Information Technology (Semester 4)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Show that $$ \int_c log z \ dz =2 \pi i, $$ where C is the circle in the z-plane.(5 marks)
1 (b) If $$ A= \begin{bmatrix}
1 &0 \\2
&4
\end{bmatrix} $$ then find the eigen values of 4A-1+3A+2I.(5 marks)
1 (c) It is given that the means of x and y are 5 and 10. If the line of regression of y on x is parallel to the line 20y=9x+40, estimate the values of y for x=30.(5 marks)
1 (d) Find the dual of the following L.P.P.
|
Maximise Z= |
2x1-x2+3x3 |
|
Subject to |
x1-2x2+x3 ? 4 |
|
|
2x1+x3 ? 10 |
|
|
x1+x2+3x3=20 |
|
|
x1, x3 ? 0, x2 unresticted |
(5 marks)
2 (a) Evaluate $$ int_c \dfrac {z+2}{z^3 -2z^2 } dz, $$ Where C is the circle |z-2=i|=2(6 marks)
2 (b) Show that $$ A= \begin{bmatrix}
7 &4 &-1 \\4
&7 &-1 \\-4
&-4 &4
\end{bmatrix} $$ is derogatory.(6 marks)
2 (c) In a distribution exactly normal 7% of items are under 35 and 89% o the items are under 63. Find the probability that an item selected at random lies between 45 & 56.(8 marks)
3 (a) A continuous random variable has probability density function f(x)=6 (x-x2)3. 0?x?1. Find (i) mean (ii) variance.(6 marks)
3 (b) Solve the following L.P.P. by simplex method
|
Maximise Z= |
4x1-3x2+6x3 |
|
Subject to |
2x1-3x2+2x3 ? 440 |
|
|
4x1+3x3 ? 470 |
|
|
2x1+5x2 ? 430 |
|
|
x1, x2, x3 ? 0 |
(6 marks)
3 (c) Find all possible Laurent's expansions of the function. $$ f(z) = \dfrac {7z-2}{z(z-2)(z+1)} \ about \ z=-1 $$(8 marks)
4 (a) Find the moment generating function of Binomial distribution & hence find mean and variance.(6 marks)
4 (b) Calculate the correlation coefficient from the following data:
|
X: |
100 |
200 |
300 |
400 |
500 |
|
Y: |
30 |
40 |
50 |
60 |
70 |
(6 marks)
4 (c) Show that the matrix $$ A= \begin{bmatrix}
8 &-6 &2 \\-6
&7 &-4 \\2
&-4 &3
\end{bmatrix} $$ is diagonalisable. Find the transforming matrix and the diagonal matrix.(8 marks)
5 (a) Ten individuals are chosen at random from a population and their heights are found to be 63, 63, 64, 65, 66, 69, 69, 70, 70, 71 inches. Discuss the suggestion that the mean height of the universe is 65 inches.(6 marks)
5 (b) Evaluate $$ \int^\infty_0 \dfrac {dx}{(x^2+a^2)^3 , \ a>0 $$ using contour integration.(6 marks)
5 (c) Use Kuhn-Tucker conditions to solve the following N.L.P.P.
|
Maximize Z= |
8x1+10x2-x12-x22 |
|
subject to |
3x1+2x2 ? 6 |
|
|
x1, x2 ? 0 |
(8 marks)
6 (a) A die was thrown 132 times and the following frequencies were observed
|
No. Obtained: |
1 |
2 |
3 |
4 |
5 |
6 |
Total |
|
Frequency: |
15 |
20 |
25 |
15 |
29 |
28 |
132 |
fit the binomial distribution.(6 marks)
6 (b) Using duality solve the following L.P.P.
|
Maximize Z= |
5x1+2x2+3x3 |
|
subject to |
2x1+2x2 - x3 ? 2 |
|
|
3x1- 4x2 ? 3 |
|
|
x1+3x3 ? 5 |
|
|
x1, x2, x3 ? 0 |
(6 marks)
6 (c) (i) A random sample of 50 items gives the mean 6.2 and standard deviation 10.24, can it be regarded as drawn from a normal population with mean 5.4 and 5% level of significance?(4 marks)
6 (c) (ii) Find the M.G.F. of the following distribution.
|
X: |
-2 |
3 |
1 |
|
P(X=x) |
1/3 |
1/2 |
1/6 |
Hence find first four central moments.(4 marks)
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