1.a. Find the extremal of $\int_{x_0}^{x_1}\frac{1+y^2}{y^{'2}} \ dx$

1.b. Evaluate $\int_c\frac {sin^2z}{(z-\frac{\pi}{6})^3} \ dz$ Where c is the circle |z|=1

1.c. If $ \begin{equation} \\A = \begin{bmatrix} \pi & \pi / 4 \\ 0 & \pi / 2 \\ \end{bmatrix} \end{equation}$ Find Cos A

1.d. The number of messages sent per hour over a computer network has the following probability distribution.

x	10	11	12	13	14	15
P(X=x)	.08	3k	6k	4k	4k	.07

Find the mean and variance of number of message sent per hour.

2.a. Construct an orthonormal basis of $R^3$ using Gram Schmidt process to S= {(1,0,0),(3,7,-2),(0,4,1)}

2.b. Evaluate $\int_0^{2+i}{(\overline{z})^2}dz $ along

i. Y = x/2

ii. The real axis to 2 and then vertically to 2 + i

2.c. i. An underground mine has 5 pumps installed for pumping out storm water. The probability of any one of the pumps failing during the storm is 1/8.What is the probability that at least 2 pumps will be working.

ii. Let W be the set of 2x2 matrice of the form $ \begin{bmatrix} \ a & 0 \\ \ 0 & b \\ \end{bmatrix}$

Show that W is a subspace of space V of all 2X2 matrices.

3.a. Calculate karl pearson's coefficient of correlation between expenditure and sales from given data.

Adevertising Expenses (000 Rs)	39	65	62	90	82	75	25	98	36	78
Sales (Lakhs of Rupees)	47	53	58	86	62	68	60	91	51	84

3.b. Show that the matrix $ \begin{equation} \\ A = \begin{bmatrix} -2 & 2 & -3\\ 2 & 1 & -6\\ -1 & -2 & 0\\ \end{bmatrix} \end{equation}$ is derogatory

3.c. Evaluate $\int_0^{2\pi}{\frac{d\theta}{13+12cos\theta}}$

4.a. Using Cauchy residue Theorem evaluate

$\int_c{\frac{sin\pi z^2+cos\pi z^2}{(z-1)^2 (z-2)}dz}$

where c is the circle ${z} = 3 $

4.b. Find the extremals of the functional $\int_{x_0}^{x_1}(y^{"2}-y^2+x^2)dx$

4.c. i. Assume that probability of an individual coal miner being injured in a mine accident during a year is 1/2400 calculate the probability that in mine employing 200 miners there will be at least 1 year ii. If x denotes the outcome when a fair die is tossed. find the M.G.F. of X about the origin. Hence find first two moments about the origin.

5.a The IQ's of army volunteers in a given year are normally distributed with mean 110 and standard deviation 10.The army wants to give advanced training to 20% of those recruits with highest scores . What is the lowest scores acceptance for advanced training.

5.b. Solve by Rayleigh-Ritz method the boundary value problem

I=$\int_0^1(y^{'2}-y^2-2xy)dx$ given y(0)=0 and y(1)=0

5.c. Show that the matrix $ \begin{equation} \\ A = \begin{bmatrix} 11 & -4 & -7 \\ 7 & -2 & -5 \\ 10 & -4 & -6 \\ \end{bmatrix} \end{equation}$ is similar to diagonal matrix. Find the diagonal matrix and transforming matrix.

6.a Verify Cayley Hamilton Theorem for $ \begin{equation} \\ A = \begin{bmatrix} 4 & 3 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 1 \\ \end{bmatrix} \end{equation}$ Hence find $A^{-1}$

6.b. Obtain Taylor's and laurent series expression for $f(z) = \frac{z-1}{(z^2-z-3)}$ indicating region of convergence.

6.c. i. The lines of regression of bivariate population are

8x-10y+66=0 and 40x-18y = 214. The variance of x is 9.

Find,

a. coefficient of correlation r

b. the standard deviation y

ii.If a,b,c are three positive numbers then using cauchy schwertz inequality prove that $(a+b+c)(\frac1a+\frac1b+\frac1c) \geq 3^2$