Osmania University (OU) 2007-2nd Sem B.E 2/4 main maths-4 - Question Paper
FACULTY OF ENGINEERING B.E. 2/4 (ECE/M/P/CSE) II Semester Main Examination, April/May-2007
MATHEMATICSIV
Time : Three Hours] [Maximum Marks : 75
Note :Answer ALL the questions from Part-A and any FIVE from Part-B.
PARTA
Choose the correct answer from the following :
1. The value of a so that u (x, y) = ax2 + y2 + xy is harmonic :
(a) -1 (b) 1
(c) 2 (d) none [ ]
l+i
2. The value of J (x2 + iy)dz along the path y = x is :
o
w M <b> H
(c) + 1 (d) none [ ]
1 f z2 + 5
3. Using Cauchys integral formula, the value of 2 7ti J z - 3 were C : | z | = 2 is :
c
(a) 28 ni (b) 14 m
(c) 14 (d) 0 [ ]
4. The zero of f(t) = "3 is :
z - 1
(a) -1 (b) -l+i
(c) 1 i (d) none [ ]
1 + ez
(ii) The residue of-: at z = 0 is
zcost + sinz ~
6. A random variable X has the following probability function :
Values of X (x) . P(x)
0 0
1 k
2 2k
3 2k
4 3 k
5 k2
6 2 k2
7 7 k2 + k
(i) Then the value of k
(ii) The value of P (x < 6)
(iii) P(x>6).
7. Indicate whether the following statements are true or false :
(i) f(t) = | z |2 is analytic function in every point of region.
(ii) If a is constant and x be a random variable then v(ax) = av(x).
13 11
(iii) If A and B are two events such that P(A) = y; P(B) = and P(A u B) = then
8. The first four moments of a distribution, about the value 5 of the variable are 2, 20, 40 and 50. Find the moments about mean.
9. Find the mean and variance of Poisson distribution.
10. Is the following statement true ? Give reasons. 40x 18y = 5 and 8x - 1 Oy + 6 = 0 are respectively the regression equation of y on x is x on y.
PARTB
11. (a) Derive Cauchy-Riemann equations in Cartesian form.
(b) Prove that u = 2x - x3 + 3xy2 is harmonic and find its harmonic conjugate.
12. State Cauchys residue theorem and hence evaluate using theorem :
(i) |
de
dx
5 + 4 sin 9
+ x
(ii) J
13. (a) Prove that:
(ii) nr = - rc, M-r-1 +
(0 a2 = H2
(b) A continuous random variable X defined by :
if -3 < x <-1
f(x) =
if -1 <x< 1
if 1 <x<3.
(3 + x) 16
6-2 x2 16
(3-x)2
16
Verify that: (i) The area under the curve is unity (ii) Also show that the mean is zero.
14. (a) In a normal distribution exactly 7% of the items are under 35 and 89% are under 63. Find the
mean and standard deviation of the distribution.
(b) Write the properties of Normal distribution.
15. A survey of 320 families with 5 children each revealed the following information :
No. of boys No. of girls No. of families
5 0 14
4 1 56
3 2 110
2 3 88
1 4 40
0 5 12
Is this result consistent with the hypothesis that male and female births are equally probable ?
(Table value of X2 - for 5 d.f. is 11 07).
16. (a) State and prove Cauchys integral formula. x + 1
if -1 < x < 1 otherwise
2
0
(b) Iff(x) =
where 'x' have Probability density function. Find the mean and standard deviation of x.
17. (a) State and prove Bayes theorem.
1
is Laurents series valid with in the region 1 < | z | < 2.
(b) Expand ~
z - 3z + 2
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Attachment: |
| Earning: Approval pending. |
