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6 Engineering Sample Midterm Problems University of California, Davis (Solutions provided at the end) NOTE: Some of the equations displayed below may not be printed correctly by all printers. This page may also not be displayed correctly by some old versions of Netscape Navigator and Internet Explorer. These sample problems are just that sample problems. They do not necessarily cover all the topics that might appear on the exam. (See Midterm Study Guide for topics that are covered.) When you are asked to solve a problem and write a script, you are to first develop a mathematical algorithm, then write a Matlab script to compute and display the specified values. You must first show your work in developing the algorithm, either mathematically or in script comments. All computation is to be done in the script, not in your head or by calculator. Matlab variable names should correspond to the chosen mathematical variables. Comments must be included in the script to define variables and to explain sections of the code. The output displayed by the script should clearly identify the variable whose value is displayed and specify its units. 1. (20 points) (a) Explain the meaning of the following Matlab commands, identifying special variables and describing what actions Matlab would take in processing the commands: (i) s = sign(x) (ii) y = imag(z) (iii) clear (iv) save session2 (v) length_a = input('Length of side A: ') (b) Evaluate the following Matlab expressions, giving the format short display of the results (not necessary to include the ans = ): (i) 4^2 (ii) rem(15,2) (iii) log10(100) + log10(0.001) (iv) conj(1 j*3) (v) angle(-1) 2. (24 points) (a) Convert the following to Matlab assignment statements. Use built-in functions where possible. f= (i) 1 2 c L p+ w u+v E= w p+ u v (ii) g y= (iii) d= e 2h (ln h) d s C (2.15 + 0.35F )1.8 T (1 C ) F (iv) (b) Give the Matlab response (i.e., what it would display) to the following sequences of commands: (i) x = 1008.0; format short e y = 10 * x a = 2; b = 4; c = a + b; b = 5; disp(c) theta = 0; f = sin(theta)/theta phi = 0; g = cos(phi)/phi (ii) (iii) (iv) 3. (8 points) Reduce the following results to the proper number of significant digits, rounding as needed and providing the correct units. (a) 4.636 + 3.64 = 8.276 --> ? (b) (2.43)(17.675) = 42.95025 --> ? (c) (4.011 103 seconds)/(3600 seconds/hour) = 1.11417 hours --> ? (d) (0.12760 0.0732)/14.06 = 0.003869132 --> ? 4. (24 points) Using the figure below, develop an algorithm and write a Matlab script to compute and display the length FA (in meters) and the angle D1 (in degrees). Define variables as indicated in the figure. 5. (24 points) (a) Write the Matlab commands to do the following: (i) Assign the complex number z = 2 + j3 and display its real part x and imaginary part y. (ii) Assign the complex number z = 4 j3 and determine and display the components needed to express the number in polar form. (iii) Compute and display the following complex number in rectangular form. z = 3 2 e j 4 / 5 (b) Given the following complex numbers z1 and z2, evaluate the expressions below. You may use either rectangular or exponential form. z1 = 2 + j 2 = 2 2e j 3 / 4 z 2 = 4 + j 4 = 4 2e j / 4 (i) z 2 z1 (ii) z1 (iii) jz1 z2 (iv) z1 6. (20 points) (a) Write the commands to create the vector of values of v(t ) = 5e 2 t cos( t / 8 + / 4) for t varying from 0 to 1 in increments of (b) 0.02. For the phasor X = 2e j / 4 write the commands to compute the vector of values of the corresponding real sinusoidal signal x(t), with frequency = 200 , for 401 values of t, with 0 t 5T, where T is the fundamental period of the sinusoidal signal. 7. (20 points) For matrix G: G= 0.6 8.2 5.7 0.5 1.2 1.5 2.3 0.5 0.5 0.1 2.0 8.2 9.0 1.5 0.5 2.4 0.5 2.3 4.5 0.5 (a) Determine the array defined by the command: A = G(: , 2) (b) Determine the array defined by the command: B = G(4:5, 1:3) (c) Determine the array defined by the command: C = G(1:2:5, : ) (d) Write the command to create a row vector v consisting of the elements in the third column of G. 8. (20 points) (a) Write the commands needed to compute the coefficients of the polynomial resulting from the product A(s) = B(s)C(s), where: B ( s ) = 3s 3 5 s 2 + 6 s 2 C ( s ) = s 5 + 3s 4 2 s 2 + 5 (b) Write the commands needed to compute the coefficient vector b of the polynomial having roots 13 + 17j and 13 17j. (c) Consider the following sequence of commands and displayed results: q n = [1 5 11 13]; d = [1 2 4]; [q r] = deconv(n,d) = 1 3 r= 0 0 1 1 Write the mathematical expressions for the original polynomial ratio and for the result of the deconv operation. Solutions 1. (a) (i) s = +1 if x > 0, -1 if x < 0, and 0 if x = 0 (ii) y is set equal to the imaginary part of the complex number z (iii) Removes all variables from workspace (iv) Saves all workspace variables to the binary MAT-file named session2 (v) Displays the prompt Length of side A: and then waits for input from the keyboard; the input is evaluated and assigned to the variable length_a. (b) (i) 16 (Note: -16.0000 acceptable because lecture notes are vague.) (ii) 1 (Note: -1.0000 acceptable because lecture notes are vague.) (iii) 1.0000 (iv) 1.000 + 3.0000i (v) 3.1416 2. (a) (i) f = 1/sqrt(2*pi*c/L) (ii) E = (p + (w/(u + v))) / (p + (w/(u v))) (iii) y = exp(-g/(2*h)) / (log(h) * sqrt(d*s)) (iv) d = (C * (2.15 + 0.35*F)^1.8) / (T*(1 C)^F) (b) (i) y = 1.0080e+004 [Note: the actual display is 10080, due to a bug in Matlab; either answer is acceptable.] (ii) 6 (iii) f = NaN (iv) g = Inf 3. (a) 8.28 (b) 43.0 (c) 1.114 hours (d) 0.00387 4. %Using sum of angles in a triangle = 180 degrees: D2 = 180 30 45; D1 = 180 D2; B = 180 30 30 45; F = 180 B D1; %By right triangle: AC = 100/sin(30*pi/180); %By the law of sines: FA = AC*sin(45*pi/180) / sin(F*pi/180) 5. (a) (i) z = 2 + j*3 x = real(z) y = imag(z) z = 4 j*3 mag = abs(z) ang = angle(z) z = 3*sqrt(2*pi)*exp(j*4*pi/5) (ii) (iii) (b) (i) 6 + j2 (ii) (iii) (iv) 6. (a) (b) -2 j2 (or, in exponential form, 2 2 e-j3 /4 or 2 2 ej5 /4) -2 j2 (or 2 2 e-j3 /4 or 2 2 ej5 /4) 2 e-j /2 = -2j t = 0:0.02:1; v = 5*exp(-2*t).*cos(pi*t/8 + pi/4); T t X x = = = = (2*pi)/(200*pi); linspace(0, 5*T, 401); 2*exp(-j*pi/4); real(X*exp(j*200*pi*t)); 7. (a) A= 1.5000 0.5000 8.2000 0.5000 -2.3000 (b) B= 0.5000 1.2000 0.5000 -2.3000 2.4000 -4.5000 (c) C= 0.6000 5.7000 1.2000 1.5000 8.2000 -2.3000 2.3000 9.0000 -4.5000 -0.5000 1.5000 0.5000 (d) v = G(:, 3)' [Note the transpose operator] 8. (a) b = [3 5 6 2]; c = [1 3 0 2 0 5]; a = conv(b,c) r = [-13+17j, -13-17j]; b = poly(r) (b) (c) s +1 N ( s) s 3 + 5s 2 + 11s + 13 = s +3+ 2 = 2 D( s ) s + 2s + 4 s + 2s + 4

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