NSCI0006 Mathematical Methods B 2023/24
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NSCI0006 Mathematical Methods B
Take-home paper — Academic Year 2023/24
Start time: 11 Jan 2024, 14:00 (2pm)
General instructions
Read the following instructions carefully before starting the assessment.
Assessment duration
The standard duration of this test is 24 hours. You are allowed to start the paper at any point in the timetabled duration but you must submit your work within 24 hours of the scheduled start time.
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The paper should take no more than a few hours to complete. It is essential that you set yourself a workplan which includes time for rest breaks, screen breaks, staying hydrated, exercising, getting some sleep and any other activities that are central to your wellbeing. Working for excessive hours can be extremely detrimental to your physical and mental health and is unlikely to improve your final mark. |
Submitting your work
You will submit your work on Moodle, which MUST be in pdf format. You can use the MS Office Lensapp to scan pages on your phone or tablet and then convert it to pdf.
Do not write your name or any other identifying information such as your student number on your script. Moodle will keep track of your submission.
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Preparing your submission and uploading it to Moodle will take some time. You must account for this to avoid late penalties. You will be given a practice opportunity to try out the steps involved before the date of the exam. |
Self-declaration
When you submit your work you will be asked to declare that it is your own. You should not discuss your work with any other person, or use AI tools such as Chat GPT. You are allowed to use your lecture notes.
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Any indication of plagiarism or collusion will be taken very seriously and subject to UCL’s academic misconduct penalities |
SORAs and extensions
• If your SORA states that you should be allowed additional time and/or rest breaks, these will be rolled into a single extension of 2 hours.
• If you have more complex needs which mean the standard amount of extra time is not suitable, you can contact the Disability, Mental Health & Wellbeing team.
• If you have been granted an extension viaExtenuating Circumstancesthen the amount of additional time that you will be allowed is 2 hours.
Late submission penalties
If your work is ...
• less than 1 hour late you will receive a deduction of 10 percentage points, but no lower than the pass mark.
• 1 to 2 hours late then it will be capped at 40.00%
• more than 2 hours late you will receive a mark of no more than 1.00%
Further information
For more information about the policies for take home papers, see the academic manual Section 6.3
Answer all questions
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Warning The maximum number of marks available for each question is indicated in the right margin. There are 25 marks available. You must write down all of the steps involved in your answers clearly. Results that are written down without explanation of justification will not be awarded marks. |
1. A mechanism describing the transport and metabolism of lactose in E. coli is given by the following equation in which X(t) represents lactose and a, b are constants:
(a) Could this problem be solved by separation of variables or integrating factor technique? Explain your answer for each method. [4]
Note: you do not have to solve the problem to obtain the marks for this question.
(b) Write down a discrete equation that could be used to obtain a numerical solution to the given problem, and explain (briefly) how you could implement this method in Python. [4]
2. A model for the volume of a tumour x(t) is given by the following equation, where α, k are parameters:
(a) Rewrite the problem as a first order ODE for y(˙) , such that [4]
(b) Solve the ODE for y(˙) that you found, and hence show that the solution for the tumour volume can be written as [3]
(c) The plot below shows 
as a function of x. The curve meets the axis at x = 0, x = x∞ and has a maximum at x = xc . Identify any equilibrium points on the plot and say whether they are stable or unstable. Explain how you determined their stability. [4]
(d) Find the value x∞ in terms of x0, α, k [1]
(e) Show that the maximum growth rate occurs at the point [3]
(f) Produce a rough sketch of how you think the solution x(t) looks for this problem. Your sketch must clearly show the shape of the curve, and the points x0 , xc , x∞ must be labelled. [2]
2024-01-16