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

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

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110014 Date: 06/06/2019

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)
State Euler?s theorem on homogeneous function. If
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
3 3
1
tan , then
prove that u u
y
u
y
y x
u
xy
x
u
x 3 cos sin 2 2
2
2
2
2
2
2
2
?
?
?
?
? ?
?
?
?
?

05
(b) (i)
If
3 3 3 2
z x y xy u ? ? ? ? ,show that u
z
u
z
y
u
y
x
u
x 3 ?
?
?
?
?
?
?
?
?

03
(ii)
If ? ? xyz z y x u 3 ln
3 3 3
? ? ? ? , prove that
? ?
2
2
9
z y x
u
z y x
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?

03
(c)
Determine whether
? ? ? ?
2 4
2
0 , 0 ,
lim
y x
y x
y x
?
?
exist or not? If they exist find the value of
the limit.
03

Q.2 (a) Find maxima and minima of the function
? ? 20 12 3 ,
3 3
? ? ? ? ? y x y x y x f
05
(b)
Expand y e
x 1
tan
?
about ? ? 1 , 1 up to second degree in ? ? 1 ? x and ? ? 1 ? y .
05
(c)
If ? ? sin , cos r y r x ? ? , find
? ?
? ? ? ,
,
r
y x
?
?
and
? ?
? ? y x
r
,
,
?
? ?

04

FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110014 Date: 06/06/2019

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)
State Euler?s theorem on homogeneous function. If
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
3 3
1
tan , then
prove that u u
y
u
y
y x
u
xy
x
u
x 3 cos sin 2 2
2
2
2
2
2
2
2
?
?
?
?
? ?
?
?
?
?

05
(b) (i)
If
3 3 3 2
z x y xy u ? ? ? ? ,show that u
z
u
z
y
u
y
x
u
x 3 ?
?
?
?
?
?
?
?
?

03
(ii)
If ? ? xyz z y x u 3 ln
3 3 3
? ? ? ? , prove that
? ?
2
2
9
z y x
u
z y x
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?

03
(c)
Determine whether
? ? ? ?
2 4
2
0 , 0 ,
lim
y x
y x
y x
?
?
exist or not? If they exist find the value of
the limit.
03

Q.2 (a) Find maxima and minima of the function
? ? 20 12 3 ,
3 3
? ? ? ? ? y x y x y x f
05
(b)
Expand y e
x 1
tan
?
about ? ? 1 , 1 up to second degree in ? ? 1 ? x and ? ? 1 ? y .
05
(c)
If ? ? sin , cos r y r x ? ? , find
? ?
? ? ? ,
,
r
y x
?
?
and
? ?
? ? y x
r
,
,
?
? ?

04

Q.3 (a)
Expand
x
e
x
cos
in Maclaurin?s series.
05
(b) (i)
Evaluate
2
0
sin tan
lim
x
x x
x
?
?
.
02
(ii)
Find the values of a and b such that,
? ?
1
sin cos 1
lim
3
0
?
? ?
?
x
x b x a x
x

03
(c)
Using Taylor?s series find
3
12 . 27 correct to four decimal places.
04
Q.4 (a)
Trace the curve ? ? ? ? x a x x a y ? ? ? 3
2 2

05
(b) Trace the curve ? ? ? cos 1 ? ?a r 05
(c)
Using reduction formula evaluate ? ?
?
2
0
5
cos
?
dx i and ? ?
?
?
0
6
sin dx ii
04

Q.5 (a) (i)
Test the convergence of
?
?
?
?
1
2
1
1
n
n

02
(ii)
Test the convergence of
?
?
?1
2
3
n
n
n

03
(b)
Test the convergence of
?
?
?
?
1
2
n
n
ne
04
(c)
Obtain the reduction formula for
?
2
0
cos
?
xdx
n

05
Q.6 (a)
Evaluate
?
R
sin dA ? , where R the region is in the 1
st
quadrant. i.e. outside the
circle 2 ? r and inside the cardioids ? ? ? cos 1 2 ? ? r .
05
(b)
Evaluate by changing the order of integration
? ?
? 1
0
2
2

x
x
dydx xy
05
(c)
Find the volume bounded by cylinder 4
2 2
? ? y x and the planes 4 ? ? z y ,
0 ? z .
04

FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110014 Date: 06/06/2019

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)
State Euler?s theorem on homogeneous function. If
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
3 3
1
tan , then
prove that u u
y
u
y
y x
u
xy
x
u
x 3 cos sin 2 2
2
2
2
2
2
2
2
?
?
?
?
? ?
?
?
?
?

05
(b) (i)
If
3 3 3 2
z x y xy u ? ? ? ? ,show that u
z
u
z
y
u
y
x
u
x 3 ?
?
?
?
?
?
?
?
?

03
(ii)
If ? ? xyz z y x u 3 ln
3 3 3
? ? ? ? , prove that
? ?
2
2
9
z y x
u
z y x
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?

03
(c)
Determine whether
? ? ? ?
2 4
2
0 , 0 ,
lim
y x
y x
y x
?
?
exist or not? If they exist find the value of
the limit.
03

Q.2 (a) Find maxima and minima of the function
? ? 20 12 3 ,
3 3
? ? ? ? ? y x y x y x f
05
(b)
Expand y e
x 1
tan
?
about ? ? 1 , 1 up to second degree in ? ? 1 ? x and ? ? 1 ? y .
05
(c)
If ? ? sin , cos r y r x ? ? , find
? ?
? ? ? ,
,
r
y x
?
?
and
? ?
? ? y x
r
,
,
?
? ?

04

Q.3 (a)
Expand
x
e
x
cos
in Maclaurin?s series.
05
(b) (i)
Evaluate
2
0
sin tan
lim
x
x x
x
?
?
.
02
(ii)
Find the values of a and b such that,
? ?
1
sin cos 1
lim
3
0
?
? ?
?
x
x b x a x
x

03
(c)
Using Taylor?s series find
3
12 . 27 correct to four decimal places.
04
Q.4 (a)
Trace the curve ? ? ? ? x a x x a y ? ? ? 3
2 2

05
(b) Trace the curve ? ? ? cos 1 ? ?a r 05
(c)
Using reduction formula evaluate ? ?
?
2
0
5
cos
?
dx i and ? ?
?
?
0
6
sin dx ii
04

Q.5 (a) (i)
Test the convergence of
?
?
?
?
1
2
1
1
n
n

02
(ii)
Test the convergence of
?
?
?1
2
3
n
n
n

03
(b)
Test the convergence of
?
?
?
?
1
2
n
n
ne
04
(c)
Obtain the reduction formula for
?
2
0
cos
?
xdx
n

05
Q.6 (a)
Evaluate
?
R
sin dA ? , where R the region is in the 1
st
quadrant. i.e. outside the
circle 2 ? r and inside the cardioids ? ? ? cos 1 2 ? ? r .
05
(b)
Evaluate by changing the order of integration
? ?
? 1
0
2
2

x
x
dydx xy
05
(c)
Find the volume bounded by cylinder 4
2 2
? ? y x and the planes 4 ? ? z y ,
0 ? z .
04

Q.7 (a)
Prove that
?
?
1
1
dx
x
p
, converges when 1 ? p and diverges when 1 ? p
05
(b) Use triple integral in cylindrical co-ordinate to find the volume of solid, bounded
above the hemisphere
2 2
25 y x z ? ? ? , below by plane xy ? and laterally by
the cylinder 9
2 2
? ? y x
05
(c) Find the volume of a cone with height cm 4 and radius of base cm 4 . Use the
method of slicing.
04

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