Where S, T and U are double types.

Note: if an output returns an error, write down “error” .

(a) Provide the syntax to create the U vector

(b) Provide the syntax to extract the 3rd column of matrix S

(c) Provide the output of transpose(T)

(d) Provide the output of C = [T; U]

(e) Provide the output of D = eye(size(S))

(f)  Provide the outputs of [F1, F2] = max(S)

(g) Provide a one-line syntax that modifies T to the following matrix without addressing

Answer the following short questions:

Note: if an output returns an error, write down “error” .

(a) Consider result = [1 5 7 9]. Provide the output of result = [result 7].

(b) Consider D = 4:3:571. Provide the output of B = D(91) – D(85)

(c) Provide the function that computes the logarithmic value of X to the base 10

(d) Provide the output of imag(5 + 9i)

(e) Consider x, y and z to be vectors of the same size. State which of the following variables [A, B, C, D] contain unnecessary element-by-element operation(s):

A = x.*y +5*sin(z);

B = z./y - 7.*pi;

C = 10.^x + 4./y

D = exp(z./y) + 2/9 – cos(5.*x)

(f)  Describe the line specifications of the following function:

plot(k,m,'px')

The distance between two points in polar coordinates can be calculated using the Law of

Cosines equation          d = rA2 + rB2 − rArB cos(eA − eB )        where rA, rB, θA and θB are

defined in the following figure:

(a) Provide  MATLAB syntax to create a function file which calculates the distance (d) between two points in polar coordinates given the radii (rA and rB) and angles (θA and θB) of the points. You are not required to document the function and all inputs should be considered as scalars. It should NOT work with vectors. Use the following information for your function file:

• Function name: LOC

• Input variables: ra, rb, tha, thb

• Output variables: d

(b) You  are  now working  in  a  separate  m-file.  Provide  MATLAB  code  to  calculate the distance between point A(3, π/3) and point B(4, 3π/4).

Consider the following MATLAB function:

function [souls, HF] = DS3(nameless,king,ng)

king = round(king);

if ng < 5

HF = nameless + ng - king;

else

HF = nameless - ng + king;

end

souls = nameless .*(3/2) + ng ./king;

Gael = sin(souls) .^5;

end

Note: if an output returns an error, write down “error” .

(a) Provide the name and extension format of this function file

(b) Provide the output of [souls, HF] = DS3(1, 3.5, 5)

(c) Provide the output of [in, valid] = DS3([1 1], [4.3 4.8], [4 6])

(d) Provide the output of Y = DS3(1, 2, 3).

The figure below depicts a gear represented by an x,y curve and the location of its defects as represented by the asterisk markers. The equations for x and y are given as:

= T cos(t)

y = T sin(t)

where

T = a + tanh(b sin(nt))

with a=1, b=10, n=9 and t is a vector from 0 to 2π (inclusive) with 7000 points in total. The coordinates of the defects correspond to the first (x,y) point and then every 600th

(x,y) point thereafter, i.e. defects occur at the 1st , 601st , 1201st , 1801st , … points.

Write MATLAB code in the following parts to successfully reproduce the figure.

Figure 1: Gear with defect locations.

(a) This is the start of the m-file. Clear all variables, close all figure windows and clear the command window.

(b) Create all relevant variables for plotting. Use element-by-element operators where appropriate.

(c) Plot y against x and  label  the  plot  accordingly.  The  line  specification  is  a  black continuous line.

(d) Create variables which define the x and y coordinates of the defect locations.

(a) Provide the output of A = (X > 50)

(b) Provide the output of B = (X == 95) | (Y < 50)

(c) Provide the output of C = (Z == 1) & (X < Y)

(d) Provide the output of [D1, D2] = find(Y==75)

(e) Provide the output of E = X(Z)

(f)  Provide the output of F = ~logical(-Z)

PART B: ATTEMPT ALL QUESTIONS

Question B1 (15 marks)

Figure 2 shows a plot of the function  Y = 3 X5 − 12.5 X3  + 8 X + 12  over the range of

0 ≤ X ≤  2.

Figure 2

(a) Write the equation whose root must be found to determine the value of X which has a Y- value equal to 10.

(b) Use the method of False Position to calculate X  at Y = 10. You should use a precision of 0.05 and employ initial guesses for the lower limit to be 0 and the upper limit to be

1.5.  Fill in the details of each iteration in the table below, provide 3 decimal places. Note: You might need less rows, but should not need more. Workings for this part are not required)

(c) Write an M-file that uses the Newton-Raphson method to locate the local maximum seen  in  Figure  2.  Use  a  precision  of  10-6   and  determine  an  initial  guess  that  will converge to the local maximum of the function (but NOT the local minimum seen Figure 2).    Complete the M-file below by filling in the answer boxes with the missing code, making sure to follow the instructions in the comments.

%  Define the anonymous functions to find the maximum of the function  using the Newton-Raphson method

% Define initial guess and precision

% Calculate initial values for the functions