For what values of k, the system of equations kx+ y+z = 1, x+ ky+z= 1, x+ y+kz = 1 have
(I) Unique solution
(ii) More than one solution
(iii) no solution

Evaluate \(\int { \frac { { x }^{ 7 } }{ { x }^{ 5 }+1 } } dx\)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin } 3xsin\ 2x\ dx\)

Find the area of the region bounded by the parabola y² = 4x and the line 2x - y = 4.

Solve: (D² + 14D + 49)y = e^-7x + 4.

The probability distribution of the discrete random variables X and Y are given below

X	0	1	2	3
P(X)	\(\frac{1}{5}\)	\(\frac{2}{5}\)	\(\frac{1}{5}\)	\(\frac{1}{5}\)

Y	0	1	2	3
P(Y)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)	\(\frac{1}{10}\)

Prove that E(Y²) = 2E(X).

Fit a straight line trend to the following data using the method of least square. Estimate the trend for 2007.

year	2000	2001	2002	2003	2004
Sales (in tonnes)	1	1.8	3.3	4.5	6.3