Download PTU B.Sc (Non-Medical) 2020 March 3rd Sem 76904 Analysis I Question Paper

Download PTU (I.K. Gujral Punjab Technical University Jalandhar (IKGPTU) B.Sc (Non-Medical) 2020 March Previous Question Papers

1 | M-76904 (S105)-1836
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
B.Sc. (Non Medical) (2018 Batch) (Sem.?3)
ANALYSIS-I
Subject Code : BSNM-305-18
M.Code : 76904
Time : 3 Hrs. Max. Marks : 50
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE marks
each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students
have to attempt any TWO questions.

SECTION-A
1. Write briefly :
a) Prove that
1
n
n
n
? ?
? ?
?
? ?
?
is divergent.
b) Define Absolute and Conditional Convergence.
c) Define lower & upper Riemann integral.
d) If f ? R [0, a], then prove that
0 0
( ) ( ) , 0
a a
f x dx f a x dx a ? ? ?
? ?
.
e) Define improper integral of first kind.
f) Examine the convergence of
0
4x
e dx
? ?
?
.
g) Define Gamma Function.
h) Prove symmetry of Beta Function.
i) State comparison test for series.
j) Write the relation between Beta & Gamma function.
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1 | M-76904 (S105)-1836
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
B.Sc. (Non Medical) (2018 Batch) (Sem.?3)
ANALYSIS-I
Subject Code : BSNM-305-18
M.Code : 76904
Time : 3 Hrs. Max. Marks : 50
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE marks
each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students
have to attempt any TWO questions.

SECTION-A
1. Write briefly :
a) Prove that
1
n
n
n
? ?
? ?
?
? ?
?
is divergent.
b) Define Absolute and Conditional Convergence.
c) Define lower & upper Riemann integral.
d) If f ? R [0, a], then prove that
0 0
( ) ( ) , 0
a a
f x dx f a x dx a ? ? ?
? ?
.
e) Define improper integral of first kind.
f) Examine the convergence of
0
4x
e dx
? ?
?
.
g) Define Gamma Function.
h) Prove symmetry of Beta Function.
i) State comparison test for series.
j) Write the relation between Beta & Gamma function.
2 | M-76904 (S105)-1836
SECTION-B
2. Test the following series for convergence.
3 3 3
1 2 3
........
2 3 4
? ? ?
3. State and prove Necessary & Sufficient condition for a bounded function to be R-
integrable on [a, b].
4. State and prove Abel?s Test.
5. Prove that B(m, n)
1 1
0 0
; , , 0
(1 ) ( )
m m
m n m n
x x
dx dx m n
x x
? ?
? ?
? ?
? ?
?
? ?
.
6. If a function f is R-integrable on [a, b] then f
2
is also R-integrable on [a, b]

SECTION-C
7. a) Show that the series
( 1) ( 2)
2 5
n
n
n ? ?
?
is absolutely convergent.
b) If a function f is integrable on [a, b] then m (b ? a) ? ( ) ( )
b
a
f x dx M b a ? ?
?
.
8. a) Show that
1
1
0
sin
, 0
x
p
dx p
x
?
?
converges absolutely for p < 1.
b) Prove that
2
1 (2 1)
2 2 ( 1)
n
n
n
? ? ? ? ?
? ? ?
? ?
? ?
? ?
.
9. a) If f
1
& f
2
? R (a, b) & f
1
(x) ? f
2
(x) ? x ? [a, b] then
1 2
( ) ( )
b b
a a
f x dx f x dx ?
? ?
.
b) Prove that B (m, n) = B (m, n + 1) +B (m + 1, n)

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.
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