1
Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
BE SEMESTER-IV • EXAMINATION WINTER 2016
Subject Code: 2140001 Date: 18/11/2016
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.
MARKS
Q.1 Short Questions 14
1
For the function
2
0
z
f z z
z
to be continuous at z 0 f
should be equal to
2 log z is analytic everywhere except at z
3 If u iv is analytic then is v analytic
4
The fixed points of the mapping
3 13
3
iz
w
z i



are
5
The value of
1
2
0
i
z dz

along the line y x is
6 The function 2 2 2x x py is harmonic if p equal to
7
The residue of tan z at
2
z

is
8 The order of convergence of Newton-Raphson method is
9 equal to (In terms of E only).
10 If 3 2 y 3x 2x then 3 y
11 The n th difference of a polynomial of degree n is
12 Putting n 1 in the Newton-Cotes quadrature formula rule
is obtained.
13 The auxiliary quantity 1 k obtained by Runge-Kutta fourth order for the
differential equation 2 0 1
dy
x y y
dx
when h 0.1 is
14 As soon as a new value of a variable is found by iteration it is used
immediately in the following equations, this method is called
Q.2 Solve the equation 2 z z 0. 03
Show that 3 2 u y 2x x is harmonic in some domain and find
its harmonic conjugate.
04

Using residue theorem, evaluate
2
0
4
5 4sin
d
.
Prove that 1 tan log
2
i i z
z
i z



.
07
OR
Check whether the function defined by

0
0 0
u x y iv x y if z
f z
if z



where
3 3
2 2
x y
u x y
x y



and
07
2

3 3
2 2
x y
v x y
x y



is analytic at z 0 or not? Also check whether
Cauchy-Riemann equations are satisfied at that point?
Q.3
Find and graph the strip x 2 under the mapping
1
w
z
.
03
Evaluate 2 Re
C
z dz where C is the boundary of the square with
vertices 1 in the clockwise direction.
04
Find the Taylor and Laurent expansion for the function


2 1
2 3
z
f z
z z



in the region z 2 2 z 3 z 3
07
OR
Q.3
State Cauchy's integral formula and hence evaluate

2
4
1
z
C
e
dz
z where
C is the circle z 3 .
03
Check whether the following functions are analytic or not

5
f z z 2 f z
04
Define bilinear transformation. Also find the bilinear transformation
which maps the points z into the points . Hence
find the image of z . Also draw the regions in z -plane and w-plane.
07
Q.4
State Simpson's
3
8
rule and evaluate
1
2
0 1
dx
x using it taking
1
6
h .
03
Determine the interpolating polynomial of degree three using Lagrange's
interpolation for the table below:
x 0 1 3
f 2 1 0
04
Determine the largest eigenvalue and the corresponding eigenvector of
the matrix
2 1 0
1 2 1
0 1 2
A





using
1
0
0
X





07
OR
Q.4 Solve the following linear system of equations by Gauss Elimination
method
2 3
1 2 3
1 2 3
8 2 7
3 5 2 8
6 2 8 26
x x
x x x
x x x



03
Find the positive root of x cos x correct to three decimal places by
bisection method between 0.5 and 1.
04
Use Gauss-Seidel method to determine solution of the following system
of equations using as initial approximation
2 3
2 3
3
x y
x y z
x z



07
Q.5 Solve 1 0 x xe correct to three decimal places between 0 and 1 using
secant method.
03
3
Compute from the following table
x 0.5 0.6 0.7 0.8
cosh x 1.127626 1.185464 1.255169 1.337435
04
Write formula of Runge-Kutta method for fourth order. Apply Euler's
method to find the approximate solution of
dy
x y
dx
with 0
and h 2 . Show your calculation up to five iterations.
07
OR
Q.5 The speed v meters per second of a car, t seconds after it starts is shown
in the following table
t 0 12 24 36 48 60 72 84 96 108 120
v 0 3.6 10.08 18.9 21.6 18.54 10.26 4.5 4.5 5.4 9.0
Using Simpson's
1
3
rule, find the distance travelled by the car in 2
minutes.
03
Develop a recurrence formula for finding N using Newton
Raphson method and hence compute 27 correct up to three decimal
places.
04

Evaluate
1
0 1
dt
by the Gaussian formula with one point, two points and
three points.
07