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 ഽ x³ sin (x⁴) dx

The marginal cost function of a commodity in a firm is 2 + e^3x where X is the output. Find the total cost and average cost function if the fixed cost is Rs. 500.

The rate of increase in the cost Cof ordering holding as the size q of the order increases is given by the differential equation \(\frac { dc }{ dq } =\frac { { c }^{ 2 }+2cq }{ { q }^{ 2 } } \). Find the relationship between c and q if c = 1 when q = 1.

From the data, find the number of students whose height is between 80 cm and 90 cm

Height in cm (x)	40-60	60-80	80 - 100	100-120	120-140
No. of. students (y)	250	120	100	70	50

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).

Four coins are tossed simultaneously. What is the probability of getting
a) atleast 2 heads
b) atmost 2 heads.

A sample poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favour of a particular candidate. Find
(a) 95% confidence limits
(b) 99% confidence limits for the proportion to all voters in favour of this candidate.

Calculate Fisher's ideal index from the following data and verify that it satisfies both time reversal and factor reversal test

Commodity	Price		Quantity
	1985	1986	1985	1986
A	8	20	50	60
B	2	6	15	10
C	1	2	20	25
D	2	5	10	8
E	1	5	40	30

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment.