2020 Semester Two (November-December 2020) Examination Period Faculty of Science EXAM CODES: MTH1030 TITLE OF PAPER: Techniques for Modelling EXAM DURATION: 2 hours 10 mins Rules During
an exam, you must not have in your possession any item/material that
has not been authorised for your exam. This includes books, notes,
paper, electronic device/s, mobile phone, smart watch/device,
calculator, pencil case, or writing on any part of your body. Any
authorised items are listed above. Items/materials on your desk, chair,
in your clothing or otherwise on your person will be deemed to be in your possession. You must not retain, copy, memorise or note down any exam content for personal use or to share with any other person by any means following your exam. You must comply with any instructions given to you by an exam supervisor. As a student, and under Monash University’s Student Academic Integrity procedure, you must undertake your in-semester tasks, and end-of-semester tasks, including exams, with honesty and integrity. In exams, you must not allow anyone else to do work for you and you must not do any work for others. You must not contact, or attempt to contact, another person in an attempt to gain unfair advantage during your exam session. Assessors may take reasonable steps to check that your work displays the expected standards of academic integrity. Failure
to comply with the above instructions, or attempting to cheat or
cheating in an exam may constitute a breach of instructions under regulation 23 of the Monash University (Academic Board) Regulations or may constitute an act of academic misconduct under Part 7 of the Monash University (Council) Regulations. Authorised Materials CALCULATORS YES NO DICTIONARIES YES NO NOTES YES NO PERMITTED ITEM YES NO if yes, items permitted are: Page 1 of 21 Instructions ALL answers in this exam that are not multiple choice are integers or fractions. INTEGERS: You must enter integers in their simplest form. For example, enter the number twenty-three as 23. Extra
spaces before, in the middle or following the integer, or inputs such
as 22.999… or 23.000 or 24-1, etc. will ALL result in an answer to be counted as incorrect. FRACTIONS: Similarly, you must enter non-integer fractions in lowest terms and in their simplest form possible. So, enter three halves as 3/2 Extra
spaces before, in the middle or following a fraction, or input as 6/4,
1.5, etc. will ALL result in an answer to be counted as incorrect. NEGATIVE
SIGN: If you want to enter a negative integer or fraction you must
enter the negative sign immediately before the number, e.g. -23 or -3/2. Again, extra spaces in between the minus sign and the number or anywhere else will result in an answer to be counted as incorrect. ATTEMPT ALL QUESTIONS: There are no negative marks for incorrect answers. Page 2 of 21 Instructions Information ALL answers in this exam that are not multiple choice are integers or fractions: You MUST enter integers in their simplest form. For example, enter the number twenty-three as 23. Extra spaces before, in the middle or following the integer, or inputs such as 22.999… or 23.000 or 24-1, etc. will ALL result in an answer to be counted as incorrect. FRACTIONS: Similarly, you must enter non-integer fractions in lowest terms and in their simplest form possible. So, enter three halves as 3/2 Extra spaces before, in the middle or following a fraction, or input as 6/4, 1.5, etc. will ALL result in an answer to be counted as incorrect. NEGATIVE SIGN: If you want to enter a negative integer or fraction you must enter the negative sign immediately before the number, e.g. -23 or -3/2. Again, extra spaces in between the minus sign and the number or anywhere else will result in an answer to be counted as incorrect. ATTEMPT ALL QUESTIONS: There are no negative marks for incorrect answers. Page 3 of 21 Integration by parts and substitution Question 1 Here
are a couple of integrals that can be dealt with using integration by
parts (IBP) or integration by substitution (IBS) or a combination of both. Among the given choices, make the best possible choice resulting in the least number of calculations a) This integral can be dealt with using
an IBP with f(t)=t and g'(t)=e^t an IBP with f(t)=e^t and
g'(t)=t an IBP with f(t)=te^t and g'(t)=1 an IBS with u=t an IBS with u=e^t an IBS with u=te^t b) This integral can be dealt with using an IBP with f(x)=1/x and g'(x)=(ln(x))^2 an IBP with f(x)=(ln(x))^2/x and g'(x)=1 an IBS with u=ln(x) an IBS with u=(ln(x))^2/x c) This integral can be dealt with using
an IBP with f(x)=x^3 and g'(x)=e^(x^2) followed by an IBS an
IBP with f(x)=e^(x^2) and g'(x)=x^3 followed by another IBP an IBP with f(x)=x^2e^(x^2) and g'(x)=x an IBS with u=x^2 followed by an IBP an IBS with u=x^3 followed by an IBP d) This integral can be dealt with using an IBS only an IBP only a combination of IBS and IBP e) The Integration by Parts formula results from integrating the chain rule the integrating factor the product rule the partial fraction approach none of the above f) The formula for an Integration by Substitution results from integrating 12 Marks Page 4 of 21 the chain rule the integrating factor the product rule the partial fraction approach none of the above Page 5 of 21 Sequences and series Question 2 a) For each of the following series, determine whether they are absolutely convergent, conditionally convergent, or divergent. (i) absolutely convergent conditionally convergent divergent (ii) absolutely convergent conditionally convergent divergent (iii) absolutely convergent conditionally convergent divergent (iv) absolutely convergent conditionally convergent divergent b) For each of the following series, determine whether they are absolutely convergent, conditionally convergent, or divergent. (i) absolutely convergent conditionally convergent d
ivergent (ii) absolutely convergent conditionally convergent divergent (iii) 15 Marks Page 6 of 21 absolutely convergent conditionally convergent divergent (iv) absolutely convergent conditionally convergent divergent c)
For each of the following series, determine whether they are an
alternating series, a p-series, a harmonic series, a geometric series, or none of the above. (i) alternating series p-series harmonic series geometric series none of the above (ii) alternating series p-series harmonic series geometric series none of the above (iii) alternating series p-series harmonic series geometric series none of the above d) Choose the correct value for the sum of the following series. (i) 3/4 5/4 4/5 2 4/3 (ii) 1/3 12/5 3 5/3 4/3 Page 7 of 21 Power series Question 3 Let a) What is the radius of convergence R of the power series Answer: R = b) What is the interval of convergence of the power series Answer: (-R,R) [-R,R) (-R,R] [-R,R] None of the above c) Suppose a function f admits the following Maclaurin series near 0. What is f'(1) f'(1) = d) Suppose f is a function with f(0) = 0, and f’ admits the following Maclaurin series near 0. What is f(1) f(1) = 11 Marks Page 8 of 21 Differential equations Question 4 a) One of the following differential equations is separable. Determine which it is. (i) (ii) (iii) (iv) b) Suppose we wish to solve the following differential equation using the method of the integrating factor. What should we choose as our integrating factor 1/x^2 1/x e^x ln(x) x c) The differential equation has a general solution of the form where a, b, c, and d are constants. Let us choose a and b so that a>0 and b<0. What are a and b Answers: a = b = 12 Marks Page 9 of 21 Vectors Question 5 Fill in the correct numbers or choose the correct answer. Let u=(1,0,1), v=(0,2,0) and w=(2,1,-2) be vectors in R . Then: a) (2v) · (3u) = b) The x-coordinate of (2v) × (3u) is c) u × (-2v) · w = d) The angle between v and w is 0 larger than pi/2 less than pi/2 pi/2 pi/4 pi/6 pi/3 e) The square of the area of the parallelogram spanned by u and v is f) The volume of the parallelepiped spanned by u, v and w is g) u, v, w (in this order) form a left-handed system a right-handed system neither a left-handed nor a right-handed system 8 Marks 3 Page 10 of 21 Points, lines, and planes Question 6 a) A plane is given by the following equation 2x-y+3z=2. This plane has a normal vector with x-coordinate -4. What is its z- coordinate Answer: b) There is a point with x-coordinate 1 and z-coordinate 1 on this plane. What is its y-coordinate Answer: c) Find the y-coordinate of the point of intersection of the line with equation (1,12,3) + t(1,2,1) and this plane. Answer: d) L is a line with equation (1,3,1)+t(2,1,d). If L is contained in the plane, what is d Answer: e) The closest point on the line L to the point (2,15,3) is given by the parameter t. Then t = Answer: f) What is the distance between the two lines with equations (1,0,-1)+t(1,2,0) and (-1,-1,4)+s(3,-2,0) Answer: 10 Marks Page 11 of 21 Reduced row echelon form Question 7 a) The augmented matrix of a linear system is given by The number of equations is The number of variables is b) Here are reduced row echelon forms of three linear systems. How many solutions do these systems have Answer: 0 solutions exactly 1 solution exactly 2 solutions infinitely many solutions this question makes no sense Answer: 0 solutions exactly 1 solution exactly 2 solutions infinitely many solutions this question makes no sense Answer: 0 solutions exactly 1 solution exactly 2 solutions infinitely many solutions this question makes no sense c) Let's say you are dealing with a linear system with 1000 unknowns and 1000 equations with randomly generated coefficients. What is most likely Answer: Most likely this system has 0 solutions exactly 1 solution exactly 2 solutions infinitely many solutions this question makes no sense d) The reduced row echelon form of an augmented matrix is 9 Marks Page 12 of 21 How many free variables does this reduced row echelon form have Answer: The
unknowns in the linear system are x, y, z, w, from left to right. There
is a solution to the linear system in which z = -2 and w = -1. In this solutions what are x and y Answers: x = y = Page 13 of 21 Determinants Question 8 Calculate the following determinants a) Answer: b) Answer: c) Answer: d) Answer: e) Answer: 9 Marks Page 14 of 21 Linear transformations Question 9 The reflection across the plane with equation 2x-2y+z=0 is a linear transformation of R with matrix Then: a) The determinant of the matrix A is b) The matrix has two eigenvalues, one is 1, the other one is c) The dimension of the eigenspace with respect to 1 is d) If an eigenvector of A has x-coordinate -4, y-coordinate 4 and negative z-coordinate, then its z-coordinate is e) If an eigenvector of A has x-coordinate 1 and y-coordinate 2, then its z-coordinate is f) What are the entries a, b and c Answers: a = b = c = 8 Marks 3 Page 15 of 21 Find the inverse Question 10 Let a) The determinant of A is equal to 0 a negative number a positive number nonsense: the matrix does not have a determinant b) Let k be the smallest positive integer such that A = O. Then k is 1 2 3 4 5 nonsense: k does not exist c)
Recall the geometric series (1–x) = 1 + x + x + x + ... for |x| < 1.
This can be used to compute the inverse of B = I –A (Hint: formally 1 = I, and let x = A). The first row of B is (a, b, c). Then a = b = c = 6 Marks k –1 2 3 –1 Page 16 of 21 Some useful facts Information Page 17 of 21 Information Page 18 of 21 Information Page 19 of 21 Information Page 20 of 21 Information Page 21 of 21