Geometrically interpret the growth rate corresponding to the population growth model

where r₁ , k, k are constants, k being the carrying capacity of the population x(t) . Hence find when the growth is maximum.

a) Consider the cubic total cost function

C = 0.004q³-0.8q³ +10q+5.

Assume that the price of q is 13 per unit. Find the output which yields maximum profit.

Return distribution of the two securities are as given in the table below:

Find the correlation coefficient of the two securities. What does this value of suggest about the shape of the curve representing the set of portfolios of the two securities?

A company manufactures an item at a rate of 12 items per day (following an exponential distribution). The service time distribution is also exponential with an average of 60 minutes

i) Calculate the utilization factor.

ii) Find the average number of items in the queue.

iii) What is the probability that the queue size is greater than or equal to 5?

Suppose that a given population can be divided into parts, those who have a given disease and can infect others and those who do not have it, but are susceptibles. Let x be the proportion of susceptible individuals and y the proportion of infectious individuals, then x + y = 1. Assume that the disease spreads by contact between sick and well members of the population and the rate of spread is proportional to the number of such contacts. If yo is the initial proportion of infectious individuals then 

i) Formulate a mathematical model for the given problem and write a differential equation governing it.

ii) Find the equilibrium points of the equation obtained in (i).

iii) Solve the given problem. What happens to the spread of the disease as t→∞?

A lake is initially stocked with 100 fishes and 1000 zooplanktons on which the fishes prey. There is ample food for the zooplanktons. Because fishes prey on zooplanktons the population of fishes will increase at a rate proportional to the number of encounters between the species. Fishes will also die at a rate

proportional to the fish population. Also fishing is permitted at the rate of  of  the existing population at any time. Zooplanktons multiply at a rate proportional to the population and die off at a rate proportional to the number of encounters between the two species. Model this situation.

A spherical cell of radius a is taking in a nutrient from its surroundings and metabolizing it. Assume that the concentration of nutrient in the cell at r = a is zero, for t > 0, and the initial concentration is C_o for r

Find the concentration C(r,t) at any instant of time. Also write the steady-state solution.

Discuss the static stability and dynamic stability for the following demand and supply functions

A patient is given a dose Q mg/ml of a drug at regular interval of time t. The concentration C, of the drug in the blood has been shown experimentally to obey the law 

i) if the first dose is administered at t = 0 hr. then find the concentration after hr. have elapsed.

ii) assuming an instantaneous rise in concentration whenever the drug is administered, find the concentration after the second dose and T hr. have elapsed again.

iii) show that the limiting value R of the concentration for doses of Qmg/ml repeated at interval T hr. is given by the formula

A cup of coffee at 95°C is kept in a room at 25°C . Two minutes later, the temperature of the coffee is 70°C . When will the temperature of the coffee reach 50°C ?

Consider the problem of heat conduction in a one dimensional slab (0

where u(x, 1) is the temperature.

Using the separation of variable method seek the solution ) u(x,t) = T t) X (x) and find

i) equation for T (t), X (x) and corresponding conditions on X (x) to solve these equations

ii) the eigenvalues and the corresponding eigenfunctions of the bvp obtained in i) above for ) X (x)

iii) the solution u(x,t) .

A stone is dropped vertically from a tower of height h. At the same time, another stone is thrown vertically upwards from the base of the tower with a velocity u. What is the minimum value of u so that the two stones will meet each other mid- air?

A raindrop starts falling from the clouds at a considerable height above the surface of the earth. During the fall, the raindrop experiences retardation due to air resistance, which is directly proportional to the instantaneous speed v(t) of the drop.

i) Write the model equations

ii) Is this system static or dynamic? Why?

iii) Obtain an expression for the speed v(t).

iv) Discuss the behaviour of v(t) as the time t changes.

Using dimensional analysis, show that the planets obey Kepler's third law.

Find the Boolean expression in the DNF form for the function defined in tabular form below:

Is there a 4-regular graph on 7 vertices? Justify your answer.

Solve the recurrence relation:

using generating function technique. Also find a₅ using your answer.

How many numbers from 0 to 759 are not divisible by either 3 or 7?

Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph:

A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green?

Show that if 7 colours are used to paint 50 bicycles and each bicycle is coloured with a single colour, at least 8 bicycles will have the same colour.