Final-exam-revision-exam-preparations apm1513
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University of South Africa**We aren't endorsed by this school
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APM 1513
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Mathematics
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Dec 30, 2024
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Final-Exam revision - Exam preparationsApplied Linear Algebra (University of South Africa)Scan to open on StudocuStudocu is not sponsored or endorsed by any college or universityFinal-Exam revision - Exam preparationsApplied Linear Algebra (University of South Africa)Scan to open on StudocuStudocu is not sponsored or endorsed by any college or universityDownloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
Final examination revision - 202 CSc.Question 1Use MATLAB to evaluate the following expressions. 1.1000(1 + 0.15/12)602.(0.0000123+5.678×10−3)×0.4567×10−4Solution :1.1000*(1+0.15/12)^60 2.(0.0000123+5.678*10^(-3))*0.4567*10^(-4) Question 2Set up a vector n with elements 1, 2, 3, 4, 5. Use MATLAB array operations on the vector n to set up the following four vectors, each with five elements: a. 2, 4, 6, 8, 10 b. 1/2, 1, 3/2, 2, 5/2 c. 1, 1/2, 1/3, 1/4, 1/5 d. 1, 1/22, 1/32, 1/42, 1/52Solution :a = 1:5 b = a*2 c = a/2 d = a.\1 e = a.^2.\1 Question 3Write MATLAB programs to find the following sums (a)with for loops, and (b)by vectorization. 1- 1/3 +1/5 -1/7 + 1/9 - ... -1/1003 Solution :sign = -1; % with for loop s = 0; for i = 1:2:1003 sign = -sign; s = s + sign/i; end disp(s); Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
a = 1:4:1001; % by vectorization b = 3:4:1003; s = sum(a.\1) - sum(b.\1); Question 4The probability density function of the random variable Tdescribed by the normal distribution is given as 22221)(tetf1.Write a program to compute and display the values of Tand f(t) over the range (-3,3) with step 0.01. 2.Plot a graph of f(t), consider the following: write the text 'The probability density function of the normal distribution ' as a title on top of the graph. label the x-axis as tlabel the y-axis as f(t). Solution :m = 0; % mean of probability distribution s = 1; % standard deviation t = -3 : 0.01 : 3; % random variable T f = 1/(sqrt(2*pi)*s) * exp(-0.5*(t-m).^2/s^2); plot(t, f), title('The probability density function of the normal distribution'); xlabel('t'), ylabel('f(t)') grid disp([t' f']) %display a table Question 5It has been suggested that the population of the United States may be modeled by the formula )25.1913(03134.01197273000)(tetPwhere t is the date in years. Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
Write a program to compute and display the population every ten years from 1790 to 2000. Plot a graph of the population against time Solution :t = 1790: 10: 2000; % date T in years P = (1+exp(-0.03134 *(t-1913.25))).\ 197273000; plot(t, P), title(' Model of the United states population '); disp([t' P']) %display a table Question 6Write a script m-file to compute the solution of a quadratic equation. check whether a = 0, to prevent a division by zero. Solution :% Step 1: Start format compact disp(' Solution of Quadratic Equation ') % Step 2: Input data A = input(' Specify coefficients as follows: [a, b, c] = '); a = A(1);b = A(2);c = A(3); % Step 3: Evaluation of roots if a==0 & b==0 & c==0 disp(' Solution is indeterminate ') elseif a==0 & b==0 disp(' There is no solution ') elseif a==0 x = -c/b; else if (b ˆ2 - 4*a*c == 0) x = –b / (2*a); elseif (bˆ2 –4*a*c < 0) disp( ’Complex roots’) else x1 = (–b + sqrt(bˆ2 –4*a*c)) / (2*a); x2 = (–b –sqrt(bˆ2 –4*a*c)) / (2*a); end end % Step 4: Stop Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
Question 7Write a script which inputs any three numbers (which may be equal) and displays the larger one with a suitable message, or if they are equal, displays a message to that effect. Solution :format compact disp(' Find maximum number ') % Step 1: Input data A = input(' Enter three integers as follows [a,b,c]= '); a = A(1);b = A(2);c = A(3); % Step 3: Evaluation of roots max =a; if max==b & max==c disp(' all numbers are equal ') else if max<b max=b; , end if max<c max=c; , end end disp(['Maximum number is: ',num2str(max)]) Question 8Using fixand remmatlab commands write a script to convert seconds into hours, minutes and seconds. Try out your script on 10 000 seconds, which should convert to 2 hours 46 minutes and 40 seconds. Solution:t1 = 1000; % first total of 10 000 secondshh = fix(t1/3600) % hourst2 = rem(t1,3600) % second total of 1000 seconds mm = fix(t2/60) % minutesss = rem(t2,60) % secondsQuestion 9Design an algorithm (i.e. write the structure plan) for a machine which must give the correct amount of change from a RS100 note for any purchase costing less than RS 100. The plan must specify the number and type of notes 50, 20, 10 ,5 and 1 riyal in the change, and should in all cases give as few notes as possible. Solution:Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
P = input(' enter purchase amount: '); % purchase less than 100S = 100 - P; % change amountD = [50 20 10 5 1]; % type of notesN = [0 0 0 0 0]; % number of each type of notesfor i = 1:5 while S >= D(i) N(i) = N(i) + 1; S = S - D(i); end; end for i = 1:5 disp(['Type: ', num2str(D(i)), ', number: ',num2str(N(i))]); end Question 10Write a script that plots the graph defined by over the range 0 to 5πSolution:x = 0 : pi/20 : 5*pi; y = sin(x); y = y .* (y > 0); % set negative values of sin(x) to zeroplot(x, y) Question 11Write a script that simulates rolling a fair dice by generating a vector dof 2000 random integers in the range 1 to 6 and estimate the probability of throwing a six by dividing the number of sixes thrown by 2000. Solution:d = floor(6 * rand(1, 2000)) + 1% Generate a random vector d in the range 1 to 6 n = sum( d == 6)% Count the number of “sixes” thrownp = n/2000 % Estimate the probability of throwing a sixQuestion 12Set up any 3 × 3 matrix a. Write command-line statements to perform the following operations on a: 1.interchange columns 2 and 3; 2.add a fourth column (of 0’s);Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080
3.insert a row of 1’s as the new second row of a (i.e. move the current second and third rows down); 4.remove the second column. a = [1 2 3;4 5 6;7 8 9] results in a = 1 2 3 4 5 6 7 8 9 (1) Interchange columns 2 and 3. a(:,[2 3]) = a(:,[3 2]) results in a = 1 3 2 4 6 5 7 9 8 (2) Add a fourth column (of 0s). a(3,4) = 0 results in a = 1 3 2 0 4 6 5 0 7 9 8 0 (3) Insert a row of 1s as the new second row (i.e., move the current second and third rows down). a([3 4],:) = a([2 3],:); a(2,:) = ones results in a = 1 3 2 0 1 1 1 1 4 6 5 0 7 9 8 0 (4) Remove the second column. a(:,2) = [ ] results in a = 1 2 0 1 1 1 4 5 0 7 8 0 Downloaded by Dean Chamier (scummieo@gmail.com)lOMoARcPSD|2572080