1.a. Find the coefficient of correlation from the following data : N=10, $\sum \ X=225$ $\sum \ Y=189$, $\sum \ (X-22)^2=85$, $\sum \ (Y-19)^2=25$

1.b. Evaluate $\int_clogz \ dz$. Where c is |z|=1

1.c. Find the projection of u= (3, 1, 3) along and perpendicular to v=(4, -2, 2)

1.d. Find an eigen values of (i) Adj (A) (ii) 24$A^{-1}$+2A-I where, $ \begin{equation} \\A = \begin{bmatrix} 1 & 2 & 3 & -2 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & -5 \\ 0 & 0 & 0 & 6 \\ \end{bmatrix} \end{equation}$

2.a. Find the extremal of $\int_0^1(y^{"2}+x^2-y^2)dx$

2.b. Use Gram-schmidt process to transform the basis {$u_1$,$u_2$,$u_3$} in to orthonormal bases where $u_1$= (1,1,1), $u_2$= (0,1,1), $u_3$= (0,0,1)

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

3.a. If X is a normal variable with mean 10 and standard deviation 4, Find

(i) P[|X-4| $\lt$ 1]

(ii) P[5 $\lt$ X$\lt$ 18]

(iii) P[X$\lt$12]

3.b. Seven dice are thrown 729 times. How many times do you expect at least four dice to show 3 or 5.

3.c. Using Rayleigh-Ritz method find solution for the extremal of the functional

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

4.a. For the 50 students in the class mean of X is 62.4 and 16Var(X)=9Var(Y). Regression line of X on Y is 3Y-5X+180=0 Find

(i) Mean of Y

(ii) Correlation r between X and Y

(iii) Regression line of Y on X

4.b. Evaluate $\int_c\frac{(z+1)}{(z^3-2z^2)} \ dz$ where c is (i) |z|=1 (ii) |z-2-i|=2 (iii) |z-1-2i|=2

4.c. Check whether the set of all pairs of real number of the form (1,x) with operations (1,y)+(1,x)=(1,y+x) and k(1,y)=(1,ky) is a vector space.

5.a Using cauchy residue theorm evaluate $\int_0^{\infty}\frac{1}{(x^2+1)(x^2+9)} \ dx$

5.b. If $ \begin{equation} \\ A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} \end{equation}$

find $A^{50}$

5.c. Find M.G.F. of poisson distribution. Hence find its mean and variance.

6.a Is the matrix derogatory? Justify your answer Where

$ \begin{equation} \\ A = \begin{bmatrix} -2 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \\ \end{bmatrix} \end{equation}$

6.b. A random variable X has the following p.d.f. f(x)=$kx^2e^{-x}$ for x$\gt$0. and f(x)=0 otherwise. Find

(i) k (ii) mean (iii) variance

(iv) M.G.F. (v) c.d.f. of X (vi) P[0$\lt$X$\lt$1]

6.c. Find all possible Laurent series of f(z)= $\frac{(z^2-1)}{(z^2+5z+6)}$