Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem New And Spfu 2110014 Calculus Previous Question Paper
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110014 Date: 17/01/2020
Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Objective Question (MCQ)
Mark
(a) 07
1.
The sum of the series
1 1 1
1 ...
2 4 8
? ? ? ?
(A) 1 (B) 2 (C) 3 (D) Infinity
2.
The series
1
1
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
3.
The series
2
1
sin
n
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
4.
The curve
? ?
2
2
91 y x x ?? is symmetric about
(A) x-axis (B) y-axis (C) Line y=x (D) origin
5.
A point ? ? , ab is said to be a saddle point if at ? ? , ab
(A)
2
0 rt s ??
(B)
2
0 rt s ??
(C)
2
0 rt s ??
(D)
2
0 rt s ??
6.
The volume of solid generated by revolving a circle
22
9 xy ?? about x-
axis
(A) 4
3
?
(B)
36
(C)
36
3
?
(D)
36 ?
7.
The value of
sin
lim
x
x
x
??
??
??
??
(A) 1 (B) 0 (C) 2 (D) Infinity
(b) 07
1. Which of the following is homogeneous function of degree one?
(A)
2
x
y
(B)
2
2
x
y
(C) 2 x
y
?
(D) yx
yx xy ?
2.
The value of
? ? ? ?
22
, 1,1
lim
xy
xy
xy
?
?
?
(A) 2 (B) 0 (C) Infinity (D) 1
2
FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110014 Date: 17/01/2020
Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Objective Question (MCQ)
Mark
(a) 07
1.
The sum of the series
1 1 1
1 ...
2 4 8
? ? ? ?
(A) 1 (B) 2 (C) 3 (D) Infinity
2.
The series
1
1
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
3.
The series
2
1
sin
n
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
4.
The curve
? ?
2
2
91 y x x ?? is symmetric about
(A) x-axis (B) y-axis (C) Line y=x (D) origin
5.
A point ? ? , ab is said to be a saddle point if at ? ? , ab
(A)
2
0 rt s ??
(B)
2
0 rt s ??
(C)
2
0 rt s ??
(D)
2
0 rt s ??
6.
The volume of solid generated by revolving a circle
22
9 xy ?? about x-
axis
(A) 4
3
?
(B)
36
(C)
36
3
?
(D)
36 ?
7.
The value of
sin
lim
x
x
x
??
??
??
??
(A) 1 (B) 0 (C) 2 (D) Infinity
(b) 07
1. Which of the following is homogeneous function of degree one?
(A)
2
x
y
(B)
2
2
x
y
(C) 2 x
y
?
(D) yx
yx xy ?
2.
The value of
? ? ? ?
22
, 1,1
lim
xy
xy
xy
?
?
?
(A) 2 (B) 0 (C) Infinity (D) 1
2
3.
If cos , sin x r y r ?? ?? then the value of
x
r
?
?
(A) cos ? (B) sec ? (C) cosec ? (D) sin ?
4.
The value of
2
10
ln
x
x
dxdy
x
??
(A) 2ln2-2 (B) 2ln2-1 (C) ln2 (D) 0
5.
The value of
0
lim
x
x
x
?
(A) 1 (B) e (C) x (D) 0
6.
The value of
1
000
y x
dxdydz
???
(A) 1 (B) 1
2
(C) 1
3
(D) 1
6
7.
The value of
sin 2
3
00
r drd
?
?
?
??
(A)
32
?
(B)
2
?
(C) 3
64
?
(D) None
Q.2 (a)
Define Jacobian and show that ?? ? ?? ? = 1 .
.
03
(b)
Find the equations of tangent plane and normal line to
2 2 2
81 x y z ? ? ?
at the point ? ? 1, 4,8 ??
04
(c) A rectangular box open the top is to have a volume of 108 c.c. find the
dimension of the box requiring least material for its construction.
07
Q.3 (a)
Show that
22
u v v u
? ? ? ?
?
? ? ? ?
where
y
yx ? ? ?
03
(b)
Discuss the continuity of
? ?
? ? ? ?
? ? ? ?
22
44
; , 0,0
4
,
1
; , 0,0
5
xy
xy
xy
f x y
xy
?
?
?
? ?
?
?
?
?
?
?
04
(c) State and prove Euler?s Theorem for Homogeneous functions.
Also, if
22
1
33
22
sin
xy
u
xy
?
??
?
??
?
??
??
?
??
then show that
i.
1
tan
2
xy
xu yu u ??
ii.
? ?
3
1
2 tan tan
4
xx xy yy
xxu xyu yyu u u ? ? ? ?
07
FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110014 Date: 17/01/2020
Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Objective Question (MCQ)
Mark
(a) 07
1.
The sum of the series
1 1 1
1 ...
2 4 8
? ? ? ?
(A) 1 (B) 2 (C) 3 (D) Infinity
2.
The series
1
1
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
3.
The series
2
1
sin
n
n
n
?
?
?
is
(A) convergent (B) divergent (C) Oscillating (D) none
4.
The curve
? ?
2
2
91 y x x ?? is symmetric about
(A) x-axis (B) y-axis (C) Line y=x (D) origin
5.
A point ? ? , ab is said to be a saddle point if at ? ? , ab
(A)
2
0 rt s ??
(B)
2
0 rt s ??
(C)
2
0 rt s ??
(D)
2
0 rt s ??
6.
The volume of solid generated by revolving a circle
22
9 xy ?? about x-
axis
(A) 4
3
?
(B)
36
(C)
36
3
?
(D)
36 ?
7.
The value of
sin
lim
x
x
x
??
??
??
??
(A) 1 (B) 0 (C) 2 (D) Infinity
(b) 07
1. Which of the following is homogeneous function of degree one?
(A)
2
x
y
(B)
2
2
x
y
(C) 2 x
y
?
(D) yx
yx xy ?
2.
The value of
? ? ? ?
22
, 1,1
lim
xy
xy
xy
?
?
?
(A) 2 (B) 0 (C) Infinity (D) 1
2
3.
If cos , sin x r y r ?? ?? then the value of
x
r
?
?
(A) cos ? (B) sec ? (C) cosec ? (D) sin ?
4.
The value of
2
10
ln
x
x
dxdy
x
??
(A) 2ln2-2 (B) 2ln2-1 (C) ln2 (D) 0
5.
The value of
0
lim
x
x
x
?
(A) 1 (B) e (C) x (D) 0
6.
The value of
1
000
y x
dxdydz
???
(A) 1 (B) 1
2
(C) 1
3
(D) 1
6
7.
The value of
sin 2
3
00
r drd
?
?
?
??
(A)
32
?
(B)
2
?
(C) 3
64
?
(D) None
Q.2 (a)
Define Jacobian and show that ?? ? ?? ? = 1 .
.
03
(b)
Find the equations of tangent plane and normal line to
2 2 2
81 x y z ? ? ?
at the point ? ? 1, 4,8 ??
04
(c) A rectangular box open the top is to have a volume of 108 c.c. find the
dimension of the box requiring least material for its construction.
07
Q.3 (a)
Show that
22
u v v u
? ? ? ?
?
? ? ? ?
where
y
yx ? ? ?
03
(b)
Discuss the continuity of
? ?
? ? ? ?
? ? ? ?
22
44
; , 0,0
4
,
1
; , 0,0
5
xy
xy
xy
f x y
xy
?
?
?
? ?
?
?
?
?
?
?
04
(c) State and prove Euler?s Theorem for Homogeneous functions.
Also, if
22
1
33
22
sin
xy
u
xy
?
??
?
??
?
??
??
?
??
then show that
i.
1
tan
2
xy
xu yu u ??
ii.
? ?
3
1
2 tan tan
4
xx xy yy
xxu xyu yyu u u ? ? ? ?
07
Q.4 (a)
Evaluate sin
A
r drd ??
??
over the area of the curve
? ? 1 cos
2
r
? ?
? above
the initial line.
03
(b)
Evaluate the integral by the changing the order of integration,
3
82
4
0
1
y
x dxdy ?
??
04
(c)
i. Use triple integral to find the volume of the cylinder
22
1 xy ??
between the planes 1 z ? and 2 z ? .
03
ii. Evaluate
? ?
22
00
xy
e dxdy
??
??
??
by changing to polar coordinates
04
Q.5 (a)
Test the convergence of the series
? ?
1
1
1
n
nn
?
?
?
?
, if convergent then find
its value.
03
(b) Test the convergence of the series
1
1 ? 2 ? 3
+
3
2 ? 3 ? 4
+
5
3 ? 4 ? 5
+ ? ,
04
(c)
For which value of x does the series
2 3 4 5
...
2 3 4 5
x x x x
? ? ? ? is absolute
or conditionally convergent or divergent? What is the radius of
convergent of
2345
...
2 3 4 5
x x x x
? ? ? ? ?
07
Q.6 (a)
Determine the convergent of
1
2
1
tan
1
n
n
n
? ?
?
?
?
03
(b) Find the volume of the solid generated by revolving the region bounded
by
2
xy ? and the lines 0, 2 xx ?? about the x-axis.
04
(c)
Trace the curve ? ? 1 cos ; 0. r a a ? ? ? ?
07
Q.7 (a)
Expand sin
4
x
? ??
?
??
??
in powers of x by using the Taylor?s series. Also,
find the value of sin 46 .
03
(b)
Find
1
23
3
0
lim
3
x x x
x
e e e
?
?? ??
??
??
04
(c) Discuss the convergence of the following integrals:
(i)
1
2
1
1
dx
x
?
?
(ii)
2
0
x
e dx
?
?
?
07
***********
FirstRanker.com - FirstRanker's Choice
This post was last modified on 20 February 2020