ME12002
.pdf
School
University of Bath**We aren't endorsed by this school
Course
MENG ME12002
Subject
Mechanical Engineering
Date
Dec 19, 2024
Pages
8
Uploaded by SuperFinch4807
University of Bath Department of Mechanical Engineering ME12002 Engineering Mathematics 31 May 2024 9:30 to 12:30 3 hours Answer all questions Only calculators provided by the University may be used. Candidates may take one double-sided A4 crib sheet in the examination venue. This will not be collected at the end of the examination. During this exam you are not permitted to cornmunicate with any person(s) except an invigilator or an assigned support worker. You must not have any unauthorised devices or materials with you. You must keep your Library card on your desk at all times. Please fill in the details on the front of your answer book/cover and sign in the section on the right of your answer book/cover, peel away adhesive strip and seal. Take care to enter the correct candidate number as detailed on your desk label. Do not turn over your question paper until instructed to by the chief invigilator. ME12002
1 , Q1 a) Write the complex number, QL_Q,Z, in Cartesian form and then find its complex conhjugate. [t mark] b) Convert the complex number 2¢7/, to Cartesian form. [1 mark] ¢} Find all possible values of 4/, [2 marks] d) Use de Moivre’s theorem to find an expression for cos 3¢ in terms of sin 6 and cosf. [2 marks] Q2 a) Find the derivatives of the following functions with respect to ¢ () 5, [1 mark} (iiy te; [1 mark]} (”) etsinZt. [1 mark] b) Find an expression for j—i when 43z +yaz® = 1. [2 marks] ¢) Find all the critical points of the function, y = z* — 22%, and classify them. [2 marks] Q3 a) Find both fi and @i when dx Sy 4 fle,y) = 2y [3 marks] b) Find the critical points of the function, (@, y) = aye™> 7Y, and classify them. [4 marks] ME12002 Page 2 of 7 kS
Q4 a) Evaluate the definite integral, 27 / 3 cost dt. 0 [2 marks] b) Use a suitable substitution to find the value of the integral, / 3 da. 0 [3 marks] ¢) Using a suitable substitution, determine the indefinite integral, z—1 [ [2 marks] Q5 The following are the position vectors of the points A, B and C, respectively 1 2 3 OA=a=|1]|, OB=ph=|1], and OC=c=|2], 1 0 : 3 a) Write the equation of the line which passes through points A and B. How close does this line pass to the origin? [2 mark] b) A plane passes through A, B and C; find both b — 2 and ¢ — a and hence write the equation of that plane using two free parameters. Determine the distance of this plane from the origin. : [3 marks] Page 3 of 7 ME12002
Q6 The following data have been obtained from an experiment %l 0 1 2 38 4 yi | 0.02 121 398 917 1571 It is proposed to fit the quadratic curve y = az? + b to this data using the method of least squares. Develop the theory, and then find the line of best fit. [5 marks] Suppose now that the fiited curve must pass through the origin. Give the equation of this curve. [2 marks] Q7 Matrix A is given by 310 A=12 3 2]). 013 a) Find all eigenvalues and eigenvectors of A [4 marks] b) Hence write the general solution of the system of ordinary differential equations a(® 310 T ZHyl=1{232]|v]. - o\, 013/ \z [t mark] ¢) What is the solution of the system of equations given in part Q7b) if the initial condition is that (z,y,z) = (0,4,0) at ¢ = 0? [1 mark] ME12002 Page 4 of 7
Q8 a) Find the determinant of the following matrix: 1100 1210 0131 0014 [2 marks] b) Use the method of Gaussian Elimination to solve the following system of simul- taneous equations: O O N [l I == O D= OO a0 o R e [5 marks] [Note: Full marks will only be awarded for a correct answer which has used the Gaussian Elimination method in its precise form, i.e. where the matrix has been reduced to upper triangutar form.] Q9 The Laplace Transform of f(¢) is defined according to F(s) = LIf(#) /f(t Jestat, a) Use the definition of the Laplace Transform to find the Laplace Transform of the function e~* where b is a constant. [2 marks] b) Use the definition of the Laplace Transform to find £[f/(t)] in terms of F(s) and F(0). [2 marks] ¢) Hence solve the equation Y+3y=e™ subject to the initial condition, y(0) = 2. [4 marks] Page 5 of 7 ME12002
Q10 a) The function f(¢) has a period equal to 2 and it is given by f(t) = * within the range —1 < ¢ < 1. Sketch the function in the range —3 < ¢ < 3 and find its Fourier series representation. [6 marks] b) Hence find the particular integral of the ordinary differential equation, d*y —Z 4y = . T T =10 Clearly, y(¢) will be continuous at ¢ = 1, but how many of its derivatives will also be continuous at that point? [4 marks] Q11 The definition of the Laplace Transform is given in Q9 a) Use the definition of the Laplace Transform to find both £[e*] and L{¢]. [2 marks] b) Use the definition of the Laplace Transform to prove the s-shift theorem L[f{t)e ] = F(s + a) where L[f(t)] = F(s). [2 marks] ¢) Use the s-shift theorem to determine the inverse Laplace Transform of L s2+6s+9 [3 marks] d) The convolution theorem is L[f x g] = F(s)G(s), where LIf#)] = F(s), Llgt)]=Gls), and [xg= [ f(r)g(t - 7)dr. Use the convolution theorem as an alternative way to find the inverse Laplace Transform of 1 s2+6s+9 [3 marks} ME12002 Page 6 of 7
Q12 a) Find the binomial series representation of the function, y=(1+2)7% and write it in summation form. [2 marks] b} By finding its derivatives, find the Taylor’s series about z = 0 of the function cos z, and write the answer in summation form. [3 marks] ¢) Use D’Alembert’s test to determine the radius of convergence of the series 2. z? z8 L 1x2 2x3 3x4 4x5 4 yle) =1~ [2 marks] d) Use LHépital’s rule to determine, ; . cosaxr—1 i Im ———— z=0 sin®w ! [3 marks] : Q13 Steady two-dimensional heat conduction occurs in a bar of unit width in the z-direction, ; but which is semi-infinite in the y-direction (i.e. 0 < y < oo). The temperature field is ; governed by Laplace’e equation Pr o ox? a2 where T is the temperature. If T is set to be zero at both z = 0 and = = 1, then use the technique of separation of variables to show a physically meaningful solution in the form of an infinite sum of terms. [10 marks] SG Page 7 of 7 ME12002