Download GTU BE/B.Tech 2019 Winter 4th Sem New 2140001 Mathematics 4 Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 4th Sem New 2140001 Mathematics 4 Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2140001 Date: 07/12/2019

Subject Name: Mathematics-4

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.
Q-1 (a)
Find the principal argument of ? =
1?7? (2+? )
2

3
(b)
If ? (? ) =
? 3
? 3
? 3
+? 3

? (? , ? ) ? (0,0)
= 0 ? (? , ? ) = (0,0)
Show that ? (? ) is not continuous at the origin.

4
(c ) Solve the following system of linear equations by Gauss-elimination
method.? + ? + ? = 9, 2? ? 3? + 4? = 13, 3? + 4? + 5? = 40 .
7

Q-2 (a) Check whether the function? (? ) = ? is analytic or not? 3
(b) Show that ? (? , ? ) = 2? ? ? 3
+ 3? ? 2
is harmonic in some domain and
find a harmonic conjugate ? (? , ? ).
4
(c ) Determine the mobius transformation that maps ? 1
= 0, ? 2
= 1, ? 3
= ?
onto ? 1
= ?1, ? 2
= ? , ? 3
= 1 respectively.
7
OR

(c )

Find real and imaginary parts of (?1 ? ? )
7
+ (?1 + ? )
7

7

Q-3 (a)
Prove that ?
? 3? ? +
? 2
? = 2?
? , where C is the circle |? | = 5.
3
(b)
Expand ? (? ) =
1? ? ? in Laurent?s series about ? = 0 and identify
singularity.
4

(c )
Use residues to evaluate ?
? 2
?
(? 2
+1)(? 2
+4)
?
0

7
OR

(a)
Find the radius of convergence of ? (1 +
1
? 2
)
? 3
? ? .
?
? =1

3
(b)
Evaluate ?
(? 2
? ? ? 2
)? ? along the parabola ? = 2? 2
from (1,2) to (2,8).
4
(c )
Expand ? (? ) =
1
(? +2)(? +4)
valid for the regions (i) |? | < 2,
(? )2 < |? | < 4 , (? )|? | > 4.
7
Q-4 (a)
Prove that :? (? ) = ? [1 +
? (? )
? (? )
]
3
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2140001 Date: 07/12/2019

Subject Name: Mathematics-4

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.
Q-1 (a)
Find the principal argument of ? =
1?7? (2+? )
2

3
(b)
If ? (? ) =
? 3
? 3
? 3
+? 3

? (? , ? ) ? (0,0)
= 0 ? (? , ? ) = (0,0)
Show that ? (? ) is not continuous at the origin.

4
(c ) Solve the following system of linear equations by Gauss-elimination
method.? + ? + ? = 9, 2? ? 3? + 4? = 13, 3? + 4? + 5? = 40 .
7

Q-2 (a) Check whether the function? (? ) = ? is analytic or not? 3
(b) Show that ? (? , ? ) = 2? ? ? 3
+ 3? ? 2
is harmonic in some domain and
find a harmonic conjugate ? (? , ? ).
4
(c ) Determine the mobius transformation that maps ? 1
= 0, ? 2
= 1, ? 3
= ?
onto ? 1
= ?1, ? 2
= ? , ? 3
= 1 respectively.
7
OR

(c )

Find real and imaginary parts of (?1 ? ? )
7
+ (?1 + ? )
7

7

Q-3 (a)
Prove that ?
? 3? ? +
? 2
? = 2?
? , where C is the circle |? | = 5.
3
(b)
Expand ? (? ) =
1? ? ? in Laurent?s series about ? = 0 and identify
singularity.
4

(c )
Use residues to evaluate ?
? 2
?
(? 2
+1)(? 2
+4)
?
0

7
OR

(a)
Find the radius of convergence of ? (1 +
1
? 2
)
? 3
? ? .
?
? =1

3
(b)
Evaluate ?
(? 2
? ? ? 2
)? ? along the parabola ? = 2? 2
from (1,2) to (2,8).
4
(c )
Expand ? (? ) =
1
(? +2)(? +4)
valid for the regions (i) |? | < 2,
(? )2 < |? | < 4 , (? )|? | > 4.
7
Q-4 (a)
Prove that :? (? ) = ? [1 +
? (? )
? (? )
]
3
(b) Find a real root of the equation ? 3
+ 4? 2
? 1 = 0 by using bisection
method correct up to two decimal places.
4
(c ) Determine the interpolating polynomial of degree three using Lagrange?s
interpolation formula for the table below.
x -1 0 1 3
y 2 1 0 -1

7
OR
(a)
Use trapezoidal rule to estimate ? ? ? 2
? 1.3
0.5
using a strip of width 0.2.
.
3
(b)
Evaluate ? = ?
?
1+? 1
0
by one point, Gaussian formula.
4
(c) Solve the following equations by Gauss-Seidel method correct up to two
decimal places 20? + 2? + ? = 30, ? ? 40? ? 3? = ?75, 2? ? ? +
10? = 30.
7

Q5 (a) Compute ? ?(0.56) using Newton?s forward difference formula for the
following table.
X 0.5 06 0.7 0.8
F(X) 1.127626 1.185465 1.255169 1.337435

3
(b) Using Newton?s divided difference interpolation formula compute? (9.2)
from the following data.
x 8.0 9.0 9.5 11.0
f(x) 2.079442 2.197225 2.251292 2.397895

4
(c) Using improved Euler?s method, solve ? ?
= 1 ? ? with the initial
condition ? (0) = 0 and tabulate the solutions at ? = 0.1, 0.2. compare the
answer with the exact solution.
7
OR
(a)
Using N-R method find an iterative formula to find? ( where N is
positive number) and hence find?5.
3
(b)
Evaluate the integral ? log
? ? ? ,
5.2
4
using Simpson?s
3
8
? ? rule.
4

(c) Find the largest eigen value and the corresponding eigenvector for ? =
[
1 6 1
1 2 0
0 0 3
]
7

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