A new transit system has just gone into operation in a city. Of those who use the transit system this year, 10% will switch over to using their own car next year and 90% will continue to use the transit system. Of those who use their cars this year, 80% will continue to use their cars next year and 20% will switch over to the transit system. Suppose the population of the city remains constant and that 50% of the commuters use the transit system and 50% of the commuters use their own car this year,
(i) What percent of commuters will be using the transit system after one year?
(ii) What percent of commuters will be using the transit system in the long run?

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

The marginal revenue function (in thousands of rupees) of a commodity is 7+^e-0.05x where x is the number of units sold. Find the total revenue from the sale of 100 units (e^-5 = 0.0067)

The total cost of production y and the level of output x are related to the marginal cost of production by the equation (6x²+2y²)dx-(x²+4xy)dy = 0. What is the relation between total cost and output if y = 2 when x = 1?

Using Lagrange's formula find the value of y when x = 4 from the following table.

x	0	3	5	6	8
y	276	460	414	343	110

The random variable X tan take only the values 0,1,2. Given that P(X = 0) = P(X = 1) = P and E(X²) = E(X), find the value of p.

If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches find the height below which 99% of the student lie?

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.

The followingdata relateto the life(inhours) of 10 samples of 6 electricbulbs each drawn at an intervalof one hour from a production process.Draw the controlchart for \(\overline { X } \) and \(\overline { R } \) and comment.

Sample No	Lifetime (inhour)
	1	2	3	4	5	6
1	620	687	666	689	738	686
2	501	585	524	585	653	668
3	673	701	686	567	619	660
4	646	626	572	628	631	743
5	494	984	659	643	660	640
6	634	755	625	582	683	555
7	619	710	664	693	770	534
8	630	723	614	535	550	570
9	482	791	533	612	497	499
10	706	524	626	503	661	754

(For n = 6,A₂= 0.483,D₃ = 0,D₄ = 2.004)

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