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 \(\int { \frac { 1 }{ { 3x }^{ 2 }+13x-10 } } dx\)

The elasticity of demand with respect to price for a commodity-is a constant and is equal to 2. Find the demand function and hence the total revenue function, given that when the price is 1, the demand is 4.

Solve: (y-x)\(\frac { dy }{ dx } \) = a²

From the following data, calculate the value of e^1.75

x	1.7	1.8	1.9	2.0	2.1
ex	5.474	6.050	6.686	7.386	8.166

The probability distribution of a random variation X is given below.

X	0	1	2	3	4
P(X)	0.1	0.25	0.3	0.2	0.15

Find
(i) V(X)
ii) V\((\frac{X}{2})\)

20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using
(i) Binomial distribution
(ii) Poisson distribution (e^-2 = 0.1353)

Measurements of the weights of a random sample of 200 ball bearings made by certain machine during one week showed a mean of 0.824 newtons and a S.D. of 0.042 newton's. Find
a) 95% and
b) 99% confidence limits for the mean weight of all the ball bearings.

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

Obtain an initial basic feasible solution to the following transportation problem using Vogels' approximation method.