Download GTU BE/B.Tech 2019 Summer 4th Sem Old 140001 Mathematics Iv Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 4th Sem Old 140001 Mathematics Iv Previous Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(OLD) ? EXAMINATION ? SUMMER 2019
Subject Code:140001 Date:09/05/2019
Subject Name: Mathematics-IV
Time:02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
Find all roots of
3
8i .
07
(b)
1) Find real and imaginary part of
? ? . 4
2
z z z f ? ?
Also, calculate the value
of f at . 1 i z ? ?
04
2) Show that
? ?
? ?
0 ; 0
0 ;
Im
?
?
?
z
z
z
z
z f

is not continuous at the origin.
03

Q.2 (a)
Find the image of the region
1 ? z
under the transformation
. 2 i z w ? ?

Sketch the region and its image.
07
(b)
Show that
? ?
2 3
3 2 , xy x x y x u ? ? ?
is harmonic in some domain D and find a
harmonic conjugate of
? ?. , y x u

07
OR
(b) If ?? ( ?? ) is an analytic function of z, show that
(
?? ????
| ?? ( ?? ) | )
2
+ (
?? ????
| ?? ( ?? ) | )
2
= | ?? ?
( ?? ) |
2

07

Q.3 (a)
Evaluate ? ?? 2
???? 2 + ?? 0
along the line ?? =
?? 2
?
07
(b) Evaluate:
1. ?
?? ?? ? 3
???? , over the contour ?? , where ?? is the circle | ?? | = 1 .
2. ?
?? ?? ?? ( 1 ? ?? )
3
???? , counterclockwise over C, where C: | ?? | = 2
3. ?
?? ?? ( ?? ? 1 ) ( ?? ? 3 )
???? , counterclockwise over C, where C: | ?? | = 2
07
OR
Q.3 (a)
Determine the Laurent series expansion of ?? ( ?? ) =
1
( ?? + 1 ) ( ?? + 3 )
valid for
a) | ?? | < 1 b) 1 < | ?? | < 3
07
(b) Using Newton?s divided difference formula, compute ?? ( 10 . 5 ) from the
following data:
x: 10 11 13 17
f(x): 2.3026 2.3979 2.5649 2.8332

07

Q.4 (a) Find a real root of the equation ?? 3
+ 4 ?? 2
? 1 = 0, lies between 0 and 1 by using
bisection method correct to decimal places.
07
(b)
Evaluate
? ?
?
?
3
0
1 x
dx
with n=6 by using Simpson?s 3/8 rule and hence calculate
ln 2.

07
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(OLD) ? EXAMINATION ? SUMMER 2019
Subject Code:140001 Date:09/05/2019
Subject Name: Mathematics-IV
Time:02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
Find all roots of
3
8i .
07
(b)
1) Find real and imaginary part of
? ? . 4
2
z z z f ? ?
Also, calculate the value
of f at . 1 i z ? ?
04
2) Show that
? ?
? ?
0 ; 0
0 ;
Im
?
?
?
z
z
z
z
z f

is not continuous at the origin.
03

Q.2 (a)
Find the image of the region
1 ? z
under the transformation
. 2 i z w ? ?

Sketch the region and its image.
07
(b)
Show that
? ?
2 3
3 2 , xy x x y x u ? ? ?
is harmonic in some domain D and find a
harmonic conjugate of
? ?. , y x u

07
OR
(b) If ?? ( ?? ) is an analytic function of z, show that
(
?? ????
| ?? ( ?? ) | )
2
+ (
?? ????
| ?? ( ?? ) | )
2
= | ?? ?
( ?? ) |
2

07

Q.3 (a)
Evaluate ? ?? 2
???? 2 + ?? 0
along the line ?? =
?? 2
?
07
(b) Evaluate:
1. ?
?? ?? ? 3
???? , over the contour ?? , where ?? is the circle | ?? | = 1 .
2. ?
?? ?? ?? ( 1 ? ?? )
3
???? , counterclockwise over C, where C: | ?? | = 2
3. ?
?? ?? ( ?? ? 1 ) ( ?? ? 3 )
???? , counterclockwise over C, where C: | ?? | = 2
07
OR
Q.3 (a)
Determine the Laurent series expansion of ?? ( ?? ) =
1
( ?? + 1 ) ( ?? + 3 )
valid for
a) | ?? | < 1 b) 1 < | ?? | < 3
07
(b) Using Newton?s divided difference formula, compute ?? ( 10 . 5 ) from the
following data:
x: 10 11 13 17
f(x): 2.3026 2.3979 2.5649 2.8332

07

Q.4 (a) Find a real root of the equation ?? 3
+ 4 ?? 2
? 1 = 0, lies between 0 and 1 by using
bisection method correct to decimal places.
07
(b)
Evaluate
? ?
?
?
3
0
1 x
dx
with n=6 by using Simpson?s 3/8 rule and hence calculate
ln 2.

07
2
OR
Q.4 (a) Solve the following system of equation using partial pivoting by Gauss
Elimination method.
26 8 2 6
8 2 5 3
7 2 8
3 2 1
3 2 1
3 2
? ? ?
? ? ?
? ? ?
x x x
x x x
x x

07
(b) Solve the following system of equations by using Gauss-Seidel method.
6 10 ; 6 10 ; 6 10 ? ? ? ? ? ? ? ? ? z y x z y x z y x

07

Q.5 (a) Using the power method, find the largest eigenvalue of the matrix
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
2 1 0
1 2 1
0 1 2
A

07
(b) Apply Runge-Kutta fourth order method to find an approximation value of y
when x=0.1 in step of 0.1 if ? ? 1 0 ,
2
? ? ? y y x
dx
dy

07
OR
Q.5 (a)
Evaluate the integral
? ?
?
?
1
0
1 x
dx
, by Gauss three point quadrature formula.
07
(b)
Solve the differential equation ? ? , 1 0 ; 0 ? ? ? y xy
dx
dy
from 0 ? x to 25 . 0 ? x using
Euler?s method taking step size 0.05.
07

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