No. of Printed Pages : 12 MTE-14 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-14 : MATHEMATICAL MODELLING Time : 2 hours Maximum Marks : 50 Weightage 70% Note : Answer any five questions. All questions carry equal marks. Use of calculator is not allowed. 1. (a) A particle moving in a straight line is 4 subjected to a resistance which produces the retardation kv3 where v is the velocity of the particle at any time t and k is a constant. Show that the velocity v and the time t are given in terms of the distance s by the equations : v 1 ksu and t= u + ks 2 2 (b) Return distribution of two securities are as given in the table below : Event Chance Return Pij = p2j Rii R2j 1 0.33 16 14 2 0.25 12 8 3 0.17 8 5 4 0.25 11 9 MTE-14 1 P.T.O.
Find the correlation coefficient p12 of the two securities. What does the value of 1)12 suggest about the shape of the curve representing the set of portfolios of the two securities? 2. (a) For steady state assuming the wind velocity 5 and diffusion coefficient to be in x - direction only, derive an expression for the dispersion of a contaminant using Fick's law of diffusion. (b) The model for the number of infectives y of 5 a population affected by the spread of a non-fatal disease results in a differential equation dy dt =y (NR -- Y Ry), y (0) = Yo Where N the total population, yo the initial infected population, y the recovery rate and R the contact rate are all constants. Solve for y and show that the epidemic converges exponentially to the stable state. 3. (a) Determine whether the equation s = so + vot 0.5 gt2 is dimensionally compatible if s is the position of the body at time t, So is the position at t = 0, v0 the initial velocity and g is the acceleration due to gravity. Give reasons for your answer. MTE-14 2
(b) Find the range of values of p and q which 3 will render the entry (2, 2) a saddle point for the following game : Player B Player A 2 4 5 10 7 4 p 6 (c) Suppose you are driving a van down a 3 highway. Use dimensional analysis to find the wind force you are experiencing, assuming that the force is affected by the wind density, the speed of the van and its surface area exposed to the wind direction. (d) The average number of new cases of disease 2 is proportional to : (i) (ii) (number of susceptibles)2 number of infectives Set up the model equation for the average number of new cases in an initial population of n individuals. 4. (a) A stone is dropped vertically from a tower 5 of height h. At the same time another stone in 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? MTE-14 3 P.T.O.
(b) The cost of production of a substance per 5 unit is given by the formula C = (12-4q + 1, where q is the material cost. Find the selling price per unit, so that the profit on 100 units will be Rs. 200, if q =15. Also calculate the cost of material per unit so that profit on 100 units can be maximised, if the selling price is Rs. 200. 5. (a) Give one example each from the real world 3 for the following, along with justification, for your example : (i) (ii) A non-linear model A stochastic model (iii) A linear deterministic model (b) Consider the following system of differential equations representing a prey and predator population model : dx 2 dt = x y dy = x+y dt Identify all the real critical points of the system of equations given above. Obtain the type and stability of these critical points. w ftt -1 A
6. (a) Formulate the initial boundary value 5 problem for the temperature T in a cylindrical rod with insulated sides and with flat ends at x = 0 and x = L, the end at x =0 is kept at 60 C and the end at x = L is insulated. At time t = 0, the temperature distribution through out the rod is f(x), 0 < x < L. Assume no internal heat generation. [Hint : Heat flux across an insulated boundary is zero, that is grad T.n =0, where n is the normal to the boundary]. (b) Find the steady state solution of the problem 3 formulated in Q 6 (a) above. (c) Define discrete and continuous models, 2 giving an example of each. 7. (a) The mean arrival rate to a service centre is 3 per hour. The mean service time is found to be 10 minutes for service. Assuming Poisson arrival and exponential service time, find : (i) (ii) the utilisation factor for this service facility. the probability of two units in the system. (iii) the expected number of units in the system. (iv) the expected time in hours that a customer has to spend in the system. MTE-14 5 P.T.O.
(b) Consider arterial blood viscosity 11= 0.025 5 poise. If the length of the artery is 1.5 cm, radius 8 x 10 3cm and P = Pi P2 = 4 x 103 dyne/ crn2 then find the (i) (ii) maximum peak velocity of blood, and the shear stress at the wall. MTE-14 6
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