Computer Engineering (Semester 4)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from Q.2 to Q.6.
(3) Use of statistical table permitted.
(4) Figures to the right indicate full marks.
Q1
a)
Evaluate $\int_{c}^{} log z dz$ , where C is the unit circle in the z-plane.
(5 marks)
00
b)
Find the eigen values of the adjoint of A = $ \begin{bmatrix}
2 & 0 & -1 \\
0 & 2 & 0 \\
-1 & 0 & 2
\end{bmatrix}. $ .
(05 marks)
00
c)
If the arithmetic mean of regression coffecient is p and their difference is 2p. find the correlation coeffecient.
(05 marks)
00
d)
Write dual of the following L.P.P .
(05 marks)
00
Maximize Z = 2$X_1$ - $X_2$ + 4$X_3$
Subject to $X_1$ + 2$X_2$ - $X_3$ < 5.
2$X_1$ - $X_2$ + $X_3$ < 6
$X_1$ + $X_2$ + 3$X_3$ < 10
4$X_1$ + $X_3$ < 12
$X_1$, $X_2$, $X_3$ > 0.
Q2
a)
Evaluate $\int_{c}^{} $/frac{cotz}{z}$ dz$, where C s the ellipse 9$x^2$ + 4$y^2$ = 1.
(06 marks)
00
b)
Show that A = $ \begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & 4 \\
3 & 4 & 5
\end{bmatrix} $ is non-derogatory.
(06 marks)
00
c)
If X is a normal variable with mean 10 and standard deviation 4. find
(06 marks)
00
i) P(|X-14|), ii P(5<X<18), iii) P(X<15)</p>
Q3
a)
Find the expectation of number of failures preceeding the first success in an infinite series of independent trials with constant probablities p & q of success and failure respectively.
(06 marks)
00
b)
Using simplex Method solve the following L.P.P
(06 marks)
00
Maximize Z - 10$x_1$ + $x_2$ + $x_3$
subejct to $x_1$ + $x_2$ - 3$x_3$ < 10
4$x_1$ + $x_2$ + $x_3$ < 20
$X_1$, $X_2$, $X_3$ > 0.
c)
Expand f(z) = $\frac{1}{z(z+1)(z-2)}$
(08 marks)
00
i) Within the unit circle about the origin.
ii) with in the anulus region between the concebtric circles about the origin having radii 1 and 2 respectively.
iii) In the exterior of the circle with centre at the origin and radius 2.
Q4
a)
If X is a Binomial distributed with mean = 2 and variance 4.3 . find the probablity distribution of X.
(06 marks)
00
b)
Calculate the value of rank correlation coefficient from the following data regrading score of 6 students in physics & chemistry test.
(06 marks)
00
Marks in Physics : 40, 42, 45, 35, 36, 39
Marks in chemistry : 46, 43, 44, 39, 40, 43.
c)
Is the matrix A = $ \begin{bmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
0 & 0 & 1
\end{bmatrix} $ diagonalisable? If so find the diagonal form and the transforming matrix.
(08 marks)
00
Q5
a)
A random sample of 50 items gives the mean 6.2 and standard deviation 10.24. can it be regraded as drawn from a normal population with mean 5.4 at 5% level of significance.
(06 marks)
00
b)
Evaluate $\int_{0}^{oo} \frac{dx}{(x^2 + a^2)^3} $, a>0. Using Cauchy's residue theorem.
(06 marks)
00
c)
Using Kuhn-Tucker condition to solve the following NLPP
(08 marks)
00
Maximize Z = 8$x_1$ + 10$x_2$ - $(x_1)^2$ - $(x_2)^2$
Subject to 3$x_1$ + 2$x_2$ < 6
$x_1$>,$x_2$ > 0
Q6
a)
The following table gives the number of accidents in a ity during a week. Find wheter the accidents are uniformly distributed over a week.
(06 marks)
00
Day |
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
Total |
No of acc |
13 |
15 |
9 |
11 |
12 |
10 |
14 |
84 |
b)
If two independent random samples of sizes 15 & 8 have respectively the following means and population standard deviations,
(06 marks)
00
$X_1$ = 980 $X_2$ = 1012
$a_1$ = 75 $a_2$ = 80
c)
Using Penally (Big M) method solve the following L.P.P
(08 marks)
00
Maximize Z = 2$x_1$ + $x_2$
subject to 3$x_1$ + $x_2$
4$x_1$ + 3$x_2$ > 6
$x_1$ + 2$x_2$ < 3
$x_1$,$x_2$ > 0