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 zplane.(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= 
2x_{1}x_{2}+3x_{3} 
Subject to 
x_{1}2x_{2}+x_{3} ? 4 

2x_{1}+x_{3} ? 10 

x_{1}+x_{2}+3x_{3}=20 

x_{1}, x_{3} ? 0, x_{2} unresticted 
(5 marks)
2 (a) Evaluate $$ int_c \dfrac {z+2}{z^3 2z^2 } dz, $$ Where C is the circle z2=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 (xx^{2})^{3}. 0?x?1. Find (i) mean (ii) variance.(6 marks)
3 (b) Solve the following L.P.P. by simplex method
Maximise Z= 
4x_{1}3x_{2}+6x_{3} 
Subject to 
2x_{1}3x_{2}+2x_{3} ? 440 

4x_{1}+3x_{3} ? 470 

2x_{1}+5x_{2} ? 430 

x_{1}, x_{2}, x_{3} ? 0 
(6 marks)
3 (c) Find all possible Laurent's expansions of the function. $$ f(z) = \dfrac {7z2}{z(z2)(z+1)} \ about \ z=1 $$(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 KuhnTucker conditions to solve the following N.L.P.P.
Maximize Z= 
8x_{1}+10x_{2}x_{1}^{2}x_{2}^{2} 
subject to 
3x_{1}+2x_{2} ? 6 

x_{1}, x_{2} ? 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= 
5x_{1}+2x_{2}+3x3 
subject to 
2x_{1}+2x_{2  }x_{3} ? 2 

3x_{1} 4x_{2} ? 3 

x_{1}+3x_{3} ? 5 

x_{1}, x_{2}, x_{3} ? 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)