Kerala University 2009-5th Sem B.Tech Mechanical Engineering Mathematics - Question Paper
Fifth Semester B.Tech. Degree Examination, June 2009
(2003 Scheme)
03.501: ENGINEERING MATHEMATICS – IV (CMNPHETARUFB)
Reg. No. :....................................
(Pages : 3) 3143
Name
Fifth Semester B.Tech. Degree Examination, June 2009
(2003 Scheme)
03.501: ENGINEERING MATHEMATICS - IV (CMNPHETARUFB)
Time : 3 Hours Max. Marks : 100
Instruction: Answer all questions from Part - A and one question from each Module.
PART - A
1. Using Cauchy Reimann Equations show that f (z) = |z|2 is not analytic at any point.
2. Show that f(z) is analytic and
i) Real f (z) is constant
ii) Im.f(z) is constant, then f(z) is a constant.
1
3. Show that under the transformation w = all circles in the z plane is transformed in
z
to circles or straight lines in the w plane.
z
f e
4. Show that J dz = 2ni, c :| z |= 1.
C z
1
5. Expand -21-- the region 0 < |z - 1| < 1.
z 3z + 2
6. Define fixed point and critical point of a bilinear transformation. Find the fixed 5 4z
point of w =-.
F 4z 2
7. Evaluate Jtanz dz where c is the circle |z| = 2.
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C
8. A random sample of 500 apples was taken from a large consignment and 60 were found to be bad. Obtain a 95% limits for percentage of bad apples in the consignment.
9. A random variable X has the following probability function :
Values of X x : 0 1 23456 7 p(x) : 0 k 2k 2k 3k k2 2k2 7k2+k
(1) find k
(2) evaluate p[X < 6], p[X > 6], p[3 < X < 6].
10. During war, 1 ship out of 9 was sunk of on an average in making a certain voyage. What was the probability that exactly 3 out of a convoy of 6 ships would arrive safely ?
PART -B MODULE - I
11. a) Determine an Analytic function whose real part is e2x (x cos 2y - y sin 2y).
b) If f (z) is an Analytic function prove that |Re f(z)|2 = 2|f'(z)f
+
dx2 dy2 v J J
c) Determine the region in the w plane into which the region < x < 1 and < y < 1 is mapped by the transformation w = z2.
12. a) If f (z) = u + iv is an analytic function and find f (z) if u + v = x when f (1) = 1.
x + y
b) Find the bilinear transformation which maps the point z = 1, i, -1 on to the points w = i, 0, - i. Hence find the image of |z| < 1.
c) Find the image of the circle |z - 3| = 5 under the transformation w = .
z
13. a) Integrate f (z) = x2 + ixy from A (1, 1) to B (2, 4) along the curve x = t, y = t2.
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1
b) Expand 2--- as a Laurents series in 1< |z|< 3.
z 4z + 3
C sin nz2 + cos nz2dz
c) Evaluate using Residue theorem I ; where c : |z| = 3.
c (z 1) (z 2)
2n
14. a) Show that I
d0 _ 5n
o(5 3cos 0)2 _ 12
b) Evaluate ? dx .
01 + x4
MODULE - III
15. a) Find the mean and variance of the Binomial distribution.
b) Fit a parabola to the data :
x : 123 4 5 6 7 89
y : 2 6 7 8 10 11 11 10 9
c) For a normally distributed variate x with mean 1 and S.D. 3, find the probability that 3.43 < x < 6.19.
16. a) In two colleges affiliated to a university 64 out of200 and 48 out of250 candidates
failed in an examination.
If the percentage failure in the university is 18%, examine whether the colleges differ significantly.
b) Out of 800 families of 5 children each, how many would you expect to have
1) 3 boys 2) 5 girls ?
c) If X is a Poisson variate such that P[X = 2] = 2P [X = 4] + 90 P [X = 6 ] find the
S.D.
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Attachment: |
| Earning: Approval pending. |
