Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem Old 110014 Calculus Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem Old 110014 Calculus Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110014 Date: 17/01/2020

Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
(i) Test the convergence of the sequence
2
2
2
nn
nn
? ?
?
?
?
.

03
(ii) Expand logx in powers of ( 1) x ? . 04
(b)
(i) Evaluate ? ?
cot
0
lim cos
x
x
x
?

03

(ii) Test the convergence of the series
1
1
sin
n
n
?
?
?


04

Q.2 (a)
(i) Evaluate
2
11
lim
2 log( 1)
x
xx
?
?
?
?
?
?


03

(ii) Determine the interval of convergence for the series
1
,0
2
n
n
n
x
x
?
?
?
?

04
(b)
Expand log cos
4
x
? ?
?
?
?
using Taylor`s theorem in ascending powers of x and
hence find the value of
0
log(cos48 ) correct up to three decimal places.
07

Q.3 (a)
(i) Evaluate
1 9
2
0
1
x
dx
x ?
?


03

(ii) Test the convergence of the improper integral
4
4
35
7
x
dx
x
?
?
?
?


04
(b)
(i) Show that
22
( , ) , ( , ) (0,0)
0, ( , ) (0,0)
xy
f x y x y
xy
xy
?
?
?
is continuous at origin.

04

(ii) If ,
xyz
ue ? show that ? ?
3
2 2 2
13
xyz
u
xyz x y z e
x y z
?
? ? ?
? ? ?


03
Q.4 (a)
If () u f r ? and
2 2 2 2
r x y z ? ? ? , prove that
222
2 2 2
2
( ) ( )
uuu
f r f r
x y z r
?
? ? ? ? ? ? ?
? ? ?

07
(b)
(i) If
,,
x y z
uf
y z x
?
?
?
?
, prove that
0
u u u
x y z
x y z
? ? ?
? ? ?
? ? ?

04
(ii) Find the equations of the tangent plane and normal line to the surface
22
2 z x y ? at the point (1,1,3)
03

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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110014 Date: 17/01/2020

Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
(i) Test the convergence of the sequence
2
2
2
nn
nn
? ?
?
?
?
.

03
(ii) Expand logx in powers of ( 1) x ? . 04
(b)
(i) Evaluate ? ?
cot
0
lim cos
x
x
x
?

03

(ii) Test the convergence of the series
1
1
sin
n
n
?
?
?


04

Q.2 (a)
(i) Evaluate
2
11
lim
2 log( 1)
x
xx
?
?
?
?
?
?


03

(ii) Determine the interval of convergence for the series
1
,0
2
n
n
n
x
x
?
?
?
?

04
(b)
Expand log cos
4
x
? ?
?
?
?
using Taylor`s theorem in ascending powers of x and
hence find the value of
0
log(cos48 ) correct up to three decimal places.
07

Q.3 (a)
(i) Evaluate
1 9
2
0
1
x
dx
x ?
?


03

(ii) Test the convergence of the improper integral
4
4
35
7
x
dx
x
?
?
?
?


04
(b)
(i) Show that
22
( , ) , ( , ) (0,0)
0, ( , ) (0,0)
xy
f x y x y
xy
xy
?
?
?
is continuous at origin.

04

(ii) If ,
xyz
ue ? show that ? ?
3
2 2 2
13
xyz
u
xyz x y z e
x y z
?
? ? ?
? ? ?


03
Q.4 (a)
If () u f r ? and
2 2 2 2
r x y z ? ? ? , prove that
222
2 2 2
2
( ) ( )
uuu
f r f r
x y z r
?
? ? ? ? ? ? ?
? ? ?

07
(b)
(i) If
,,
x y z
uf
y z x
?
?
?
?
, prove that
0
u u u
x y z
x y z
? ? ?
? ? ?
? ? ?

04
(ii) Find the equations of the tangent plane and normal line to the surface
22
2 z x y ? at the point (1,1,3)
03

2
Q.5 (a)
If
33
1
cos
xy
u
xy
?
? ?
?
?
?
?
,show that
(i)
2cot
uu
x y u
xy
?
? ? ?
?

(ii)
2 2 2
22
2 2 3
sin 2 sin 4cos
2
sin
u u u u u u
x xy y
x x y y u
? ? ? ?
? ? ?
? ? ? ?

07
(b)
(i) Trace the curve
33
3 , 0 x y axy a ? ? ?
04

(ii) Discuss the maxima and minima of the function
22
6 12 x y x ? ? ?
03
Q.6 (a)
(i) Evaluate
2
11
22
00
1
x
dxdy
xy
?
?
?


03

(ii) Evaluate
2
0
y
x
e dydx
?
?
?
,by changing the order of integration.

04
(b)
The temperature at any point ( , , ) x y z in space is
2
400 T xyz ? .Find the
highest temperature on the surface of unit sphere
2 2 2
1 x y z ? ? ? by the
method of Lagrange`s multipliers.

07

Q.7 (a)
Evaluate
22
( ) ,
R
x y dA ?
?
by changing the variables, where R is the region
lying in the first quadrant and bounded by the hyperbolas
2 2 2 2
1, 9, 2 x y x y xy ? ? ? ? ? and 4 xy ?

07
(b)
Evaluate
22
2
3
2 2 2
2
1
11
()
1 0 0
xy
x
x y z
e dxdydz
?
?
? ? ?
?
? ? ?


07

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