Mathematical Modelling MCQs

1

Mathematics • Mathematical Modelling

If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate): If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate):

A. dx/dt = beta x(n - 1 + x) dx/dt = beta x(n - 1 + x) B. dx/dt = -beta x(n + 1 - x) dx/dt = -beta x(n + 1 - x) C. dx/dt = -beta x(n - 1 + x) dx/dt = -beta x(n - 1 + x) D. dx/dt = beta x(n + 1 - x) dx/dt = beta x(n + 1 - x)

सही उत्तर: B Correct Answer: B

With no removal, x + y = n + 1 and susceptibles decrease at rate dx/dt = -beta xy. Substituting y = n + 1 - x gives dx/dt = -beta x(n + 1 - x).

2

Mathematics • Mathematical Modelling

The size of a population of small rodents was recorded as follows: Month: 0, 2, 6, 10; Population: 2, 5, 20, 109. The growth rate as a percentage per month and the population at the end of 12 months are respectively: The size of a population of small rodents was recorded as follows: Month: 0, 2, 6, 10; Population: 2, 5, 20, 109. The growth rate as a percentage per month and the population at the end of 12 months are respectively:

A. 37.2428% and 234 37.2428% and 234 B. 38.2428% and 235 38.2428% and 235 C. 32.2428% and 131 32.2428% and 131 D. 39.2428% and 231 39.2428% and 231

सही उत्तर: D Correct Answer: D

Fitting the exponential population-growth model to the given observations gives the monthly growth rate and projected 12-month population shown in option D.

3

Mathematics • Mathematical Modelling

Which of the following is NOT a purpose of a model from political science? Which of the following is NOT a purpose of a model from political science?

A. Providing insights into a general class of problems. Providing insights into a general class of problems. B. Produce obvious directions for further study. Produce obvious directions for further study. C. Forecast events or outcomes. Forecast events or outcomes. D. Organizing empirical generalizations or known facts. Organizing empirical generalizations or known facts.

सही उत्तर: B Correct Answer: B

Models are used to organize facts, provide insight and help forecast outcomes. Producing obvious directions for further study is marked as not being a purpose in the source paper.

4

Mathematics • Mathematical Modelling

Single-species population model is given by the equation (where the symbols have their usual meaning): Single-species population model is given by the equation (where the symbols have their usual meaning):

A. dN/dt = B(t)D(t) + D(t)N(t) dN/dt = B(t)D(t) + D(t)N(t) B. dN/dt = B(t)N(t) + D(t)N(t) dN/dt = B(t)N(t) + D(t)N(t) C. dN/dt = B(t)/N(t) + N(t)/D(t) dN/dt = B(t)/N(t) + N(t)/D(t) D. dN/dt = B(t)N(t) - D(t)N(t) dN/dt = B(t)N(t) - D(t)N(t)

सही उत्तर: D Correct Answer: D

The rate of population change equals births minus deaths. With per-capita birth and death rates B(t) and D(t), dN/dt = B(t)N(t) - D(t)N(t).

5

Mathematics • Mathematical Modelling

Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees? Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees?

A. 12.08 12.08 B. 15.09 15.09 C. 14.09 14.09 D. 13.09 13.09

सही उत्तर: C Correct Answer: C

By Newton's law of cooling, V - 60 = 40e^(-kt). The first 10 minutes give e^(-10k) = 3/4. Solving for the additional time needed to reach V = 80 gives approximately 14.09 minutes.

6

Mathematics • Mathematical Modelling

Which of the following statements is NOT correct? Which of the following statements is NOT correct?

A. Urban water uses involve transportation. Urban water uses involve transportation. B. Urban water uses involve water supply. Urban water uses involve water supply. C. Urban water uses involve power generation. Urban water uses involve power generation. D. Urban water uses involve snow-removal strategies. Urban water uses involve snow-removal strategies.

सही उत्तर: D Correct Answer: D

Snow-removal strategies are not classified as an urban water use in the context of the question; option D is marked as the incorrect statement.

7

Mathematics • Mathematical Modelling

A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is: A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is:

A. 28 28 B. 32 32 C. 24 24 D. 20 20

सही उत्तर: C Correct Answer: C

Using logistic growth with carrying capacity 500, annual growth at population 300 is 0.2 x 300 x (1 - 300/500) = 24. Hunting 24 deer per year maintains the population at 300.

8

Mathematics • Mathematical Modelling

In a simple epidemic model dS/dt = -beta SI and dI/dt = beta SI, where S(t) and I(t) denote the numbers of susceptible and infected persons respectively, S(t) is given by: In a simple epidemic model dS/dt = -beta SI and dI/dt = beta SI, where S(t) and I(t) denote the numbers of susceptible and infected persons respectively, S(t) is given by:

A. S(t) = n(n+1)/(n + e^((n+1)t)) S(t) = n(n+1)/(n + e^((n+1)t)) B. S(t) = n(n+1)/(n + e^((n+1)beta t)) S(t) = n(n+1)/(n + e^((n+1)beta t)) C. S(t) = (n+1)/(n e^(n beta t)) S(t) = (n+1)/(n e^(n beta t)) D. S(t) = n/(n + e^(n beta t)) S(t) = n/(n + e^(n beta t))

सही उत्तर: B Correct Answer: B

Using S+I=n+1 and separating dS/dt=-beta S(n+1-S), the solution with the stated standard initial conditions is the expression in option B.

9

Mathematics • Mathematical Modelling

A population of shrimp has unlimited resources and its population size grows at a rate of 10% per month. The initial population size is 1 million. The number of shrimp that may be caught each month without ultimately exhausting their population is: A population of shrimp has unlimited resources and its population size grows at a rate of 10% per month. The initial population size is 1 million. The number of shrimp that may be caught each month without ultimately exhausting their population is:

A. 10,000 10,000 B. 1,01,000 1,01,000 C. 1,00,000 1,00,000 D. 10,00,000 10,00,000

सही उत्तर: C Correct Answer: C

Ten percent of the initial population of 1,000,000 is 100,000. A monthly harvest equal to this growth can be sustained under the stated unlimited-resource model.

Mathematical Modelling MCQs

Mathematical Modelling MCQs

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