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IGNOU MSCMACS MMT 9 2024 Solution

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Last Date of Submission of IGNOU MMT-09 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MMT 9 2024 Solution
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 9
Subject Name	Mathematical Modeling
Year	2024
Session	-
Language	English Medium
Assignment Code	MMT-09/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 09 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-09 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMT 9/2024

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Questions Included in this Help Book

Ques 1.

A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 5 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.

Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?

Portfolio	Expected return	Standard deviation
W	9%	21%
X	5%	7%
Y	15%	36%
Z	12%	15%

Ques 2.

Let G )t( be the amount of the glucose in the bloodstream of a patient at time t. Assume that the glucose is infused into the bloodstream at a constant rate of k /g min . At the same time, the glucose is converted and removed from the bloodstream at a rate proportional to the amount of the glucose present. If at t = ,0 G = G )0( then

i) formulate the model.

ii) find )t(g at any time t.

iii) discuss the long term behavior of the model.

Ques 3.

Tumour is developing from the organ of a human body with concentration 9 2.3 ×10 with growth and decay control parameters 9.2 and 2.7 respectively. In how many days the size of the tumor will be twice?

Ques 4.

Return distributions of the two securities are given below:

X	Y	p_xj= p_yj= p _j
0.16	0.14	0.33
0.12	0.08	0.25
0.08	0.05	0.17
0.11	0.09	0.25

Find which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

Ques 5.

Let P (w , w ) = 1 2 be a portfolio of two securities X and Y. Find the values of w1 and w2 in the following situations: i) ρxy = −1 and P is risk free. ii) σx = σy and variance P is minimum.

Variance P is minimum and ρxy = − ,5.0 σx = 2 and σy = 3

Ques 6.

Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:

Customer Orders (in thousands)	Add-delete items (in thousands)	CPU time (in hours)
123.5	2.108	141.5
146.1	9.213	168.9
133.9	1.905	154.8
128.5	0.815	146.5
151.5	1.061	172.8
136.2	8.603	160.1
92.0	1.125	108.5

i) Find a linear regression equation that best fit the data.

ii) Estimate the error variance for the regression model obtained in i) above.

Ques 7.

The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:

Husband’s age (years)	Wife’s age (years)	Husband’s age (years)	Wife’s age (years)
30	27	51	50
29	20	48	46
36	34	37	36
72	67	50	42
37	35	51	46
36	37	36	35

i) Draw a scatter plot of the data

ii) Write two important characteristics of the data that emerge from the scatter plot.

iii) Fit a linear regression model to the data and interpret the result in terms of the comparative change in the age of husband and wife.

iv) Calculate the standard error of regression and the coefficient of determination for the data.

Ques 8.

Consider a discrete model given by

$N_{t+1}=\frac{rN_{t}}{1+bN^{2}_{t-1}}=f(N_{t})r> 1.$

Investigate the linear stability about the positive steady state $N^{*}$ by setting

$N_{t=}N^{*}+n_{t.}$ Show that $n_{t}$ satisfies the equation

$n_{t+1-n_{t+}}2(r-1)r^{-}n_{t-1}=0.$

Hence show that r = 2 is a bifurcation value and that as r → 2 the steady state bifurcates to a periodic solution of period 6.

Ques 9.

a) The population dynamics of a species is governed by the discrete model $x_{a+1=x_{a}}exp\left [ r\left ( 1-\frac{x_{n}}{K} \right ) \right ],$ where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population fo $r=2+\varepsilon ,$ where $0< \varepsilon < < 1.$ Since a species becomes extinct if $x_{n\leq 1} n> 1,$ $x_{n\leq 1}$ for any $n> 1,$ show using iterations, that irrespective of the size of $r > 1$ the species could become extinct if the carrying capacity $k <$ r exp $\left [ 1+e^{r-1} -2r\right ].$

b) Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.

$\frac{dn_{t}}{dt}=r_{0N_{1}}-r_{1C_{O}}N_{1}-bN_{1}N_{2}$

$\frac{dn_{2}}{dt}=d_{0}N_{2}-d_{1}V_{0}N_{2+}\beta _{0}bN_{1}N_{2}$

$\frac{dC_{0}}{dt}=k_{1}P-g_{1}C_{0}-m_{1}C_{0}$ $\frac{dV_{0}}{dt}=k_{2}P-g_{2}V_{0}-m_{2}V_{0}$

$\frac{dP}{dt}=Q-hP-kP(N_{1}+N_{2})+gC_{0}N_{1}+lV_{0}N_{2 .}$

Where all the variables and constants are same as defined in the system (32)-(35) except for the following

Q = constant input rate h = decay rate )t

(P = environmental toxicant concentration

k = ingestion rate of toxicant by the populations

$g,l=$ return rate of toxicant in the environment after the death of the populations, assuming that toxicant is non-degradable

$Q> 0,h,k,g,l$ are positive constants.

Do the stability analysis of the following competing species system of equations with diffusion and advection

$\frac{\partial N_{1}}{\partial t}=a_{1}N_{1}-b_{1}N_{1}N_{1}+D_{1}\frac{\partial ^{2}N_{1}}{\partial X^{2}}-V_{1}\frac{\partial N_{1}}{\partial X}$

$\frac{\partial N_{2}}{\partial t}=-d_{1}N_{2}+C_{1}N_{1}N_{2}+D_{2}\frac{\partial ^{2}N_{2}}{\partial X}, 0\leq X\leq L$

where $V_{1}$ and $V_{2}$ are advection velocities in x direction of the two populations with densities $N_{1}$ and $N_{2}$ respectively. $a_{1}$ is the growth rate, $b_{1}$ is the predation rate $d_{1}$ is the death rate $c_{1}$ $C_{1}$ is the conversion rate. $D_{1}$ and $D_{2}$ are diffusion coefficients. The initial and boundary conditions are:

$N_i$ $N_i(X,0))=f_i(X)> 0,0\leq X\leq L,i=1,2.$

$N_i=\bar{N_{1}}$ at $X=0L\forall t,i=2.$

where $N_i$ are the equilibrium solutions of the given system of equations.

Interpret the solution obtained and also write the limitations of the model.

Maximize $Z=3X+_{1+X_{2}}+3X_{3,}$ subject to the constraints

$-X_{1}+2X_{2}+X_{3}\leq 4,4X_{2}-3X_{2}+2X_{3}\leq 3$ and $(X_{1,}X_{2,}X_{3})\geq 0$ and are integers.

b) Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of interarrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?

c) A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

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