GTU BE 1st and 2nd Semester 110008 MATHS-I Summer 2018 Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE-I – SEMESTER ? 1
st
– EXAMINATION ? SUMMER 2018
Subject Code: 110008 Date: 21-05-2018
Subject Name: MATHS-I
Time: 02:30 pm to 05:30 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) (1) Find the value of k so that the function given below is continuous at 0 ? x

?
?
?
?
?
?
?
?
0
0
2
3 sin
) (
x k
x
x
x
x f (2) State Sandwich theorem on limit of sequences and using it find
) ( lim
0
x g
x ?
, if x x g x sec 3 ) ( 3
3
? ? ? , R x ? ?
03

04
(b)If ), ( r f u ? where
2 2 2
y x r ? ? , prove that ) (
1
) (
2
2
2
2
r f
r
r f
y
u
x
u
? ? ? ? ?
?
?
?
?
?
07

Q.2 (a) (1) Evaluate
0
lim
? x
2
1
x
x e
x
? ?
(2) Find the area of the region between the x-axis and the graph of

2 1 , 2 ) (
2 3
? ? ? ? ? ? x x x x x f

03
04
(b)State (1) Rolle?s Theorem
(2) The Mean Value Theorem
Find the value of c using Mean Value Theorem , for the function

2
1 ) ( x x f ? ? , in 2 0 ? ? x
07

Q.3 (a) (1) Find the gradient of ) ( 3 2 ) , , (
2 2 3
y x z z y x f ? ? ? at the point (1,1,1). (2) Change the order of integration in the integral
dx dy
y
e
x
y
? ?
? ? ?
0
and evaluate it.
03
04
(b) Trace the curve 0 ); cos 1 ( ? ? ? a a r ? 07

Q.4 (a) (1) Find the curl of k xy j xe i z x F
z
? ? ?
) (
2
? ? ? ?
?(2) If ? ?
m
z y x u
2 2 2( ? ? ? then find .
2
2
2
2
2
2
z
u
y
u
x
u
?
?
?
?
?
?
?
?
03
04
(b) Find the local extreme values of the function
4 2 2 ) , (
2 2
? ? ? ? ? ? y x y x xy y x f
07
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE-I – SEMESTER ? 1
st
– EXAMINATION ? SUMMER 2018
Subject Code: 110008 Date: 21-05-2018
Subject Name: MATHS-I
Time: 02:30 pm to 05:30 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) (1) Find the value of k so that the function given below is continuous at 0 ? x

?
?
?
?
?
?
?
?
0
0
2
3 sin
) (
x k
x
x
x
x f (2) State Sandwich theorem on limit of sequences and using it find
) ( lim
0
x g
x ?
, if x x g x sec 3 ) ( 3
3
? ? ? , R x ? ?
03

04
(b)If ), ( r f u ? where
2 2 2
y x r ? ? , prove that ) (
1
) (
2
2
2
2
r f
r
r f
y
u
x
u
? ? ? ? ?
?
?
?
?
?
07

Q.2 (a) (1) Evaluate
0
lim
? x
2
1
x
x e
x
? ?
(2) Find the area of the region between the x-axis and the graph of

2 1 , 2 ) (
2 3
? ? ? ? ? ? x x x x x f

03
04
(b)State (1) Rolle?s Theorem
(2) The Mean Value Theorem
Find the value of c using Mean Value Theorem , for the function

2
1 ) ( x x f ? ? , in 2 0 ? ? x
07

Q.3 (a) (1) Find the gradient of ) ( 3 2 ) , , (
2 2 3
y x z z y x f ? ? ? at the point (1,1,1). (2) Change the order of integration in the integral
dx dy
y
e
x
y
? ?
? ? ?
0
and evaluate it.
03
04
(b) Trace the curve 0 ); cos 1 ( ? ? ? a a r ? 07

Q.4 (a) (1) Find the curl of k xy j xe i z x F
z
? ? ?
) (
2
? ? ? ?
?(2) If ? ?
m
z y x u
2 2 2( ? ? ? then find .
2
2
2
2
2
2
z
u
y
u
x
u
?
?
?
?
?
?
?
?
03
04
(b) Find the local extreme values of the function
4 2 2 ) , (
2 2
? ? ? ? ? ? y x y x xy y x f
07

2
Q.5 (a) (1) A fluid motion is given by
k y z xy j yz z x i x z y v
?
) cos (
?
) 2 sin (
?
) sin sin (
2
? ? ? ? ? ?
?
.
Show that the motion is irrotational. (2) Expand )
4( sin x ?
?
in powers of . x Hence find the value of . 46 sin
?
03

04
(b) Find the point on the plane 13 3 2 ? ? ? z y x closest to the point (1,1,1). 07

Q.6 (a)State modified Euler?s Theorem. Show that, if
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
2 2
1
tan
u u u y xyu u x
yy xy xx
2 sin
2
1
4 sin
4
1
2
2 2
? ? ? ?
07
(b) State Green?s Theorem. Using Green?s theorem, evaluate
) cos (sin ydy dx y e
C
x
?
?
?
where C is the rectangle with vertices
).
2
, 0 ( ),
2
, ( ), 0 , ( ), 0 , 0 (
? ?
? ?
07

Q.7 (a) Write the statement of Cauchy?s integral test. Test the convergence of the series
,
) (log
1
2
?
?
? n
a
n n

for 1 0 ? ? a
.
07
(b) Find the volume of the solid of revolution of the area bounded by the curve
x
xe y ? and the straight lines 0 , 1 ? ? y x
07
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