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MU Computer Engineering (Semester 4)
Applied Mathematics 4
December 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Find the Eigenvalues and eigenvectors of the matrix.
A=$$\begin{bmatrix} 2 & 2& 0\\ 0 & 2& 1\\ 0& 0& 2 \end{bmatrix}$$/
5 M
1(b) Evaluate the line integral $\int_{0}^{l+i}\left ( x^2+iy \right )$ dz along the path y=x
5 M
1(c) Find k and then E (x) for the p.d.f.
$$f(x)=\left\{\begin{matrix} k(x-x^2),0\leq x\leq 1,k> 0& \\ 0, & otherwise \end{matrix}\right.$$/
5 M
1(d) Calculate Karl person's coefficient of correlation from the following data.
 x 100 200 300 400 500 y 30 40 50 60 70
5 M

2(a) Show that the matrix $$A=\begin{bmatrix} 2 & -2& 3\\ 1& 1& 1\\ 1& 3& -1 \end{bmatrix}$$/ is non-derogatory.
6 M
2(b) Evaluate $\int \frac{e^2^z}{\left ( z+1 \right )^4}$ dz where C is the circle |z-1|=3
6 M
2(c) If x is a normal variate with mean 10 and standard deviation 4 find
i) P(|x-14|<1)
ii) P(5≤x≤18)
iii) P(x≤12)
8 M

3(a) Find the relative maximum of minimum (if any) of the $Z=X_{1}^{2}+X_{2}^{2}+X_{3}^{2}-4X_1-8X_2-12X_3+100$
6 M
3(b) If x is Binomial distributed with E(x)=2 and V(x)=4/3,find the probability distribution of x.
6 M
3(c) If $$A=\begin{bmatrix} 2& 1\\ 1 & 2 \end{bmatrix}$$/,
find A50.
8 M

4(a) Solve the following L.P.P by simplex method Minimize
z=3x1+2x2 Subject to 3x1+2x2≤18
0≤x1≤4
0≤x2≤6
x1,x2≥0.
6 M
4(b) The average of marks scored by 32 boys is 72 with statndard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test 1% level significance whether the boys perform better than the girls.
6 M
4(c) Find Laurent's series which represents the function
$f(z)=\frac{2}{\left ( Z-1 \right )\left ( z-2 \right )}$ When
i) |z| <1,
ii) 1<|z|<2
iii) |z|>2
8 M

5(a) Evaluate $\int \frac{Z^2}c_{\left ( z-1 \right )^2\left (z+1 \right )}$ dz where C is|z| =2 using residue theorem
6 M
5(b) The regression lines of a sample are x+6y=6 and 3x+2y=10 Find
i) Sample means
$\bar{x} \ \text{and}\ \bar{y}$
ii) Correlation coefficient between x ad y. Also estimate y When x=12
6 M
5(c) 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
Using χ2-test examine the hypothesis that the die is unbiased.
8 M

6(a) Evaluate $\int ^\infty _\\-\infty\frac{x^2+x+2}{x^4+10x^2+9}$ dx using contour integration.
6 M
6(b) If a random variable x follows Poisson distribution such that P(x-1)=2(x=2) Find the mean the variance of the distribution Also find P(x=3).
6 M
6(c) Use Penalty method to solve the following L.P.P. Minimize
z=2x,sub>1+3x2
x1+x2≥5
x1+2x2≥6 x1, x2≥0.
8 M

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