Osmania University (OU) 2009 B.E Computer Science and Engineering 2/4 cse maths iv 09 - Question Paper
BE 2/4 maths-iv
Computer Science & Engineering
December09
Osmania university
Code No.: 121/N
FACULTY OF ENGINEERING
B.E. II/IV Year (ECE/Mech/Prod/CSE) II Semester (Supplementary)
Examination, December 2008
MATHEMATICS IV
Time : 3 Hours] [Max. Marks : 75
Answer all questions of Part A.
Answer any five questions from Part B.
Part A - (Marks : 25)
Choose the correct answer from the following.
1. If f(z) = x + ay + i (bx + cy) is analytic the values of a =-and c =-.
(a) a = b, c = 1 (b) a = c, b = 1 (c) a = -1, b = 1 (d) a = - b, c = 1
1+1
2. The value of J (x2 - iy) dz along the path y = x? is
o
5-i 5 + i i- 5
(a) (b) -g- (c) (d) None
r z3 - z
3. The value of J " rry Q2: where C is | z| = 3, using Cauchys Integral formula is
c [z -2)
(a) 12 7U (b) - 12 7ii (c) tci (d) None
1 - e2x
4. Residue of /(z) =-j at z - 0 is
z
. (a) | (b) (c) | (d) None
r (z - 1) dz
5. Using Cauchys Residue theorem the value of (z + i)2 (2 _ 2) were tie circle | z - i | = 2 is
7tl 7Xt -2711
(a) (b) (c) (d) None
6. The Mean of Poisson distribution (A, as a parameter) is-.
(a) X2 (b) X (c) 2X (d) None
7. If x and y are random variables then v (x - y) is-.
(a) v (x) + v (y) (b) v [x) - v {y)
(c) v {x) + v {y) + 2 cov (x, y) (d) v [x) + v [y) - 2 cov [x, y)
8. Correlation coefficient is independent of change of scale and change of origin. (TRUE/FALSE)
9. Find the correlation coefficient (r) for the following data:
x : 1 2 3 4 5 6 y : 6 5 4 3 2 1
10. Derive Mean and Variance for the normal distribution.
Part B - (Marks : 50)
/*%
A
| Re /(z) j2= 2 | f\z) |2
+
dx2 dy2
11. (a) If f(z) - u + iv is analytic, show that
(b) Find the bilinear transformation that maps z = 0, - i, - 1 onto w= i, 1,0.
rZ -zdz
12. (a) Using Cauchys integral formula evaluate J (z-2)
3 where C is | z - 2 | = 1.
dx
00
(b) Evaluate J
[x2 + a2) (x2 + b2) a>0>b>0 usin residue theorem.
13. (a) If X is a random variable with the following distribution
|
X |
13-45 |
|
/(*) |
0.4 0.1 0.2 0.3 |
Find the mean, variance and standard deviation of X.
(b) A pair of dice is thrown. Let X denote the minimum of the two numbers which occur. Find the distribution and expectation of X.
14. (a) Let X be normally distributed with mean ja = 8 and standard deviation a = 4. Find (i) P (5 < X< 10), (ii) P(10 <X< 15), (iii) P{X> 15).
(b) Suppose 1 per cent of the items made by a machine are defective, find the probability that 3 or more items are defective in a sample of 100 items.
15. (a) Calculate the first four moments of the following distribution about the mean
and hence find Pi and P2.
x : 0 1 2 3 4 5 678 / : 1 8 28 56 70 56 28 8 1
(b) Derive M.G.F. and C.G.F. of chi-square distribution.
16. The no. of automobile accidents for week in a certain community are as follows:
12, 8, 20, 2, 14, 10, 15, 6, 9, 4
Are these frequences in agreement with the belief that accidents condition were the
r\
same during this 10 week period (Table value of % - 16.9).
17. (a) State and prove Taylors theorem.
1
(b) Expand f(z) =
z2 - 3z + 2
in the region \z\ < 1.
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| Earning: Approval pending. |
