TMA 02 MST224 Mathematical methods

TMA 02 2024J

Covers Units 4, 5 and 6

Cut-off date 14 January 2025

You should submit this TMA electronically as a PDF file by using the University’s online TMA/EMA service. Before starting work on it, please read the guidance for preparing and submitting TMAs, available from the ‘Assessment’ tab of the module website.

The marks allocated to each part of each question are indicated in brackets in the margin.

You may use a computer algebra system to check answers, but it should not be used otherwise. TMA answers must show full working to attract full marks.

In order to encourage you to present your solutions to the TMA questions in a good mathematical style, your tutor will comment on how you:

• use correct mathematical notation;
• define any symbols that you introduce in formulating and solving a problem;
• give references for standard formulas and derivations;
• include comments and explanations within your mathematics;
• explicitly state results and conclusions, giving answers to an appropriate degree of accuracy and interpreting answers in the context of the question;
• draw diagrams and graphs;

These features are seen as being essential to complementing your mathematical skills. Your tutor will make comments on how well you achieve these objectives and give you guidance on how to satisfy the threshold requirement.

Five of the marks for each TMA will be allocated to the way you write your solutions. It is expected that most students will receive the majority of these presentation marks; such marks are included in TMAs to encourage, and emphasise the need for, thinking about how you present your mathematics.

PLAGIARISM WARNING – the use of assessment help services and websites

The work that you submit for any assessment/examination on any module should be your own. Submitting work produced by or with another person, or a web service or an automated system, as if it is your own is cheating. It is strictly forbidden by the University.

You should not:

• provide any assessment question to a website, online service, social media platform or any individual or organisation, as this is an infringement of copyright
• request answers or solutions to an assessment question on any website, via an online service or social media platform, or from any individual or organisation
• use an automated system (other than one prescribed by the module) to obtain answers or solutions to an assessment question and submit the output as your own work
• discuss examination questions with any other person, including your tutor.

The University actively monitors websites, online services and social media platforms for answers and solutions to assessment questions, and for assessment questions posted by students. Work submitted by students for assessment is also monitored for plagiarism.

A student who is found to have posted a question or answer to a website, online service or social media platform and/or to have used any resulting, or otherwise obtained, output as if it is their own work has committed a disciplinary offence under our Code of Practice for Student Discipline. This means the academic reputation and integrity of the University has been undermined.

The Open University’s Academic Conduct Policy defines plagiarism in part as:

• using text obtained from assignment writing sites, organisations or private individuals
• obtaining work from other sources and submitting it as your own.

If it is found that you have used the services of a website, online service or social media platform, or that you have otherwise obtained the work you submit from another person, this is considered serious academic misconduct and you will be referred to the Central Disciplinary Committee for investigation.

TMA 02

Cut-off date 14 January 2025

Question 1 (Unit 4)  – 17 marks Consider the vectors

a = −i + j + 2k,         b = 2i − j.

(a) Find the magnitude of the vector a. [2]

(b) Find a unit vector in the direction of a.  [2]

(c) Calculate the scalar product of a and b and hence calculate the angle between a and b in degrees (to the nearest degree). [8]

(d) Calculate the vector product a × b. [5]

Question 2 (Unit 4) – 4 marks

Consider a pyramid with a square base with sides of length a and sloping sides also of length a, as shown below. Choose a co-ordinate system with

−→

origin, O, at one corner, i-direction aligned with OA, j-direction aligned with

−−→

OC and k-direction perpendicular to this plane such that all the components

−−→

of OD are positive.

(a) Write down the vector equation of the line l through the centre of the square base that is perpendicular to the base. [2]

(b) Use the fact that the vertex D is on the line l together with the fact that the length of the sloping side is a to calculate the coordinates of the point D with respect to the origin O. [2]

Question 3 (Unit 4) – 7 marks

(a) Calculate the determinant

0  2  1

3  2  2 .

 2  1  3

(b) Hence, or otherwise, state whether the following simultaneous equations have a unique solution. Briefly justify your answer.

2y + z = 1,

3x + 2y + 2z = 2, [5]

2x + y + 3z = 3. [2]

Question 4 (Unit 5) – 10 marks Consider the system of equations

2x + 3y − z = 6, 4x + 7y + z = 10,

2x + 4y − 3z = 9.

(a) Use the Gaussian elimination method to reduce the system to upper triangular form. Clearly label the operations that you use. [5]

(b) If the system has no solution, then clearly state this; if it has a unique solution, then find it; and if it has an infinite number of solutions, then find the most general solution. [5]

Question 5 (Unit 5) –20 marks

(a)

Find the eigenvalues and eigenvectors of the matrix

Express the vector v =  2 / 3 in the form αv₁ + βv₂, where v₁ and v₂ are the eigenvectors that you found in part (a).  [4] (c)         Hence, or otherwise, calculate the product A¹² v. [6]

Question 6 (Unit 6) – 30 marks

(a) Express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form:

dx = x + 2y + 2e^−t, dt

dy = −x + 4y + 11e^−t. dt

Write down, also in matrix form, the corresponding homogeneous system of equations. [2]

(b) Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. [10]

(c) Hence write down the complementary function for the system of equations. [2]

(d) Find a particular integral for the original inhomogeneous system. [8]

(e) Hence write down the general solution of the original inhomogeneous system.   [1]

(f) Find the particular solution of the original inhomogeneous system with x = 4 and y = −1 when t = 0. [5]

(g)  What is the long-term behaviour of this particular solution as t becomes large? Does the ratio y/x tend to a fixed number, and if so what number? [2