No. of Printed Pages : 12 MTE-14 BACHELOR'S DEGREE PROGRAMME (BDP)

Size: px

Start display at page:

Download "No. of Printed Pages : 12 MTE-14 BACHELOR'S DEGREE PROGRAMME (BDP)"

Gyles Woods
5 years ago
Views:

1 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 MTE-14 1 P.T.O.

2 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

3 (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 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.

4 (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 (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

5 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.

6 (b) Consider arterial blood viscosity 11= 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

7 chttigrol T1111.1T fit, c51st14-1 : 11 uic1 "VT:at-14 : 11 uic1 fro1 71:1*A1-14 : 50 ch/ : 70 % : 79-frmṮe 3-iT tr ch/ 3-77#77 t/ 1. (a) r -4 'I c1h1-1 chuff ITT -51-WN Z A 4 kv3 v %ft TITM t t k 31-q7 t ra-) s -crq)' v TriTzt t f c-f TNt-*--Turff t : 1 = = + ks` 1 + ksu u chut -5rrifITT dr t (b) tf -51t5P4 dirrichl -44 f-7:11 6 4iql t : w.-11 "FtzliTT 51 c-hrf Pii = P2i Rli R2j MTE-14 7 P.T.O.

8 q111-51thlifdt ch.01q. 11-n p12 11c1 Trf 1 1;1fd 9rEdt 4;1 fa-- FEr r ash 31-r%7 p 12 chi 2. (a) T-2.fft--3WT-2.1T t 41 Bch qpj C fa-m-ur x t, n a feft facer Th-zi-Tr chi r %Tk TkErur % 04,i4) (b) 44-)1 4-Ich %*---4 Tif9-ilf4d 5 y , c-141 chi dy dt 'y (NR Y RY), Y (0) Yo 'H 41 chi 5TRI 61c-11 t,,311 N Yo 4st, lid Trriftsa, y -81-W -T{3 R Tfcrt yt FQ-t fq-ur-7 6 Tfz 34-1-% elf n ḡrarri. 3Trim-ftff -el cu* 3. (a) q f CT.{ frifff s t t = 0, -ER ft-4fff 2 so t vo 51tdiTW fffqi-f-cd enoi s =so + vot 0.5 gt2 fq1:1171 t LI TITirrff t TIT le i atcr4 67It --d77 I cbrul MTE-14 8

9 (b) p q 14T91 r-{ 711d q)- 3 (2, 2) 4,1-.1-) yolilf4s.111 t : Pg B p 6 (c) 3iTcr 7-AArrf -cr-t 74(11 t I 3 HI-I c \T fa-) 31-rcr a7 t t t c4 r?i N-174,,1-1 cf cril fqrr -cj-e8-1-zr -Prra-- ti fqq TkqPi -q-- 71M w1-f-- (d) f*it ')'1 A trl*t '97 Afrr-z 41 A:m 4,961 2 F Ocf : (i) c r.r-nif a,1 4(2q1)2 (ii) 4941 Pi ch 061 ra-ci q). oi-c1 err I n c4irch-q 3-1-1k \31-R- 1 1 frr-zi' 3-ftu t Fri f9-q-qf T-24-rfcm 4. (a) 41 -cq oh..a.td7 14-(q7 ch) 'W-q-Td.: 5 frruzn. A-PrzrZTdZ ciq. 1=17-q7 u A **I 'lif I u 1'1 4-1c cl-q I Ifq,r-4 rch -9-c2.17 MTE-14 -r

10 (b) T."1 3fff TTT-U Tpr 5 C= q2 4q+1,,5-16q HM (1141cf t gri fq-tit +NI t I fitt"*-1 ch1 --1-Tru quo wl'r7 fqq4 ch 100 T-*--r--4-t "TT 200 T. t rl q =15 *1 qr<*q4 witru 200 T. t t 100T*-Tft fir*--o-R i;rrcl.wt4 fm M rcf 4-1 I (1 l I cr -W1f4 I 5. (a) F-H--Irciroo.1R-of-gob 71-q-ff i4tf-4fem 1->- 3 b ock4uo : (i) 311-fuw (ii) f"0 (iii) rftff, f9ilitutf f9-0 I (b) rt.a a 14) a.)kku f9mtzr (41 FA 7 MWR AT ITTIM1 t : \s1-14(s1 I fa-0 1,-1 f9tfera. dx 2 = x y dt dy = x+y dt (i) qr +IR Tr1:11-WTTri f9-t171 Trifi. f4s irr I MTE-14 10

11 (ii) m-fr-a-t f-414 *T-21-rr-4 1:1{ 6. (a) \Ich ec*-1-11chlt 14 c T m1alfq 4: r To-ur Tr-dt t fq:rktt1 t k x=03 tx=l,zrti x =0 TR -ft9 ch.). 60 C c ITT Ttgi lir t *x=ltrft,ru -TuTRWti Trwzrt=o -crilt f(x),0<x<lt clr\31 err \Arid t k61 t1 (-4k9. : t td (grad T.n =0, n 4F41-11 ITT 3Tfird*--4 tl) (b) ( -) 1471f-A- Tf1=1- T 3-111fTddl 31-4T-21T 3 4c1 (c) Trfo 3-1Tr-dd =hl 2 1-1Rwr-Em 7. (a) f- 7:fl" ifful 3TTIPT fff t I 5-1-ruq 74-al -qtrzi 100 -FIT9-z 51-rd t1 ar t I lzir:1 3TPTITff 3117 i.t-11c-11ch1h -4q1 T c1 c c4 11WT rl Forod 4c-r : (i) A-q-r Iqqr MTE P.T.O.

12 ch1 771(,1 4(9-1r, (iv) vorrr.> 1)-ffw gri e -17 f--*-zrr 311 T T mcqipid wrzt I (b) Wit trb Liteoi ti qk q` 1.5 cm, f5i- ET 8 x 10-3cm ct2tt P = - P2 =4 x103 dyne/cm2 t, : (i) Td?f fvrw21-r ta-r 1 Z 31 1'ITIT MTE-14 12

BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-10 : NUMERICAL ANALYSIS

No. of Printed Pages : 11 MTE-10 L. BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-10 : NUMERICAL ANALYSIS Time : 2 hours Maximum Marks : 50 (Weightage