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
FirstRanker.com - FirstRanker's Choice
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

FirstRanker.com - FirstRanker's Choice