Download GTU BE/B.Tech 2018 Winter 4th Sem New 2140001 Mathematics 4 Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 4th Sem New 2140001 Mathematics 4 Previous Question Paper

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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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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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