Enlist the different types of lymphocytes. Mention their function, site of differentiation, and the type of immunity imparted by them.

Detail the molecular basis of mutations.

In a randomly mating laboratory population of Drosophila, 4 per cent of the flies have black bodies (black is autosomal recessive, b) and 96 per cent have brown bodies (normal colour, B). If this population is assumed to be in Hardy-Weinberg equilibrium, what are the allelic frequencies of B and b, and the genotypic frequencies of BB and Bb?

Does the operon concept apply to eukaryotes? Describe a suggested mechanism for gene regulation in eukaryotes.

Detail the Watson and Crick model of DNA structure. How did it fit in the data provided by Chargaff?

Describe with appropriate examples the role of environment in the determination of sex.

State the type of modeling you will use for the following problems, giving reasons for your answers. Also state four-essentials for each of these problems.

i) The economic viability of an insurance company depends critically on its ability to assess risks and decide on the premium charged to cover risks. If the premium are low, then payouts can exceed revenue collected and the company can go bankrupt. On the other hand, if they are high, the number of customers will go down, thus affecting profitability. To help the insurance company decide the premium it should charge for different risks to ensure economic

State the type of modeling you will use for the following problems, giving reasons for your answers. Also state four-essentials for each of these problems.

i) The economic viability of an insurance company depends critically on its ability to assess risks and decide on the premium charged to cover risks. If the premium are low, then payouts can exceed revenue collected and the company can go bankrupt. On the other hand, if they are high, the number of customers will go down, thus affecting profitability. To help the insurance company decide the premium it should charge for different risks to ensure economic

ii) Companies located on the banks of a river and producing chemicals dispose their waste by indiscriminately dumping it into the river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeded certain specified concentration limits. To find a policy for discharging the waste so as to ensure that the concentration level never exceed the specified limits.

Television sets for repair arrive at random at an average rate of 4 per day to a single repairman who takes an average of  hours to carry out each repair. It being assumed that the repair times have an exponential distribution. What is the average number of television sets in the workshop? What is the probability that an arriving set will find at least 3 sets in front of it? The repairman works for eight hours a day.

Let P = (w₁, w₂) be a portfolio of two securities. Find the value of w₁ and w₂ in the following situations

i) = -1 and P is risk-free.

ii) σ₁ = σ₂ and variance P is minimum.

Variance on P is minimum and = -0.5, σ₁ = 2 and σ₂ = 3.

The sales of a company from 1993-1998 are given below

Fit a linear curve using the least squares method. Hence find out the company's sales in 1999.

There are N identical firms producing a particular commodity. The cost function for each firm producing q units is q³ + 2q² + 4q + 6 units of money. Obtain the supply function for each firm. The demand function is   . ( Find ) the equilibrium price.

Let the utility function of a consumer be and that the consumers income for the period is Rs.15000. Obtain the quantities required by the consumer so that his utility function is maximized by consuming this combination.

Suppose that the previous forecast was 2083 and the actual value of the variable of interest for the last period was 1975 and the oldest value of interest was 1945. Using the moving average technique based upon the most recent four observations find new forecast for the next period.

Consider the pay-off table for two players as given below

Apply dominance to find the value of the game and the optimal mixed strategy for each player.

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.