Show that the equations 2x + y = 5,4x + 2y = 10 are consistent and solve them.

Show that the equations 2x - y + z = 7, 3x + y - 5z = 13, x + y + z = 5 are consistent and have a unique solution.

Evaluate \(\int { \frac { { 8 }^{ 1+x }+{ 4 }^{ 1-x } }{ { 2 }^{ x } } } dx\)

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { log\quad x }{ { x }^{ 2 } } } dx\)

The marginal cost function is MC = 300 \({ x }^{ \frac { 2 }{ 5 } }\) and fixed cost is zero. Find out the total cost and average cost functions.

Determine the cost of producing 200 air conditioners if the marginal cost (is per unit) is C' (x) = \(\frac { { x }^{ 2 } }{ 200 } \) + 4

Find the area under the demand curve xy = 1 bounded by the ordinates x = 3, x = 9 and x-axis

Find the order and degree of the following differential equations.
\(\frac{d^{3} y}{d x^{3}}+3\left(\frac{d y}{d x}\right)^{3}+2 \frac{d y}{d x}=0\)

From the following data, estimate the population for the year 1986 graphically.

year	1960	1970	1980	1990	2000
Population (in thousands)	12	15	20	26	33

Estimate the population for the year 1995.

year (x)	1961	1971	1981	1991	2001
population in thousands (y)	46	66	81	93	101

Let X be a discrete random variable with the following p.m.f
\(p(x) = \begin{cases}0.3 & \text { for } x =3 \\ 0.2, & \text { for } x = 5 \\ 0.3, & \text { for } x = 8 \\ 0.2, & \text { for} x = 10 \\ 0, & \text { otherwise } \\ \end{cases}\)
Find and plot the c.d.f. of X.

Explain the terms
(i) probability mass function,
(ii) probability density function and
(iii) probability distribution function.

What are the properties of
(i) discrete random variable and
(ii) continuous random variable?

A fair die is thrown. Find out the expected value of its outcomes.

A random variable X can take all nonnegative integral values and the probabilities that X takes the value r is proportional to aT (0 < ∝ < 1). Find P(X = 0)

The probability that a student get the degree is 0.4 Determine the probability that out of 5 students
(i) one will be graduate
(ii) atleast one will be graduate

If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.

The probability that an event A happens in one treat of an experiment is 0.4. Three independent treats of the experiment are performed. Find the p!probability that the event A happens at least once.

A die is thrown 120 times and getting 1 or 5 is considered a success. Find the mean and variance of the number of successes.

Write a brief note on seasonal variations

The following figures relates to the profits of a commercial concern for 8 years

Year	1986	1987	1988	1989	1990	1991	1992	1993
Profit (Rs.)	15,420	15,470	15,520	21,020	26,500	31,950	35,600	34,900

Find the trend of profits by the method of three yearly moving averages.

Compute Fisher's price index number for the following data.

Commodity	Base Year		Current Year
	Price	Quantity	Price	Quantity
A	10	12	12	15
B	7	15	5	20
C	5	24	9	20
D	16	5	14	5

Determine an initial basic feasible solution of the following transportation problem by north west corner method

A person wants to invest in one of three alternative investment plans: Stock, Bonds and Debentures. It is assumed that the person wishes to invest all of the funds in a plan. The pay-off matrix based on three potential economic conditions is given in the following table:

Alternative	Economic conditions
	High growth(Rs.)	Normal growth(Rs.)	Slow growth (Rs.)s
Stocks	10000	7000	3000
Bonds	8000	6000	1000
Debentures	6000	6000	6000

Determine the best investment plan using each of following criteria i) Maxmin ii) Minimax.

For the given pay-off matrix, choose the best alternative for the given states of nature under
(i) Maximin (ii) Minimax princple

Alternative	States of Nature
	Good	Fair	Bad
A	100	60	+50
B	80	50	+10
C	40	20	+5