Gujarat Technological University 2011 Certification Computer Engieering Mathematics 4 - Question Paper
GUJARAT TECHNOLOGICAL UNIVERSITY
B.E. Sem-IV exam June- 2010
Subject : Mathematics IV
Seat No. Enrolment No.
GUJARAT TECHNOLOGICAL UNIVERSITY
B.E. Sem-IV Examination June- 2010
Total Marks: 70
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q. 1 Do as directed.
(14)
Find the value of Re (f(z)) and Im (f(z)) at the indicated point where
(a)
1
f(z) =
at 7 + 2i.
1 - z
z - i
(b)
(c)
(d)
(e)
(f)
(g)
(a)
(b)
Find the value of the derivative of- at i .
z + i
Find an upper bound for the absolute value of the integral Jezdz, where C is
C
the line segment joining the points (0,0)and (1, 242). c dz
Evaluate J- , where C is | z + i | =1, counterclockwise.
Cz +1
Develop f (z) = sin2z in a Maclaurin series and find the radius of convergence. Define : (i) Singular point (ii) Essential singularity
(iii) Removable singularity (iv) Residue of a function
If f(x) = , find the divided differences [a,b] and [a,b,c]. x
1
Evaluate Je dx by the Gauss integration formula with n=3.
0
Compute f (9.2) from the following values using Newtons divided difference formula.
(03)
(04)
Q2
|
x |
8 |
9 |
9.5 |
11.0 |
|
f((x) |
2.079442 |
2.197225 |
2.251292 |
2.397895 |
(03)
(04)
(c) (i) Find the positive root of x = cosx correct to three decimal places by bisection method.
(ii) Solve the following system of equations using partial pivoting by Gauss-elimination method.
8x2 + 2x3 = -7 3x1 + 5x2 + 2x3 = 8 6x1 + 2x2 + 8x3 = 26
OR
12 3 4
(03)
(04)
(c) (i) Find the dominant eigen value of A =
by power method and hence
find the other eigen value also. Verity your results by any other matrix theory. (ii) Solve the following system of equations by Gauss- seidal method.
10x1 + x2 + x3 = 6 x1 + 10x2 + x3 = 6 x1 + x2 + 10x3 = 6
Q.3 (a) Determine the interpolating polynomial of degree three using Lagranges (04)
interpolation for the table below :
|
x |
-1 |
0 |
1 |
3 |
|
f(x) |
2 |
1 |
0 |
-1 |
3 dx 3
(b) Evaluate J-with n=6 by using Simpsons rule and hence calculate
{ 1 + x 8
(05)
(05)
log2. Estimate the bound of error involved in the process.
Using improved Eulers method, solve + 2xy2 = 0 with the initial
dx
condition y(0)=1 and compute y (1) taking h = 0.2. Compare the answer with exact solution.
OR
Find an iterative formula to find 4n (where N is a positive number) and hence find V5.
(c)
(04)
(a)
(b)
Q.3
|
(05) | |||||||||||||||
(c) Apply Runge-Kutta method of fourth order to calculate y (0.2) given
= x+y, y (0) =1 taking h=0.1 dx
(05)
Find and plot all roots of \f&i .
Find out (and give reason) whether f (z) is continuous at z =0 if
Re( z2) f(z) = --, z 0
(a)
(b)
Q.4
(03)
(03)
1 z 1
= 0
z = 0
2
Using residue theorem, evaluate J -dz, c : | z | = 2
(c)
(d)
(04)
(02)
(02)
(03)
(03)
(04) (02)
C 4 z -1
1 - ez
(i) Expand f (z) =-in Laurents series about z =0 and identify the
z
singularity.
(ii) Find all solutions of sinz = 2.
Solve the equation z2 - (5+i) z + 8 + i = 0.
(a)
(b)
(c)
(d)
Show that if f(z) is analytic in a domain D and | f(z) | = k = const. in D, then f (z) = const. in D.
- 2 z + 3
Find all Taylor and Laurent series of f(z) = -with center 0.
z 3z + 2
(i) Find the center and the radius of convergence of the power series Z (n + 2i) "z"
Find and sketch the image of region x > 1 under the transformation w =
(03)
(03)
(04)
z
2n
dQ
Using the residue theorem, evaluate J
0
Evaluate J Re(z 2)dz, where C is the boundary of the square with vertices
C
0, i, 1 + i, 1 in the clockwise direction.
(i) State and prove cauchy integral theorem.
(ii) Determine a and b such that u = ax3 + bxy is harmonic and find a conjugate harmonic.
OR
Define Mobius transformation. Determine the mobius transformation that maps z= 0, z2 = 1, z3 = m onto w = -1, w2 = -i, w3 = 1 respectively.
Using contour integration, show that J-4 = j=
01 + x 2V2
(b)
(c)
(d)
Q.5 (a) (b)
5 - 3sin Q
(02)
(02)
(03)
(03)
- dz where C is (a) 1 z 1 = 1 (b) 1 z-1 1 = 1. Cz(1 - z) 2 2
Check whether the following functions are analytic or not.
5
. = 2 _
e
Evaluate J-
(04)
(04)
(c)
(d)
(1) f(z) = z 2 (ii) f(z) = z
3
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Attachment: |
| Earning: Approval pending. |
