Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem New And Spfu 3110015 Mathematics 2 Previous Question Paper
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 3110015 Date: 01/01/2020
Subject Name: Mathematics ?2
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Marks
Q.1 (a) Find the length of curve of the portion of the circular helix
? ??
( ) cos sin r t ti t j tk ? ? ? from 0 t to t ? ??
03
(b)
? ?
? ?
? ?
? ?
3,4
2 3 2 2
1,2
3 xy y dx x y xy dy ? ? ?
?
is independent of path joining the points
(1, 2) and (3,4). Hence, evaluate the integral.
04
(c)
Verify tangential form of Green?s theorem for ? ? ? ?
??
sin cos , F x y i y j ? ? ?
where C is the boundary of the region bounded by the lines 0, 2 yx ? ??
and yx ? .
07
Q.2 (a) Find the Laplace transform of () ft defined as
( ) 0
1
t
f t t k
k
tk
? ? ?
??
03
(b)
Find the inverse Laplace transform of
? ? ? ?
2
2 2 2 2
s
s a s b ??
04
(c)
(i) Calculate the curl of the vector
? ?
2 2 2
? ??
3 xyzi x y j xz y z k ? ? ?
(ii) The temperature at any point in space is given by T xy yz zx ? ? ? .
Determine the derivative of T in the direction of the vector
? ?
34 ik ? at the
point (1, 1, 1).
07
OR
(c)
Let
? ??
r xi y j zk ? ? ? , rr ? , and a is a constant vector. Find the value of
n
ar
div
r
? ??
??
??
07
Q.3 (a) Find constants a, b and c such that
? ? ? ? ? ?
? ??
2 3 4 2 V x y az i bx y z j x cy z k ? ? ? ? ? ? ? ? ? is irrotational.
03
(b) Using Fourier cosine integral representation show that
22
0
cos
2
kx
xe
d
kk
??
?
?
? ?
?
?
?
04
(c) Solve the following differential equations:
(i) cos( ) x y dy dx ??
(ii)
23
sec tan
dy
y x y x
dx
??
07
OR
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Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 3110015 Date: 01/01/2020
Subject Name: Mathematics ?2
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Marks
Q.1 (a) Find the length of curve of the portion of the circular helix
? ??
( ) cos sin r t ti t j tk ? ? ? from 0 t to t ? ??
03
(b)
? ?
? ?
? ?
? ?
3,4
2 3 2 2
1,2
3 xy y dx x y xy dy ? ? ?
?
is independent of path joining the points
(1, 2) and (3,4). Hence, evaluate the integral.
04
(c)
Verify tangential form of Green?s theorem for ? ? ? ?
??
sin cos , F x y i y j ? ? ?
where C is the boundary of the region bounded by the lines 0, 2 yx ? ??
and yx ? .
07
Q.2 (a) Find the Laplace transform of () ft defined as
( ) 0
1
t
f t t k
k
tk
? ? ?
??
03
(b)
Find the inverse Laplace transform of
? ? ? ?
2
2 2 2 2
s
s a s b ??
04
(c)
(i) Calculate the curl of the vector
? ?
2 2 2
? ??
3 xyzi x y j xz y z k ? ? ?
(ii) The temperature at any point in space is given by T xy yz zx ? ? ? .
Determine the derivative of T in the direction of the vector
? ?
34 ik ? at the
point (1, 1, 1).
07
OR
(c)
Let
? ??
r xi y j zk ? ? ? , rr ? , and a is a constant vector. Find the value of
n
ar
div
r
? ??
??
??
07
Q.3 (a) Find constants a, b and c such that
? ? ? ? ? ?
? ??
2 3 4 2 V x y az i bx y z j x cy z k ? ? ? ? ? ? ? ? ? is irrotational.
03
(b) Using Fourier cosine integral representation show that
22
0
cos
2
kx
xe
d
kk
??
?
?
? ?
?
?
?
04
(c) Solve the following differential equations:
(i) cos( ) x y dy dx ??
(ii)
23
sec tan
dy
y x y x
dx
??
07
OR
2
Q.3 (a)
Find the Laplace transform of (i)
0
sin
t
t
dt
t
?
(ii) ? ?
2
3 t u t ?
03
(b) Using Convolution theorem obtain
? ?
1
22
1
L
s s a
?
??
??
??
?
??
04
(c)
Find the power series solution of
2
2
0
dy
xy
dx
??
07
Q.4 (a) Find the Laplace transform of the waveform
2
( ) ,0 3
3
t
f t t
??
? ? ?
??
??
03
(b) Using the Laplace transforms, find the solution of the initial value problem
25 10cos5 y y t ???? (0) 2, (0) 0 yy ? ??
04
(c)
Using variation of parameter method solve
? ?
2
1 sin D y x x ??
07
OR
Q.4 (a)
Solve
2
2
2
d y dy dy
y
dx dx dx
??
??
??
??
03
(b)
Solve 3 3 4
t
y y y y e ? ? ? ? ? ? ? ? ? ?
04
(c)
Solve
2
2
2
2 4 2 3
x
d y dy
y x e
dx dx
?
? ? ? ? using method of undetermined
coefficients.
07
Q.5 (a)
Classify the singular points of the equation ? ?
33
2 6 0 x x y x y y ? ? ? ? ? ? ?
03
(b)
Solve
? ?
2
4 cos2 D y x ??
04
(c)
Solve (i)
? ?
20
xx
ye dx y e dy ? ? ? (ii) 2 tan sin
dy
y x x
dx
??
07
OR
Q.5 (a)
Solve
3
22 x
dy y
dx e y
?
?
03
(b)
If
1
yx ? is one of solution of
2
0 x y xy y ? ? ? ? ? ? find the second solution.
04
(c)
Using Frobenius method solve
? ?
22
4 2 0 x y xy x y ? ? ? ? ? ? ?
07
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This post was last modified on 20 February 2020