Find the rank of the matrix \(\left( \begin{matrix} 1 & 2 & -1 \\ 2 & 4 & 1 \\ 3 & 6 & 3 \end{matrix}\begin{matrix} 3 \\ -2 \\ -7 \end{matrix} \right) \)

Evaluate \(\int { \frac { x+2 }{ \sqrt { 2x+3 } } } dx\)

Using integration, find the area of the region bounded by the line y −1 = x, the x axis and the ordinates x = –2, x = 3.

Find the differential equation of the following
y = cx + c − c³

Construct a forward difference table for y = f(x) = x³+2x+1 for x = 1,2,3,4,5

Let X be a discrete random variable with the following p.m.f
\(p(x) = \begin{cases}0.3 & \text { for } x =3 \\ 0.2, & \text { for } x = 5 \\ 0.3, & \text { for } x = 8 \\ 0.2, & \text { for} x = 10 \\ 0, & \text { otherwise } \\ \end{cases}\)
Find and plot the c.d.f. of X.

Suppose A and B are two equally strong table tennis players. Which of the following two events is more probable:
(a) A beats B exactly in 3 games out of 4 or
(b) A beats B exactly in 5 games out of 8 ?

Find the sample size for the given standard deviation 10 and the standard error with respect of sample mean is 3.

Obtain an initial basic feasible solution to the following transportation problem using least cost method.

Here O_i and D_j denote i^th origin and j^th destination respectively.

A person wants to invest in one of three alternative investment plans: Stock, Bonds and Debentures. It is assumed that the person wishes to invest all of the funds in a plan. The pay-off matrix based on three potential economic conditions is given in the following table:

Alternative	Economic conditions
	High growth(Rs.)	Normal growth(Rs.)	Slow growth (Rs.)s
Stocks	10000	7000	3000
Bonds	8000	6000	1000
Debentures	6000	6000	6000

Determine the best investment plan using each of following criteria i) Maxmin ii) Minimax.