Find the rank of the matrix \(A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right) \)

Evaluate \(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)

Find the area under the demand curve xy = 1 bounded by the ordinates x = 3, x = 9 and x-axis

Form the differential equation for y = (A + Bx)e^3x where A and B are constants.

Using graphic method, find the value of y when x=27.

x	10	15	20	25	30
y	35	32	29	26	23

A random variable. X has following distribution

X	-1	0	1	2
P(X=x)	\(\frac{1}{3}\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{3}\)

Find E(2X+3)²

If a random variable X follows Poisson distribution such that P(X = 2) = 9. P(X = 4) + 90 P(X = 6) then find the mean and variance.

A sample of five measurements of the diameter of a sphere were recorded by a scientist as 6.33, 6.37,6.36,6.32 and 6.37 mm. Determine the point estimate of
(a) mean
(b) variance.

Fit a trend line to the following data by graphic  method.

Year	1978	1979	1980	1981	1982	1983	1984	1985	1986
Production of steel	20	22	24	21	23	25	23	26	25

For the given pay-off matrix, choose the best alternative for the given states of nature under
(i) Maximin (ii) Minimax princple

Alternative	States of Nature
	Good	Fair	Bad
A	100	60	+50
B	80	50	+10
C	40	20	+5