UFMFJ9-30-1 Page 1 of 5

Faculty of Environment & Technology

Academic Year: 16/17 Examination Period: Summer

Module Leader: Alison Hooper Module Code: UFMFJ9-30-1 Title of Module: Engineering Mathematics Work Item Code: Duration: 2 hours Standard materials required for this examination:

Examination Answer Booklet Yes

Multiple Choice Answer Sheet No

Graph Paper Type of paper e.g. G3, G14

Number of sheets per student Additional materials required for this examination:

Details of additional material supplied by UWE: Additional Specialised Material : UFMFJ9 Formula Sheet (10 pages) To be collected with Answer Booklet (please delete as appropriate) No

Details of approved material supplied by Student: To be collected with Answer Booklet (please delete as appropriate)

University approved Calculator Yes

Candidates permitted to keep Examination Question Paper Yes

Candidates are NOT permitted to turn the page over until the exam starts

UFMFJ9-30-1 Page 2 of 5

Instructions to Candidates:

1. You should attempt all questions. 2. Show all working and express answers in as simple a form as possible.

Question 1 (12 marks) Let 𝒑 = −2𝒊 + 3𝒋 + 𝒌 𝒒 = 𝒊 − 5𝒋 + 𝒌

𝒓 = 4𝒊 + 2𝒋 + 2𝒌

(i) Find |𝒑 − 𝒒| [3 marks]

(ii) Show that 𝒑 and 𝒓 are perpendicular. [3 marks]

(iii) Find 𝒑 × 𝒒 . [3 marks]

(iv) Find a unit vector whose direction is perpendicular to the directions

of both that 𝒑 and 𝒒. [3 marks]

Question 2 (10 marks) Consider the points with co-ordinates 𝐴 at (−1, 2, 2), 𝐵 at (−2, −1, 0) and 𝐶 at (0, 1, 4)

(i) Find the vectors 𝐴𝐵⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ and𝐵𝐶⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . [2 marks]

(ii) Find (in degrees) the angle between 𝐴𝐵⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ and 𝐵𝐶⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , [5 marks]

(iii) Find the work done moving the force 𝑭 = 𝒊 − 𝒋 + 2𝒌 from point A to point C [3 marks]

UFMFJ9-30-1 Page 3 of 5

Question 3 (15 marks)

Consider the matrices 𝐴 =(2 −35 2 ) ,𝐵 =(0 1 26 −2 3), 𝐶 =(78)

(i) Compute matrix 𝐶𝑇𝐴𝐵 . [4 marks]

(ii) Find 𝐴−1 and use it to solve the system of equations

2𝑥 − 3𝑦 = 7, 5𝑥 + 2𝑦 = 8 . [5 marks]

(iii) Express the following system of equations in augmented matrix form and use Gaussian elimination to find the unique solution of these equations. 𝑥 + 2𝑦 − 3𝑧= 32𝑥 − 𝑦 − 𝑧 = 113𝑥 + 2𝑦 + 𝑧= −5

[6 marks]

Question 4 (15 marks) Consider the matrix 𝐴 = (3 21 2)

(i) Show clearly that the characteristic equation of 𝐴 is 𝜆2 − 5𝜆 + 4 = 0 and use it to find the eigenvalues and corresponding eigenvectors of 𝐴.

[6 marks]

(ii) Verify that each eigenvalue of 𝐴 with their corresponding

eigenvector 𝒙 satisfies the equation

𝐴𝒙 = 𝜆𝒙 [4 marks]

(iii) Using the Cayley-Hamilton theorem, which states that 𝐴 satisfies

its characteristic equation 𝜆2 − 5𝜆 + 4 = 0 , show that 𝐴−1 = 14[5𝐼 − 𝐴] where 𝐼 is the identity matrix, and then use this to find 𝐴−1.

[5 marks]

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Question 5 (8 marks)

Find the solution to the first order differential equation 𝑑𝑦𝑑𝑡+ 2𝑦 = 𝑡 + 2

with the initial condition 𝑦(0) = 114 .

[8 marks] Question 6 (20 marks) (i) By solving its auxiliary equation, find the complementary function of the

differential equation 𝑑2𝑦𝑑𝑡2 + 5𝑑𝑦𝑑𝑡+ 6𝑦 = 0

[4 marks]

(ii) Find the particular integral for the differential equation 𝑑2𝑦𝑑𝑡2 + 5𝑑𝑦𝑑𝑡+ 6𝑦 = 3𝑡 + 1

[9 marks] (iii) Using your answers to (i) and (ii), write down the general solution to the differential equation 𝑑2𝑦𝑑𝑡2 + 5𝑑𝑦𝑑𝑡+ 6𝑦 = 3𝑡 + 1 and hence find the solution which satisfies the conditions 𝑦(0) = 34, 𝑦′(0) = 12

[7 marks]

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Question 7 (20 marks) Let 𝑦(𝑡) denote the solution of the differential equation 𝑑2𝑦𝑑𝑡2 + 2𝑑𝑦𝑑𝑡+ 4𝑦 = 4 with initial conditions 𝑦(0) = 1 and 𝑦′(0) = −3.

(i) Show that the Laplace transform of 𝑦(𝑡),denoted by 𝑌(𝑠), is given by 𝑌(𝑠) = 𝑠2 − 𝑠 + 4𝑠(𝑠2 + 2𝑠 + 4)

[8 marks]

(ii) Use partial fractions to show that 𝑌(𝑠) can be written in the form

𝑌(𝑠) = 𝐴𝑠+ 𝐵𝑠 + 𝐶𝑠2 + 2𝑠 + 4

where, 𝐴, 𝐵 and 𝐶 are constants to be found. [4 marks]

(iii) Hence take inverse Laplace transform of 𝑌(𝑠) to obtain the solution 𝑦(𝑡). [4 marks]

(iv) Verify that your solution found in (iii) satisfies the initial conditions

given.

[4 marks]

END OF QUESTION PAPER