1.a. If $ \begin{equation} \\A = \begin{bmatrix} 2 & 4 \\ 0 & 3 \\ \end{bmatrix} \end{equation}$

then find the eigenvalues of $6A^{-1}+A^3+2I$

1.b. Find a vector orthogonal to both u = (-6,4,2), v = (3,1,5)

1.c. Show that $\int_c logz \ dz \ = \ 2\pi \ i $, Where C is the unit circle in the Z plane.

1.d. Let X be continuous random variable with probability distribution

$p(X=x) = \begin{cases} \frac{x}{6}+k, \ if \ & 0 \leq x\leq3 \\ 0, & elsewhere \end{cases}$

Evaluate k and find p(1$\leq$x$\leq$2).

2.a. Show that the matrix A is diagonalizable. Also find the transforming matrix M and the diagonal matrix D where $ \begin{equation} \\ A = \begin{bmatrix} -9& 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \\ \end{bmatrix} \end{equation}$

2.b. Find the extremals of the function $\int_0^\frac{\pi}{2}(y^{'2}-y^2+2xy) \ dx$ with y(0) = 0 , y($\frac{\pi}{2}$) = 0.

2.c. Let $R^4$ have the Euclidean inner product. Use Gram-schmidt process to transform the basis {$u_1$,$u_2$,$u_3$} in to orthonormal basis where $u_1$= (1,0,1,1), $u_2$= (-1,0,-1,1), $u_3$= (0,-1,1,1)

3.a. The number of accidents in a year attributed to taxi driver in a city follows poisson distribution with mean 3. Out of 1,000 taxi drivers, find approximately the number of drivers with (i) No accident in a year (ii) more than 3 accident in a year. ( Given: $e^{-1}$ = 0.3679, $e^{-2}$ = 0.1353, $e^{-3}$ = 0.0498,

3.b. Calculate Rank Correlation co-efficient for the following data:

X	10	12	18	18	15	40
Y	12	18	25	25	50	25

3.c. Expand f(z) = $\frac{1}{z^2(z-1)(z+2)}$ about z = 0 for

(i) |z|$\leq$1 (ii) 1$\leq$|z|$\leq$2 (iii) |z|$\gt$2.

4.a. Evaluate $\oint_c \frac{sin^6z}{(z-\frac{\pi}{6})^3} \ dz$, Where c is the circle |z| = 1

4.b. Find the m.g.f of a random variable whose probability density function is

$p(X=x) = \begin{cases} (\frac{1}{2})^x, & \ x \ = 1,\ 2, \ 3... \\ 0, & elsewhere \end{cases}$

Hence, find the mean and variance.

4.c. Verify the Cayley-Hamilton Theorem for matrix A and hence find $A^{-1}$ for

$ \begin{equation} \\A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \\ \end{bmatrix} \end{equation}$ Hence, find $A^5-4A^4-7A^3+11A^2-A-10I$ in terms of A.

5.a Express $p(x) = 7+8x+9x^2$ as a linear combination of $p_1(x) = 2+x+4x^2$, $p_2(x) = 1-x+3x^2$, $p_3(x) = 2+x+5x^2$.

5.b. Using Cauchy residue theorem, evaluate $\oint_0^{2\pi}\frac{cos2\theta}{5+4cos\theta} \ d\theta$ (6 marks) 00