UFMFFT-15-1 Page 1 of 6 Faculty of Environment and TechnologyAcademic Year: 2020/2021Assessment Period or Date: 2 Module Leader: Dr Lloyd BridgeModule Code UFMFFT-15-1Module Title: Mathematical Modelling for Electronics and RoboticsExamination Duration: The assessment is available for 24 hours and you should be able tocomplete the exam in 2 hours.ONLINE EXAMInstructions to Students:• Students should attempt all five questions.• The marks available for each question and part are shown. A total of 100 marks is available.• Students must show all workings clearly and correctly to gain full credit.• Solutions exhibiting unclear, inconsistent or incomplete methodology will receive little credit.• This is an individual assessment: do not copy and paste work from any other source or work with anyother person during this exam. Text-matching software will be used on all submissions.FormattingPlease upload a single .pdf file (if possible). We cannot ensure that other formats are compatible withmarkers’ software and cannot guarantee to mark incorrect formats.Please include the module name and number and your student number (not your name).Please indicate clearly which questions you are answering.Instructions for submissionYou must submit your assignment before the stated deadline by electronic submission through Blackboard.• Multiple submissions can be made to the portal, but only the final one will be accepted. Please saveyour work frequently. •It is your responsibility to submit exam in a format stipulated above.Your marks may be affected if your tutor cannot open or properly view your submission. • Do not leave submission to the very last minute. Always allow time in case of technical issues.• The date and time of your submission is taken from the Blackboard server and is recorded when yoursubmission is complete, not when you click Submit.• It is essential that you check that you have submitted the correct file(s), and that each complete filewas received. Submission receipts are accessed from the Coursework tab.There is no late submission permitted on this timed assessment.Question 1(a) The charge q(t) stored on a capacitor having capacitance C (measured in Farads),discharging through a resistor of resistance R (measured in Ohms), is given byq(t) = Qe-RC t ,where Q is the initial charge (measured in Coulombs), and t is time (measured in seconds). Suppose Q = 50 and C = 0.25.(i) For R = 2.5, calculate the charge in the capacitor after 2 seconds have elapsed.(2 marks)(ii) Suppose that for a different resistor we are given that 40% of the initial charge islost in the first 3 seconds. Find the value of R. (4 marks)(b) A voltage of the form v = 8 sin(2t) is applied to an electrical circuit. The steady-stateoutput current i flowing through the circuit is given by the sinusoidal oscillationi = 2 sin(2t) + 5 cos(2t).The voltage is measured in volts and the current is measured in amps.(i) State the angular frequency of oscillation in radians per second and the fundamental period of oscillation in seconds of the input voltage (that is the minimum timerequired for a single cycle of oscillation). (2 marks)(ii) Express the current i in the form i = R sin(ωt + α), where α is in radians. Hencestate the amplitude of the output steady-state current.(6 marks)(Q1 total: 14 marks)UFMFFT-15-1 Page 2 of 6Question 2(a) Consider the oscillatory function y =cos(2x)x.(i) Describe what happens to the amplitude of oscillation as x increases in value.(2 marks)(ii) Show thatdydx is given bydydx = –2x sin(2x) + cos(2x)x2(3 marks)(iii) Determine the equation of the tangent line to y =cos(2x)xat the point (π4 , 0).(3 marks)(b) The function f(t) = te-at, where a > 0, rises to a peak value then decays towardszero for t > 0.It can be used to model the transient response of an electrical circuit to the suddenapplication of a voltage.For the case where a = 0.23, find the following. (i) f′(t).(ii) The maximum value of f(t) for t > 0.(3 marks)(3 marks) (c) Use integration by parts to evaluate the definite integral Z0 3π x sin x dx.(4 marks)(d) Expand the rational functionx + 1×2 + 7x + 12in terms of its partial fractions and hence evaluate the indefinite integralZ x2 + 7 x + 1 x + 12 dx .(6 marks)(Q2 total: 24 marks)UFMFFT-15-1 Page 3 of 6Question 3(a) Find the values of a for which the matrix A = -a12 5 -2 a has no inverse.(3 marks)(b) Given thatB = 43 1 -2 and C = -2 4 3 -5 ,solve the matrix equation3X + I = -BC, where I is the identity matrix, to find X.(c) Consider the following linear system:(4 marks) 3x – 4y = -27, 5x + 12y = -17.(3.1)(i) Write the system in matrix form. That is, write in the form Ax = b, wherex = xy, clearly identifying A and b.(ii) Find the inverse matrix A-1.(iii) Find the solution to the system (3.1) using the inverse matrix method.(2 marks)(3 marks) (3 marks)(d) Use Gaussian elimination to solve the following system of equations:2x – y + 6z = 54,3x + 2y – 4z = -41,8x + 5y – 2z = -16.(10 marks)(Q3 total: 25 marks)UFMFFT-15-1 Page 4 of 6Question 4(a) Find the solution y(x) of the initial value problemdydx = (x + 1)y2, y(0) = 2.(8 marks)(b) The current i(t) flowing through an RL-circuit resulting from the application of a sinusoidal voltage may be modelled by the initial value problemdi dt + 2i = cos(t),i(0) = 0.Solve the differential equation for the current i(t) and identify the natural and forcedresponse of the RL-circuit.(11 marks) (Q4 total: 19 marks)UFMFFT-15-1 Page 5 of 6Question 5Consider the “square wave” functionf(t) = 1 0 for for 0 π < t ≤ t ≤≤ π, 2π.The function f(t) can be approximated by a sum of sinusoids of different frequencies. Asimple such approximation which is smooth is given byg(t) = 12+2 πsin(t) + 23πsin(3t).(a) Using MATLAB, plot both f(t) and g(t) over the interval 0 ≤ t ≤ 2π. Include both your MATLAB code and the plot in the document that you submit.(b) Find the average (mean) value of f(t) over the interval 0 ≤ t ≤ π.(5 marks)(2 marks)(c) Find the exact average (mean) value of g(t) over the interval 0 ≤ t ≤ π. Then reportyour answer correct to 5 decimal places.(6 marks) (d) Using MATLAB, apply the trapezoidal rule with 20 equally spaced subintervals to approximate the average (mean) value of g(t) over the interval 0 ≤ t ≤ π. Include bothyour MATLAB code and your answer (correct to 5 decimal places) in the document thatyou submit.(5 marks)(Q5 total: 18 marks)END OF QUESTION PAPERUFMFFT-15-1 Page 6 of 6
