80% of students who do maths work during one study period, will do the maths work at the next study period. 30% of students who do english work during one study period, will do the english work at the next study period. Initially there were 60 students do maths work and 40 students do english work.
Calculate,
(i) The transition probability matrix
(ii) The number of students who do maths work, english work for the next subsequent 2 study periods.

Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

The sum of three numbers is 6. If we multiply the third number by 2 and add the first number to the result we get 7. By adding second and third numbers to three times the first number we get 12. Find the numbers using rank method

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 \(\int { { \left( \log x \right) }^{ 2 } } dx\)

Integrate the following with respect to x.
e^x (1+ x) log(xe^x)

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

Using integrals as limit of sums, evaluate \(\int _{ 2 }^{ 4 }{ (2x-1) } dx\)

Using integration find the area of the circle whose center is at the origin and the radius is a units.

The elasticity of demand with respect to price p for a commodity is \(\eta _{ d }=\frac { p+2{ p }^{ 2 } }{ 100-p-{ p }^{ 2 } } \).Find demand function where price is Rs. 5 and the demand is 70.

The Marginal revenue for a commodity is MR=\(\frac { { e }^{ x } }{ 100 } +x+{ x }^{ 2 }\), find the revenue function.

The marginal cost C' (x) and marginal revenue R' (x) are given by C' (x) = 20 +\(\frac{x}{20}\) and R' (x) = 30. The fixed cost is Rs.200. Determine the maximum profit.

A firm has found that the cost C of producing x tons of certain product by the equation x\(\frac { dC }{ dx } =\frac { 3 }{ x } -C\) and C = 2 when x = 1. Find the relationship between C and x.

Solve \(\frac { dy }{ dx } +ycosx+x=2cosx\).

Solve: x²\(\frac { dy }{ dx } \) = y²+2xy given that y = 1, when x = 1

Equipment maintenance and operating costs (are related to the overhaul interval x by the equation \({ x }^{ 2 }\frac { dc }{ dx } -10xc=-10\) with c = c₀ and x = x₀. Find c as a function of x.

Using interpolation, find the value of f(x) when x = 15

x	3	7	11	19
f(x)	42	43	47	60

Using Lagrange’s interpolation formula find a polynomial which passes through the points (0, –12), (1, 0), (3, 6) and (4,12).

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

From the following table, estimate the premium for a policy maturing at the age of 58.

Age (x)	40	45	50	55	60
Premium (y)	114.84	96.16	83.32	74.48	68.48

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
\(f(x)=\left\{\begin{array}{l} Ax,for \ 0≤x10 \\ A(20−x),for \ 10 ≤x< 20 \\ 0,\quad \quad \quad otherwise \end{array}\right.\)
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?

The probability density function of a continuous random variable X is
\(f(x)=\left\{\begin{array}{l} a+b x^{2}, 0 \leq x \leq 1 \\ 0, \text { otherwise } \end{array}\right.\)
where a and b are some constants. Find
(i) a and b if E(X)\(\frac{3}{5}\)
(ii) Var(X).

A discrete random variable X has the following probability distribution.

x	1	2	3	4	5	6	7
P(X)	c	2c	2c	3c	c2	2c2	7c2+c

Find the value of e. Also, find the mean of the distribution.

The probability distribution of a random variable X is

X	1	2	4	2A	3A	5A
P(X)	\(\frac{1}{2}\)	\(\frac{1}{5}\)	\(\frac{3}{25}\)	\(\frac{1}{10}\)	\(\frac{1}{25}\)	\(\frac{1}{25}\)

Calculate
(i) A if E(X) = 2.94
(ii) V(X)

One fifth percent of the the blades produced by a blade manufacturing factory turn out to be defective. The blades are supplied in packets of 10. Use Poisson distribution to calculate the approximate number of packets containing no defective, one defective and two defective blades respectively in a consignment of 1,00,000 packets (e^–0.2 =.9802)

The average daily sale of 550 branch offices was Rs.150 thousand and standard deviation is Rs. 15 thousand. Assuming the distribution to be normal, indicate how many branches have sales between
(i) Rs. 1,25,000 and Rs. 1, 45, 000
(ii) Rs. 1,40,000 and Rs. 1,60,000

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)

Marks in an aptitude test given to 800 students of a school was found to be normally distributed 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90?

A sample of 400 individuals is found to have a mean height of 67.47 inches. Can it be reasonably regarded as a sample from a large population with mean height of 67.39 inches and standard deviation 1.30 inches at 0.05 level of significance?

The average score on a nationally administered aptitude test was 76 and the corresponding standard deviation was 8. In order to evaluate a state’s education system, the scores of 100 of the state’s students were randomly selected. These students had an average score of 72. Test at a significance level of 0.05 if there is a significant difference between the state scores and the national scores.

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.

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 price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

Commodities	Price		Quandity
	2003	2009	2003	2009
Rice	10	13	4	6
Wheat	125	18	7	8
Rent	25	29	5	9
Fuel	11	14	8	10
Miscellaneous	14	17	6	7

The following are the sample means and ranges for 10 samples, each of size 5. Calculate the control limits for the mean chart and range chart and state whether the process is in control or not.

Sample number	1	2	3	4	5	6	7	8	9	10
Mean	5.10	4.98	5.02	4.96	4.96	5.04	4.94	4.92	4.92	4.98
Range	0.3	0.4	0.2	0.4	0.1	0.1	0.8	0.5	0.3	0.5

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

From the data given below, calculate seasonal indices.

Quarter	Year
	1984	1985	1986	1987	1988
I	40	42	41	45	44
II	35	37	35	36	38
III	38	39	38	36	38
IV	40	38	40	41	42

A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below :

How should the tasks be allocated to subordinates so as to minimize the total man-hours?

Determine an initial basic feasible solution to the following transportation problem by using North West Corner rule 

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

Solve the following assignment problem.