Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 4th Sem New 2141905 Complex Variables And Numerical Methods Previous Question Paper
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2141905 Date:09/05/2019
Subject Name: Complex Variables and Numerical Methods
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) State De?Movier?s. Find arg [ i / ( - 2 ? 2i) ] . 03
(b) Define the operators ?, ? and E. Prove that ?? ?= ?. 04
(c) State Cauchy ? Riemann Equations. Show that
(i) f(z) = sin z is everywhere analytic
(ii) f(z) = xy + iy is nowhere analytic.
07
Q.2 (a) State the formula for sin
? 1
z. Find
sin
? 1
( - i ) 03
(b) Find analytic function f(z) = u + iv, if u = 2x(1 ? y). 04
(c) Classify the singularities of the analytic function.
In each of the following case, identify the singular point and its type with
justification.
(i)
?? 2
(?? + 1)
? (ii)
sin ?? ?? ?
(iii)
(1 ? cosh ?? )
?? 3
?
07
OR
(c)
Use residues to evaluate the improper integral:
?
?
?? ?? ????
(?? ?? + ?? )(?? ?? + ?? )
??
07
Q.3
(a)
Evaluate the integral ? z ? dz
C
, when C is the right-hand half
z = 2e
????
(?
?? 2
?
? ?? ?
?? 2
?
), of the circle |z| = 2 from ? 2i to 2i.
03
(b) Show that the mapping by w = 1/z transforms circles and lines into circles and
lines.
04
(c) Give two Laurent series expansions in powers of z for the function
f(z) = 1/[z
2
(1 ? z)] and specify the regions in which those expansions are valid.
07
OR
Q.3 (a) Find the bilinear transformation which transforms z 1 = ?, z 2 = i, z 3 = 0 into
w 1 = 0, w 2 = i, w 3 = ? .
03
(b)
Determine and sketch the region :
(i) 0 ? arg ?? ?
?? 4
?
,(ii) | 2z + 3| > 4,
Which of them are domains?
04
(c) State the Cauchy?s Integral Formula and its extension. Hence evaluate integral
?
z+4
z
2
?2z+5
C
dz , where C is circle |z +1 + i | =2.
07
FirstRanker.com - FirstRanker's Choice
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2141905 Date:09/05/2019
Subject Name: Complex Variables and Numerical Methods
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) State De?Movier?s. Find arg [ i / ( - 2 ? 2i) ] . 03
(b) Define the operators ?, ? and E. Prove that ?? ?= ?. 04
(c) State Cauchy ? Riemann Equations. Show that
(i) f(z) = sin z is everywhere analytic
(ii) f(z) = xy + iy is nowhere analytic.
07
Q.2 (a) State the formula for sin
? 1
z. Find
sin
? 1
( - i ) 03
(b) Find analytic function f(z) = u + iv, if u = 2x(1 ? y). 04
(c) Classify the singularities of the analytic function.
In each of the following case, identify the singular point and its type with
justification.
(i)
?? 2
(?? + 1)
? (ii)
sin ?? ?? ?
(iii)
(1 ? cosh ?? )
?? 3
?
07
OR
(c)
Use residues to evaluate the improper integral:
?
?
?? ?? ????
(?? ?? + ?? )(?? ?? + ?? )
??
07
Q.3
(a)
Evaluate the integral ? z ? dz
C
, when C is the right-hand half
z = 2e
????
(?
?? 2
?
? ?? ?
?? 2
?
), of the circle |z| = 2 from ? 2i to 2i.
03
(b) Show that the mapping by w = 1/z transforms circles and lines into circles and
lines.
04
(c) Give two Laurent series expansions in powers of z for the function
f(z) = 1/[z
2
(1 ? z)] and specify the regions in which those expansions are valid.
07
OR
Q.3 (a) Find the bilinear transformation which transforms z 1 = ?, z 2 = i, z 3 = 0 into
w 1 = 0, w 2 = i, w 3 = ? .
03
(b)
Determine and sketch the region :
(i) 0 ? arg ?? ?
?? 4
?
,(ii) | 2z + 3| > 4,
Which of them are domains?
04
(c) State the Cauchy?s Integral Formula and its extension. Hence evaluate integral
?
z+4
z
2
?2z+5
C
dz , where C is circle |z +1 + i | =2.
07
2
Q.4
(a)
Evaluate ?
7
3
x
2
log x dx taking four sub-intervals by trapezoidal rule.
03
(b) Apply Bisetion method to find a real root of the equation 2x
3
? 5x + 1 = 0 correct
to 2 decimal places.
04
(c) Given f(1)=22, f(2)=30, f(4)=82, f(7)=106, f(12)=206, find f(8) using Lagrange?s
interpolation formula.
07
OR
Q.4 (a) The velocity of a car (running on a straight road) at intervals of 2 minutes are given
below.
Time (in min.): 0 2 4 6 8 10 12
Velocity (in km/hr): 0 22 30 27 18 7 0
Apply Simpson?s 1/3
rd
rule to find the distance covered by the car.
03
(b) Newton Raphson method find a root of the equation xsinx +cosx = 0 correct to
four decimal places (taking initial guess x0 = ?).
04
(c) State Striling?s Interpolation formula.
Interpolate by means of Gauss? forward formula the population for the year 1936
given the following data:
Year: 1901 1911 1921 1931 1941 1951
Population
(1000s)
12 15 20 27 39 52
07
Q.5 (a) Using secant method find a real root of the equation x
3
? 5x + 1 = 0 up to three
iterations.
03
(b) Use Runge-Kutta method to solve y ? = xy, y(0)=1, for x = 0.2, with h=0.1. 04
(c) Using Gauss-Seidel method to solve the following system correct to 3 decimal
places: 83x + 11y ? 4z = 95, 7x + 52y + 13z = 104, 3x + 8y + 29z = 71.
07
OR
Q.5 (a) Solve x log10 x = 1.2 by Regula Falsi method correct to two decimal places. 03
(b) Solve y ? = 1 ? y, y(0) = 0 in [0, 0.3] by modified Euler?s method taking h = 0.1. 04
(c) Find the largest eigenvalue and corresponding eigenvector using power method,
for
?? = [
4 4 2
4 4 1
2 1 8
] , taking ?? 0
= (
1
0
0
) .
07
*************
FirstRanker.com - FirstRanker's Choice
This post was last modified on 20 February 2020