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

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

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2140001 Date:09/05/2019
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 principal argument of ?? =
?? ?
3+??
03
(b) Check whether the following functions are analytic or not at any point:
(i) ?? (?? ) = ?? 2
+ ?????? (ii) ?? (?? ) = ?? 2


















not
04
(c)
(i) Expand ?? (?? ) = ???????? (
1
?? 3
) in Laurent?s series near ?? = 0 and
identify the singularity.
(ii) Show that if ?? is any ?? ?? ?
root of unity other than unity itself, than
1 + ?? + ?? 2
+ ? + ?? ?? ?1
= 0.
07

Q.2 (a) Find and sketch the image of the region |?? | < 1 under the
transformation 2?? ? ?? .
03
(b) Show that the function ?? (?? , ?? ) = ?? 3
? 3?? 2
?? is harmonic in some
domain ?? and find its conjugate ?? (?? , ?? ).
04
(c) Find the Mobius transformation that maps the points ?? = 1, ?? , ?1 into
the points ?? = ?? , 0, ??? . Hence find the image of |?? | = 1.
07
OR
(c)
Evaluate the integral
?
C
dz z ) Re(
2
, where C is the boundary of the
square with vertices 0, ?? , 1 + ?? , 1 in clockwise direction.
07
Q.3 (a)
Evaluate dz iy x
i
) (
1
0
2
?
?
?
along the path ?? = ?? 2
.
03
(b)
Find the residue at each pole of
9
) (
2
?
?
z
ze
z f
iz

04
(c)
Expand ?? (?? ) =
1
(?? +1)(?? ?2)
in Laurent?s series in the region
(i) |?? | < 1 (ii) 1 < |?? | < 2 (iii) |?? | > 2.
07
OR
Q.3 (a)
Write the Cauchy integral formula and using it evaluate dz
z
z
C
?
? ?
cos

where C is the circle |?? | = 4.
03
(b)
Evaluate
?
? ?
?
C
dz
z z z
z
) 3 )( 1 (
1 2
, where C is the circle |?? | = 2.
04
(c)
Using the residue theorem, evaluate
?
?
?
?
?
2
0
sin 3 5
d

07




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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2140001 Date:09/05/2019
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 principal argument of ?? =
?? ?
3+??
03
(b) Check whether the following functions are analytic or not at any point:
(i) ?? (?? ) = ?? 2
+ ?????? (ii) ?? (?? ) = ?? 2


















not
04
(c)
(i) Expand ?? (?? ) = ???????? (
1
?? 3
) in Laurent?s series near ?? = 0 and
identify the singularity.
(ii) Show that if ?? is any ?? ?? ?
root of unity other than unity itself, than
1 + ?? + ?? 2
+ ? + ?? ?? ?1
= 0.
07

Q.2 (a) Find and sketch the image of the region |?? | < 1 under the
transformation 2?? ? ?? .
03
(b) Show that the function ?? (?? , ?? ) = ?? 3
? 3?? 2
?? is harmonic in some
domain ?? and find its conjugate ?? (?? , ?? ).
04
(c) Find the Mobius transformation that maps the points ?? = 1, ?? , ?1 into
the points ?? = ?? , 0, ??? . Hence find the image of |?? | = 1.
07
OR
(c)
Evaluate the integral
?
C
dz z ) Re(
2
, where C is the boundary of the
square with vertices 0, ?? , 1 + ?? , 1 in clockwise direction.
07
Q.3 (a)
Evaluate dz iy x
i
) (
1
0
2
?
?
?
along the path ?? = ?? 2
.
03
(b)
Find the residue at each pole of
9
) (
2
?
?
z
ze
z f
iz

04
(c)
Expand ?? (?? ) =
1
(?? +1)(?? ?2)
in Laurent?s series in the region
(i) |?? | < 1 (ii) 1 < |?? | < 2 (iii) |?? | > 2.
07
OR
Q.3 (a)
Write the Cauchy integral formula and using it evaluate dz
z
z
C
?
? ?
cos

where C is the circle |?? | = 4.
03
(b)
Evaluate
?
? ?
?
C
dz
z z z
z
) 3 )( 1 (
1 2
, where C is the circle |?? | = 2.
04
(c)
Using the residue theorem, evaluate
?
?
?
?
?
2
0
sin 3 5
d

07




2


Q.4 (a) Find the positive root of the equation 2 sin ?? ? ?? = 0 using bisection
method in six stages.
03
(b) Solve the following system of equations by Gauss Seidel method:
32 4 28 ? ? ? z y x 35 4 17 2 ? ? ? z y x 24 10 3 ? ? ? z y x
Correct up to two decimal places.
04
(c) Using the power method find the largest eigenvalue of the matrix
?
?
?
?
?
?
?
?
?
?
?
?
5 3 6
1 4 4
2 3 1

07
OR
Q.4 (a) Use the secant method in three stages to find the root of the equation
cos ?? ? ?? ?? ?? = 0.
03
(b) Find an approximate value of ) 6 . 3 ( f using Newton?s backward
difference formula from the following data:
x 0 1 2 3 4
f(x) -5 1 9 25 55

04
(c) Using Lagrange?s interpolation formula find y when x = 5 from the
following table:
x 1 2 3 4 7
y 2 4 8 16 128

07



Q.5 (a)
Use Simpson?s 1/3 rule to evaluate dx e
x
?
?
2
1
2
. Take h = 0.25.
03
(b) Use Gauss elimination method to solve the system of equations
; 4 6 4 2
3 2 1
? ? ? ? x x x ; 10 3 5
3 2 1
? ? ? x x x . 5 2 3
3 2 1
? ? ? x x x
04
(c) Derive Euler?s formula to solve the initial value problem
); , ( y x f
dx
dy
? . ) (
0 0
y x y ? Find ) 1 . 0 ( y for y x
dx
dy
? ?
2
, where
1 ) 0 ( ? y using improved Euler?s method. Take . 05 . 0 ? h
07
OR

Q.5 (a) Find the real root of the equation ?? 3
? 9?? + 1 = 0 up to five decimal
places by the Newton-Raphson?s method. Take ?? 0
= 3.
03
(b) Find ) 15 ( f from the following table using Newton?s divided
difference formula:
x 4 5 7 10 11 13
f(x) 48 100 204 900 1210 2028

04
(c) Apply fourth order Runge-Kutta method to find ) 1 . 0 ( y and ) 2 . 0 ( y for
the differential equation ,
2
1
3 y x
dx
dy
? ? 1 ) 0 ( ? y . Take . 1 . 0 ? h
07



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