Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 4th Sem New 2140001 Mathematics 4 Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2140001 Date:22/11/2018
Subject Name:Mathematics-4
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 the complex conjugate of
5+2?? 1???
03
(b)
Find the locus of z given by|
?? ?1
?? +1
| = 1.
04
(c) Show that ?? = ?? 3
? 3?? 2
?? is a harmonic function. Also find its harmonic
conjugate.
07
Q.2 (a) Determine the region in the z-plane represented by 1 < |?? ? 2| < 3. 03
(b)
Show that
1+2?? ?? 2
+?? 3
=
1
?? 2
+
1
?? ? 1 + ?? ? ?? 2
+ ? ???? 0 < |?? | < 1.
04
(c) Find the roots common to the equation ?? 4
+ 1 = 0 ?????? ?? 6
? ?? = 0. 07
OR
(c)
Evaluate ? ?? ???? ?? along the straight line joining ?? = 1 ? ?? ???? ?? = 3 + 2?? .
07
Q.3 (a)
Expand ?? (?? ) =
1
?? as a Taylor?s series about the point ?? 0
= 1. Also determine
the region of convergence and radius of convergence.
03
(b) Find the bilinear transformation which maps the points ?? = 1, ?? , ?1 into the
points ?? = ?? , 0, ??? .
04
(c)
Evaluate ?
?????? 2?? 5+4???????? ???? 2?? 0
07
OR
Q.3 (a) Determine and sketch the image of |?? | = 1 under the transformation ?? = ?? +
?? .
03
(b)
Determine the poles of the equation ?? (?? ) =
?? 2
(?? ?1)
2
(?? +2)
and residue at each
pole.
04
(c)
Evaluate ? ???? (?? 2
)???? ?? , where C is the boundary of the square with vertices
0, ?? , 1 + ?? , 1 in the clockwise direction.
07
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2140001 Date:22/11/2018
Subject Name:Mathematics-4
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 the complex conjugate of
5+2?? 1???
03
(b)
Find the locus of z given by|
?? ?1
?? +1
| = 1.
04
(c) Show that ?? = ?? 3
? 3?? 2
?? is a harmonic function. Also find its harmonic
conjugate.
07
Q.2 (a) Determine the region in the z-plane represented by 1 < |?? ? 2| < 3. 03
(b)
Show that
1+2?? ?? 2
+?? 3
=
1
?? 2
+
1
?? ? 1 + ?? ? ?? 2
+ ? ???? 0 < |?? | < 1.
04
(c) Find the roots common to the equation ?? 4
+ 1 = 0 ?????? ?? 6
? ?? = 0. 07
OR
(c)
Evaluate ? ?? ???? ?? along the straight line joining ?? = 1 ? ?? ???? ?? = 3 + 2?? .
07
Q.3 (a)
Expand ?? (?? ) =
1
?? as a Taylor?s series about the point ?? 0
= 1. Also determine
the region of convergence and radius of convergence.
03
(b) Find the bilinear transformation which maps the points ?? = 1, ?? , ?1 into the
points ?? = ?? , 0, ??? .
04
(c)
Evaluate ?
?????? 2?? 5+4???????? ???? 2?? 0
07
OR
Q.3 (a) Determine and sketch the image of |?? | = 1 under the transformation ?? = ?? +
?? .
03
(b)
Determine the poles of the equation ?? (?? ) =
?? 2
(?? ?1)
2
(?? +2)
and residue at each
pole.
04
(c)
Evaluate ? ???? (?? 2
)???? ?? , where C is the boundary of the square with vertices
0, ?? , 1 + ?? , 1 in the clockwise direction.
07
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2
Q.4 (a)
Using Simpson?s
1
3
rule evaluate ? ?? (?? )????
2.5
1
from the date. Take h = 0.3
x 1 1.3 1.6 1.9 2.2 2.5
F(x) 1 1.69 2.56 3.61 4.84 6.25
03
(b) Solve the following system of equation using Gauss Elimination method with
partial pivoting
?? + ?? + ?? = 7
3?? + 3?? + 4?? = 24
2?? + ?? + 3?? = 16
04
(c) Find the values of ?? ?????? ?? = 21 ?????? ?? = 28 from the following data.
x 20 23 26 29
y 0.3420 0.3907 0.4384 0.4848
07
OR
Q.4 (a)
Find the largest eigenvalue and corresponding eigen vector for ?? = [
5 2
2 1
]
03
(b) Find the positive root of ?? = ???????? correct upto 3 decimal places, using N-R
method.
04
(c) Solve the following system by Gauss-Jacobi method.
27?? + 6?? ? ?? = 85
6?? + 15?? + 2?? = 72
?? + ?? + 54?? = 110
07
Q.5 (a) Evaluate ?
2
?????? 2?? 03
(b)
Express the function
3?? 2
?12?? +11
(?? ?1)(?? ?2)(?? ?3)
as a sum of partial fraction, using
Largrange?s formula.
04
(c)
Find the value of y for
????
????
= ?? + ?? ; ?? (0) = 1, ?? ????? ?? = 0.1, 0.2 with step
size h =0.05. Also compare with analytic solution.
07
OR
Q.5 (a) Find a root of the equation ?? 3
? ?? ? 11 = 0, using the bisection method up
to fourth approximation.
03
(b) From the following table, find ?? (?? ) using Newton?s divided difference
formula
x 1 2 7 8
f(x) 1 5 5 4
04
(c) Determine the largest eigenvalue and the corresponding eigenvector of
the matrix ?? = [
4 4 2
4 4 1
2 1 8
]
07
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This post was last modified on 20 February 2020