Term Test 1 Questions PDF

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Applied Math 1201B – Practice Term TestMultiple Choice Question BookletSunday, February 2nd, 2020 10:00am – 12:00pmInstructions: (these will be displayed on your test!) Correctly write your full name and student number on the space below. The signature line is to indicate that you are in agreement w...

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Applied Math 1201B – Practice Term Test #1 Multiple Choice Question Booklet

Sunday, February 2nd, 2020

10:00am – 12:00pm

Instructions: (these will be displayed on your test!) 1. Correctly write your full name and student number on the space below. The signature line is to indicate that you are in agreement with all instructions listed on this sheet. 2. This booklet contains 15 multiple choice questions, each worth one mark. This booklet will not be marked. Enter the best answer on the provided Multiple Choice answer sheet (the last page of the answer booklet). 3. You are more than welcome to use this booklet for rough work, or any other calculations that may be needed. However, as stated in the last instruction, this booklet will not be marked. 4. This is a closed book examination. All notes, phones or other electronic devices are prohibited. 5. For all other instructions, please read the front page of the answer booklet. 6. Marks total to 25 (short answer and multiple choice combined).

Student’s Name (Print): Student Number: By signing below, I agree to all instructions listed above. Student’s Signature:

Note Regarding This Practice Test: This practice test consists of 30 multiple choice questions, exactly double the amount you will have on your actual Term Test #1. Some questions may appear on the actual test, some questions may be similar to those on the actual test, and some are just meant as practice. Recall that one multiple choice question will also be based around the topic in Assignment #1, so make sure you study that as well! Your actual test will also consist of 10 marks for short answer questions. In lieu of posting full questions, I have listed several really good homework problems and in-lesson examples. If you can do all said problems, you will be totally fine to do the actual short answer questions. Your best bet is to attempt all problems, before looking at the correct answers (also posted on OWL). During the completion of these questions, time yourself, and see how long it takes you to work through all thirty questions. This amount of time divided in half should be a good indicator for how long it will take you to complete the actual test’s multiple choice section, and give you an idea of how much time you will have left to complete the short answer. Remember that you will have two hours to complete the entire test. Best of luck in studying! Please remember to come to office hours and Help Centre if you’re struggling with any of the questions on this test! -Tyler

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Practice Term Test #1 Multiple Choice

AM1201B

1. The eigenvector corresponding to the eigenvalue 3 + 2i for the matrix

 6 −13  1

  3 + 2i (A) 1

(B)

  3 − 2i 1

(C)



1 3 + 2i



(D)

0

is

  3 + 2i 2

For Questions 2 and 3:   3 −2 Consider the matrix A = . 1 0 2. If A = P DP −1 , what is the P matrix generated from A if λ1 = 1 and λ2 = 2?   −1 2 (A) 1 −1   −1 1 (C) 1 2

  1 −2 (B) 1 1   1 2 (D) 1 1

  −1 , what is the solution to the recursion equation, ~nt+1 = A~nt , in 3. Assuming that ~n0 = 1 terms of the eigenvalues and eigenvectors?     −1 t 1 (A) ~nt = 3 1 −2 2t 2 −1     2 t 1 t 1 −2 2 (C) ~nt = 3 1 1

    1 t 2 (B) ~nt = −2 1 − 2t 1 −1     1 t 1 (D) ~nt = −2 1 +3 2t −2 −1

4. Which of the following vectors is the longest? (A) [−1, 8, 3]

(B) [3, 7, 7]

(C) [−2, 6, −5]

(D) [2, 7, 0]

5. Suppose that a certain species of fish reproduces only during the third year of its life, then dies. We wish to construct a system of difference equations of the form ~nt+1 = Q~nt to describe the population dynamics of this species. We base our model on the following observations: • 15% of newborns survive the first year • 70% of fish in their second year of life reach sexual maturity • Each sexually mature individual produces an average of 5 offspring Which matrix  0 0.15 (A) 0 0 5 0

Q correctly captures these dynamics?      0 0.15 0 0 0 5 0 0.7  (B)  0 0.7 0 (C) 0.15 0 0  0 0 0 5 0 0 0.7

 0 0 5 (D) 0.15 0 0  0 0.7 0 

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Practice Term Test #1 Multiple Choice

AM1201B

6. Consider the following equations in matrix notation describing a disease model with susceptible S, infected I and recovered R individuals.     St+1 0.75 0 1.0 St  It+1  =  0.25 0.75 0   It  Rt+1 0 0.25 0 Rt 

Which of the following compartment diagrams corresponds to these equations?

0.25

0.75

0.25

1.0

0.25

0.75

0.75

(A) 0.25

0.25

0.75

1.0

(B) 0.75

0.25

0.75

0.25

0.75

1.0

(C)

7. Which of  1 (A) 2

0.75

0.25

1.0

(D)

the following matrices is not primitive?      0.5 0.9 0 2 1 (B) (C) 1 0.4 0 1 1

  1.2 2 (D) 0 0.5

8. If A is a 2 × 2 matrix, B is a 2 × 3 matrix and C is a 3 × 2 matrix, which of the following algebraic statements does not make mathematical sense? (A) ABC

(B) CA

(C) AB

(D) BA

9. Which of the following would be an example of empirical modelling? (A) Taking a survey of the class and fitting a curve to the data. (B) Using a biologist’s feedback on bacteria-phage interactions to develop a model, studying the evolution of both. (C) Theoretically determining a model to estimate the rate of growth of a tumour. (D) Darth Vader walking the runway on the Death Star’s Next Top Model. 10. Which of the following would best describe the use of cobweb analysis in practice? (A) (B) (C) (D)

Determining the fixed points of a discrete-time model. Determining the stability of the fixed points of a discrete-time model. Making fun pictures to show your friends that math is cool. Constructing discrete-time models.

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Practice Term Test #1 Multiple Choice

AM1201B

For Questions 11 and 12: Consider the matrix model describing how the vector ~xt representing the antigenic state of the chicken pox virus changes from one season to the next,   2 1 ~xt+1 = ~x 0 0.70 t   6 Assume the vector for the current season is given by . 0.5 11. What does the model predict will be the vector in the next       25.35 11.5 12.5 (B) (C) (D) (A) 0.35 0.245 0.5

season?   9.5 1.2

12. What does the model predict was the vector in the previous season?         6 2.6 3.5 3.7 (A) (B) (C) (D) 0.5 0.71 0.2 1.0 13. Which of the following real-world scenarios would best be modelled using a linear growth difference equation, rather than Malthusian growth? (A) (B) (C) (D)

A fishery with capped harvesting in each generation. A fishery with no harvesting that is left alone over each generation. A garden where the owner plants a set number of new flowers each generation. Both (A) and (C).

14. Consider the following matrix:  1 1 i 0 0 i 0 i 0  What is its characteristic polynomial?

(A) λ3 + λ2 + λ + 1 = 0 (C) λ3 − λ2 − λ + 1 = 0

(B) λ3 − λ2 + λ − 1 = 0 (D) λ3 − λ2 − λ − 1 = 0

15. Which of the following is not a conclusion of the Perron-Frobenius Theorem? (A) The dominant eigenvalue will specify the long-term growth rate of the population. (B) The matrix in question will be primitive. (C) The dominant eigenvalue will be real and positive. (D) The ratio of the classes will be given by the eigenvector corresponding to the dominant eigenvalue. 16. For what value of α do the matrices A and B (shown below) commute (ie. what value of α ensures that AB = BA)?     2 α 2 3 B= A= 1 4 1 3 (A) 1

(B) 2

(C) 3

(D) Matrices never commute.

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Practice Term Test #1 Multiple Choice

AM1201B

17. The body eliminates 25% of the amount of pain reliever drug present each hour. Assuming there is an initial dose of 200 mg given to a patient, how much drug do you expect to be in the body after 3 hours? (A) 84 mg

(B) 391 mg

(C) 3 mg

(D) 200 mg

18. Consider an n × n matrix A (where n ≥ 2). If AT = A−1 , what is a possible value for the determinant of A? (Hint: Check to see how the determinant of A relates to the determinant of its transpose and inverse!) (A) 1

(B) n

(C) 0

(D) Cannot be determined.

19. A population of insects is divided into two stage categories. The population can be modelled by the system of difference equations (shown in matrix form),   1 0.5 ~nt ~nt+1 = 4 0 It was found that the long-term growth rate of the population is 2. Suppose in the long-term there are 1000 total individuals in the population, approximately how many of these insects would you expect to be in the second stage of their life cycle? (A) 667

(B) 222

(C) 800

(D) 200

For Questions 20 and 21: Consider that we have developed a model describing a population of llamas over a period of time. The function that we have determined fits the real-world best is a power function given by, y(t) = y0 (1 − K)t+τ where y is the number of llamas at time t. 20. Using empirical data, we can determine values for the parameters in the model. How many parameters are present in our model of the llama population? (A) 0

(B) 1

(C) 2

(D) 3

21. To ensure dimensional consistency, what must be the units of K in the above model? 1 1 (D) (A) # of llamas (B) Dimensionless (C) time # of llamas 22. Suppose A=



 2 0 x 2

B=

and A + B = AB. Then (x, y) could be (A) (2, 2)

(B) (0, −1)

(C) (1, 0)

(D) (2, −2)



 2 0 y 2

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Practice Term Test #1 Multiple Choice

AM1201B

For Questions 23 and 24: A researcher has constructed a discrete-time model for pest control. Their model is given by the difference equation, 4Nn2 Nn+1 = (1 + Nn )(2/3 + Nn ) 23. Which of the following is not a fixed point for this model? 2 1 (D) N ∗ = 2 (C) N ∗ = (A) N ∗ = 0 (B) N ∗ = 3 3 24. Below is the (incomplete) cobweb diagram for this model (with a slightly different parameter set, as to not give away the answer to the previous question). If the initial condition is given by N0 = 1, we can use cobweb analysis to determine which fixed point is locally stable?

(A) I

(B) II

(C) III

(D) Cannot determine stability.

25. Suppose A  has eigenvalues λ1 = −1 and λ2 = 5 with corresponding eigenvectors  a matrix  that 1 −2 and ~v2 = , respectively. What is the value of A5 ? ~v1 = 1 1       2 3125 1041 2084 1 4 (B) (C) (D) Not enough information given. (A) 2 3 −1 3125 1042 2083     1 1 1 0 26. Given that A = and its diagonal matrix of eigenvalues is given by D = , which 0 2 0 2 of the following is the corresponding matrix of eigenvectors (P )?         0 1 1 1 1 1 1 1 (A) (B) (C) (D) 1 1 1 0 1 2 0 1     1 i 27. Vector ~v = is an eigenvector of matrix A, with eigenvalue 2. If w ~ = , then Aw ~ = −1 i       1 i 2i 2 (A) (B) (C) (D) all of the previous −2 2i 2 −1

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Practice Term Test #1 Multiple Choice

28. Consider the following matrices,   1 0 A= 1 1

  2 −2 B= 1 3

AM1201B



 0 1 C= −1 0

Which matrix X is the solution to the equation AX − C = B ?  2 (A) X = 2  1 (C) X = 0

 −1 2  0 1

 3 −1 (B) X = −3 3   2 −1 (D) X = −2 4 

29. Consider a lake fish population whose yearly birth rate is 1.4 and whose yearly death rate is 1.2. What is the growth rate of this population, in the absence of harvesting? (A) 0.2

(B) 1.4

(C) 1.2

(D) 1

 1 i and I is the 2 × 2 identity matrix. If AB = I, then B = 30. Suppose A = −i 2         −2 i 1 −i 2 −i −1 i (A) (B) (C) (D) −i −1 i 2 i 1 −i −2 

List of Problems to Practice for the Short Answer: You should do all the homework problems, but the concepts tested in the short answer portion of the test will more than likey (...cough, cough...) come from somewhere in this list. 1. Both examples in Section 1.1 2. Example regarding discrete logistic model in Section 1.2 (including the cobweb diagram) 3. Last example in Section 1.3 4. Last example in Section 1.4 5. Example regarding the dragonfly population in Section 1.5 6. Homework Exercises: 4, 6, 10, 20, 21, 29, 35, and 36...