Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) 3110015 Mathematics ?2 Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) 3110015 Mathematics ?2 Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110015 Date: 01/06/2019

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 Fourier integral representation of
? (? ) = {
? ; ? ? (0, ? )
0 ; ? ? (? , ?)

03
(b) Define: Unit step function. Use it to find the Laplace transform of
? (? ) = {
(? ? 1)
2
; ? ? (0, 1]
1 ; ? ? (1, ?)

04
(c) Use the method of undetermined coefficients to solve the differential
equation ? ?
? 2? ?
+ ? = ? 2
? ? .
07

Q.2 (a)
Evaluate ? ? ?
? ? ? ? ;
? where ? ?
= (? 2
? ? 2
)? ? + 2? ? ? and C is the curve
given by the parametric equation
? ? ? (? ) = ? 2
? ? + ? ? ? ; 0 ? ? ? 2 .
03
(b) Apply Green?s theorem to find the outward flux of a vector field ? ?
=
1
?
(? ? ? + ? ? ? ) across the curve bounded by ? = ? ? , 2? = 1 and ? = 1.
04
(c) Integrate ? (? , ? , ? ) = ? ? ? ? 2
over the curve ? = ? 1
+ ? 2
, where C1 is
the line segment joining (0,0,1) to (1,1,0) and C2 is the curve y=x
2
joining
(1,1,0) to (2,4,0).
07
OR
(c)
Check whether the vector field ? ?
= ? ? +2? ? ? + ? ? ? +2? ? ? + 2? ? ? +2? ? ?
is
conservative or not. If yes, find the scalar potential function ? (? , ? , ? ) such
that ? ?
= grad ? .
07

Q.3 (a) Write a necessary and sufficient condition for the differential equation
? (? , ? )? + ? (? , ? )? = 0 to be exact differential equation. Hence check
whether the differential equation
[(? + 1)? ? ? ? ? ]? ? ? ? ? ? = 0
is exact or not.
03
(b) Solve the differential equation
(1 + ? 2
)? = (? ? tan
?1
? ? ? )?
04
(c)
By using Laplace transform solve a system of differential equations
?
?
=
1 ? ? ,
?
?
= ? , where ? (0) = 1, ? (0) = 0.
07
OR
Q.3 (a) Solve the differential equation
(2? 3
+ 4? )? ? ? = 0.
03
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110015 Date: 01/06/2019

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 Fourier integral representation of
? (? ) = {
? ; ? ? (0, ? )
0 ; ? ? (? , ?)

03
(b) Define: Unit step function. Use it to find the Laplace transform of
? (? ) = {
(? ? 1)
2
; ? ? (0, 1]
1 ; ? ? (1, ?)

04
(c) Use the method of undetermined coefficients to solve the differential
equation ? ?
? 2? ?
+ ? = ? 2
? ? .
07

Q.2 (a)
Evaluate ? ? ?
? ? ? ? ;
? where ? ?
= (? 2
? ? 2
)? ? + 2? ? ? and C is the curve
given by the parametric equation
? ? ? (? ) = ? 2
? ? + ? ? ? ; 0 ? ? ? 2 .
03
(b) Apply Green?s theorem to find the outward flux of a vector field ? ?
=
1
?
(? ? ? + ? ? ? ) across the curve bounded by ? = ? ? , 2? = 1 and ? = 1.
04
(c) Integrate ? (? , ? , ? ) = ? ? ? ? 2
over the curve ? = ? 1
+ ? 2
, where C1 is
the line segment joining (0,0,1) to (1,1,0) and C2 is the curve y=x
2
joining
(1,1,0) to (2,4,0).
07
OR
(c)
Check whether the vector field ? ?
= ? ? +2? ? ? + ? ? ? +2? ? ? + 2? ? ? +2? ? ?
is
conservative or not. If yes, find the scalar potential function ? (? , ? , ? ) such
that ? ?
= grad ? .
07

Q.3 (a) Write a necessary and sufficient condition for the differential equation
? (? , ? )? + ? (? , ? )? = 0 to be exact differential equation. Hence check
whether the differential equation
[(? + 1)? ? ? ? ? ]? ? ? ? ? ? = 0
is exact or not.
03
(b) Solve the differential equation
(1 + ? 2
)? = (? ? tan
?1
? ? ? )?
04
(c)
By using Laplace transform solve a system of differential equations
?
?
=
1 ? ? ,
?
?
= ? , where ? (0) = 1, ? (0) = 0.
07
OR
Q.3 (a) Solve the differential equation
(2? 3
+ 4? )? ? ? = 0.
03
2
(b)
Solve: (? + 1)
?
?
? ? = ? 3? (? + 1)
2
.
04
(c)
By using Laplace transform solve a differential equation
? 2
? ? ? 2
+ 5
?
?
+ 6? =
? ? , where ? (0) = 0, ? ?
(0) = ?1.
07

Q.4 (a) Find the general solution of the differential equation
? ? ?
?
+
? ? ? =
1
? 2

03
(b)
Solve :
? 3
? ? ? 3
? 7
?
?
+ 6? = ? ?
04
(c) Find a power series solution of the differential equation ? ?
? ? = 0 near
an ordinary point x=0.
07
OR
Q.4 (a) Find the general solution of the differential equation

?
?
+
? ? ? ? = 0.
03
(b)
Solve : ? 3
? 3
? ? ? 3
+ 2? 2
? 2
? ? ? 2
+ 2? = ?
04
(c) Find a Frobenius series solution of the differential equation 2? 2
? ?
+ ? ? ?
?
(? + 1)? = 0 near a regular-singular point x=0.
07

Q.5 (a) Write Legendre?s polynomial ? ? (? ) of degree-n and hence obtain ? 1
(? )
and ? 2
(? ) in powers of x.
03
(b) Classify ordinary points, singular points, regular-singular points and
irregular-singular points (if exist) of the differential equation ? ?
+ ? ? ?
=
0.
04
(c) Solve the differential equation
? 2
? 2
? ? ? 2
? 2? ?
?
+ 2? = ? 3
cos ?
by using the method of variation of parameters.
07
OR
Q.5 (a) Write Bessel?s function ? ? (? ) of the first kind of order-p and hence show
that ? 1/2
(? ) = ?
2
?
sin? .
03
(b) Classify ordinary points, singular points, regular-singular points and
irregular-singular points (if exist) of the differential equation ?
?
+ ? ?
=
0.
04
(c) Solve the differential equation ? ?
+ 25? = sec 5?
by using the method of variation of parameters.
07


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