
x2 – x
n[a^m{fV \$bZ f Ho$ ñWmZr` Ma_ _mZ h¢ `m Zht & 5

13. (H$) H$moB© ^r Xmo dmñV{dH$ g§»`mAm| x Am¡a y Ho$ {bE,

{XImBE {H$ |x – y|  ||x| – |y|| hmoVm h¡ & 4

(I) {XImBE {H$ àË`oH$ A{^gmar AZwH«$_ n[a~Õ hmoVm

h¡ & Š`m BgH$m {dbmo_ ^r gË` h¡ ? AnZo CÎma H$s
nw{îQ> H$s{OE & 6

14. (H$) tan–1 x H$m nyd©J kmV H$s{OE Am¡a g_mH$b

1


0
tan – 1 x H$m _mZ kmV H$s{OE & 5

(I) ~moëOmZmo-dm`ñQ´>m©g à_o` H$m H$WZ Xr{OE Am¡a Bgo

{gÕ H$s{OE & 5

BMTC-133 12