Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) MTH001 Calculus Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) MTH001 Calculus Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (SPFU) EXAMINATION ? SUMMER-2019
Subject Code: MTH001 Date: 07/06/2019

Subject Name: Calculus

Time: 10:30 AM TO 01:00 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) Discuss the convergence of the series
? ?
1
1
1
n
n
n
n
nx
n
?
?
?
?
?

04

(ii) Test the convergence of the series
? ?
1
1
1
21
n
n
n
n
?
?
?
?
?
?

03
(b)
If
2 2 2 2
,
m
u r r x y z ? ? ? ? show that ? ?
222
2
2 2 2
1
m
uuu
m m r
x y z
?
?
? ? ? ?
? ? ?

07

Q.2 (a)
(i) Show that
22
( , ) , ( , ) (0,0)
0, ( , ) (0,0)
xy
f x y x y
xy
xy
?
?
?
is continuous at the origin.
04


(ii) If cos , sin x r y r ? ? , show that
2
2
1.
rr
xy
? ? ?
?
? ?
?
?
?

03
(b)
Determine absolute or conditional convergence of the series
? ?
2
3
1
1
1
n
n
n
n
?
?
?
?
?

07

Q.3 (a)
(i) Evaluate
1
00
x
dydx
?
.
03

(ii) Evaluate xydxdy
?
over the region enclosed by the x-axis, the line 2 xa ?
and the parabola
2
4 x ay ? .
04
(b)
If
1
sin ,
xy
u
xy
?
?
?
?
?
?
?
?
prove that
(i) 2 2 tan
uu
x y u
xy
?
?
?

(ii)
? ?
2 2 2
2 2 3
22
1
2 tan tan
4
u u u
x xy y u u
x x y y
? ? ?
? ? ? ?
? ? ? ?

07

Q.4 (a) (i) Find the equation of the tangent plane and normal line to the surface
22
2 z x y ? at the point (1, 1 ,3).
04

(ii) If
22
4 , , 2 u y ax x at y at ? ? ? ? find
du
dt
.
03
(b) Use triple integral to find the volume of the solid within the cylinder
22
9 xy ? between the planes 1 z ? and 1 xz ? .
07

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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (SPFU) EXAMINATION ? SUMMER-2019
Subject Code: MTH001 Date: 07/06/2019

Subject Name: Calculus

Time: 10:30 AM TO 01:00 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) Discuss the convergence of the series
? ?
1
1
1
n
n
n
n
nx
n
?
?
?
?
?

04

(ii) Test the convergence of the series
? ?
1
1
1
21
n
n
n
n
?
?
?
?
?
?

03
(b)
If
2 2 2 2
,
m
u r r x y z ? ? ? ? show that ? ?
222
2
2 2 2
1
m
uuu
m m r
x y z
?
?
? ? ? ?
? ? ?

07

Q.2 (a)
(i) Show that
22
( , ) , ( , ) (0,0)
0, ( , ) (0,0)
xy
f x y x y
xy
xy
?
?
?
is continuous at the origin.
04


(ii) If cos , sin x r y r ? ? , show that
2
2
1.
rr
xy
? ? ?
?
? ?
?
?
?

03
(b)
Determine absolute or conditional convergence of the series
? ?
2
3
1
1
1
n
n
n
n
?
?
?
?
?

07

Q.3 (a)
(i) Evaluate
1
00
x
dydx
?
.
03

(ii) Evaluate xydxdy
?
over the region enclosed by the x-axis, the line 2 xa ?
and the parabola
2
4 x ay ? .
04
(b)
If
1
sin ,
xy
u
xy
?
?
?
?
?
?
?
?
prove that
(i) 2 2 tan
uu
x y u
xy
?
?
?

(ii)
? ?
2 2 2
2 2 3
22
1
2 tan tan
4
u u u
x xy y u u
x x y y
? ? ?
? ? ? ?
? ? ? ?

07

Q.4 (a) (i) Find the equation of the tangent plane and normal line to the surface
22
2 z x y ? at the point (1, 1 ,3).
04

(ii) If
22
4 , , 2 u y ax x at y at ? ? ? ? find
du
dt
.
03
(b) Use triple integral to find the volume of the solid within the cylinder
22
9 xy ? between the planes 1 z ? and 1 xz ? .
07

2
Q.5 (a)
(i) Prove that
2 4 8 16
1 ........
3 9 27 81
? ? ? ? ? converges and find its term.
03

(ii)Investigate the convergence of the series
1
25
3
n
n
n
?
?
?
?

04
(b)
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

Q.6 (a)
(i) Discuss the maxima and minima of the function
2 2 3
3x y x ?
04

(ii)Test the convergence of the series
? ?
44
1
11
n
nn
?
?
? ? ?
?

03
(b)
Evaluate xyzdxdydz
?
over the positive octant of the sphere
2 2 2
4. x y z ? ? ?
07

Q.7 (a)
Evaluate
22
22 0
a a y
a
a a y
dxdy
?
?
?
by changing the order of integration.
07
(b)
(i) Find the minimum value of
22
xy ? , subject to the condition . ax by c ?
04

(ii)Expand
xy
e
?
in power of ? ? 1 x ? and ? ? 1 y ? up to first degree terms.
03

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