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**Q1:**=) Evaluate \oint_{C}(y \cos (x)-x y \sin (x)) d x+(x y+x \cos (x)) d y where C is the triangle from (0,0) to (0,4) to (2,0) to (0, 0). Show all working.See Answer**Q2:**6) Let the surface S be part of the sphere x2 + y² + z² = 4 (oriented away from the origin)that lies within the cylinder x² + y² = 1 and above the plane z = 0, and let denote the unit normal vector in the direction of the orientation. Let C be the boundary curve of S, oriented clockwise when viewed from the x-y plane. Consider the vector field F(x, y, z) = xi + yj + xyzk. \text { (a) Evaluate } \oint_{C} F \cdot d r \text { without using Stokes' Theorem. } \text { (b) Evaluate } \iint_{S}(\nabla \times \boldsymbol{F}) \cdot \boldsymbol{n} d S \text { without using Stokes' Theorem. }See Answer**Q3:**\text { Find the rank and nullity of the matrix } A=\left[\begin{array}{rrrrr} 5 & 4 & 3 & 1 & 3 \\ 6 & -4 & -2 & 2 & 2 \\ 3 & 4 & -10 & 4 & 1 \\ 9 & 10 & -18 & 20 & 21 \end{array}\right] \text { . }See Answer**Q4:**3. (30 points) Consider the characteristics of the following digitalcommunication system. a. (8 points) A signal x(t) is used to transmit bits. Shown below is theFourier Transform, X(f) of the signal shown on a frequency scale. What is the bit rate? b. (4 points) What percentage of the energy is included from the originalcomplete spectrum? c. (8 points) Show the plot of the time domain signal y(t) that wouldtransmit at twice the bit rate as x(t). d. (10 points) Now consider a different signal. What is the maximum bit rate Than can be transmitted over a channel with 100 kHz maximum frequency? 95% of the signal energy is transmitted.See Answer**Q5:**2. (20 points) Consider the characteristics of the following digitalcommunication system. a. (6 points) A signal x(t) is used to transmit bits at 1 Mbps. Show thewaveform x(t). Remember to label all plots completely. b. (6 points) Show the Fourier Transform X(@). c. (5 points) What bandwidth is needed in order to transfer 95% of the signalenergy? d. (3 points) If instead 20 MHz bandwidth was available, what data rate could be supported with sending 95% of the signal energy?See Answer**Q6:**3. (2pts) Find an example of a nonlinear equation, which is not solvable using the methods covered in Chapter 2, and which has y = x2 as one of its solutions.See Answer**Q7:**3. (20 points) Consider the characteristics of the following communicationsystem. a. (10 points) The following signal is used to transmit bits. What is the bit rate? Note the time scale is in microseconds. b. (10 points) A signal formatted as above is multiplied by a cosine signal andtransmitted at a center frequency of 2.405 GHz. Another signal formattedthe same is also transmitted with the following requirements. • Both signals require 95% of the original signal energy to be transmittedto have tolerable distortion. • All frequencies outside the 95% range are filtered out. • The signals must not overlap in the frequency. What is the next center frequency above this one that can be used to transmit the other signal? Show all significant digits of your answer inGHz.See Answer**Q8:**- Consider the Acemoglu, Johnson, and Robinson (2001) data on economic development and property rights from class. a) Regress log GDP per capita in 1995 (logpgp95) on the measure of property rights(avexpr) and report your results. Use hetereoskedasticity-robust standard errors. b) Construct 95% confidence intervals for the intercept and slope coefficients. c) Test Ho : B1 = 0 against Hị : B1 > 0. In answering, be sure to report the test statistic and p-value. d) Test Ho : B1 = 0.5 against H1 : B1 > 0.5. In answering, be sure to report the test statistic and p-value. e) Test Ho : B1 = 0 against H1 : B1 < 0. In answering, be sure to report the test statisticand p-value. f) Test Ho : B1 = 0 against H1 : B1 # 0. In answering, be sure to report the test statistic and p-value. g) Repeat parts b)–f) using homoskedasticity-only standard errors. Do your results change? h) Plot the regression residuals against avexpr. Does this explain your answer to g)?Hint: You will need to use the commandSee Answer**Q9:**a) Regress average test score (testscr) on average expenditure per student (expnstu) and report your results. Use heteroskedasticity-robust standard errors. b) Construct a 95% confidence interval for B1. c) Suppose you learned that test scores and expenditure per student were independent.Would you be surprised? Explain. d) Construct a 95% confidence interval for the predicted effect of increasing expenditure by $1000 per student. e) Test the null hypothesis that the predicted effect is 3 points against the alternative hypothesis that the predicted effect is different from 3 points.Hint: you can use your answer to part d).3 f) A local congressman claimed that increasing per-student expenditure by $1000 would increase average test scores by at least 3 points. Is the congressman's claim valid? Ex-plain.See Answer**Q10:**(5+5 pts) Suppose {an} converges to L. Using only the definition of the limit,show the following: (a) {an} is bounded. (b) {a} converges to L2.See Answer**Q11:**Consider the point A(3,2,-1) and the plane x+2y+2z =8. a) State a unit vector,n, orthogonal to the plane [2 marks] b) Find the distance, D, between the point and the plane [2 marks] c) Find the point on the plane, P, closest to the point A. (Hint, the vector AP is in the direction n) [4 marks] Enter your answer to part b) below stating your answer as a decimal correct to 2 d.p.See Answer**Q12:**Use the "guess" method to solve for x in the following equations. \begin{aligned} &\text { a) }\\ &x^{2}+6 x^{7}-\frac{6-x^{5 / 9}}{\left(x^{2}-1\right)}=47.6 \end{aligned} \begin{aligned} &\text { b) }\\ &\frac{\left(x^{2}+x^{5}\right)}{x^{3}}-1=x \end{aligned}See Answer**Q13:**Q-5: The following points are from a sine curve, what are the y-values for x = -2.35 by using Lagrange interpolation? Plot the data. х= (-5, -4, -3, -2, -1, 0, 1, 2, 3, 4), y =(0.958, 0.757, -0.141, -0.909, -0.841, 0.841, 0.909, 0.141,-0.757, -0.959)See Answer**Q14:**An Easter Egg can be modelled in polar coordinates as: r=\frac{16}{5-3 \sin \theta} b) Hence or otherwise sketch the ellipse indicating all intercepts with the axes and showing all working. [4 marks] \text { a) Convert this into the standard form for an ellipse in terms of } x \text { and } y \cdot \frac{\left(x-x_{c}\right)^{2}}{a^{2}}+\frac{\left(y-y_{c}\right)^{2}}{b^{2}}=1[4 \text { marks }]See Answer**Q15:**Use goal seeker function to solve the following equation. \begin{aligned} &\text { c) }\\ &\frac{6 x^{0.45}}{\sqrt{x}}-x e^{x / 2}+x^{2}=10 \end{aligned} \text { d) Find } \Phi \text { in degrees. } 0.5113=\frac{(\cos \phi)^{2}}{\pi \sin \phi}See Answer**Q16:**Problem 4: The time-dependent equations of motion for the mass-spring system illustrated below are: \begin{array}{l} \ddot{x}_{1}+\frac{\left(k_{1}+k_{2}\right)}{m_{1}} x_{1}-\frac{k_{2}}{m_{1}} x_{2}=0 \\ \ddot{x}_{2}-\frac{k_{2}}{m_{2}} x_{1}+\frac{\left(k_{2}+k_{3}\right)}{m_{2}} x_{2}-\frac{k_{3}}{m_{2}} x_{3}=0 \\ \ddot{x}_{3}-\frac{k_{3}}{m_{3}} x_{2}+\frac{\left(k_{3}+k_{4}\right)}{m_{3}} x_{3}=0 \end{array} Assuming solutions of the form x; = X; sin(@t), transform the above system into an eigenvalue problem; then, find the three natural frequencies (w1, w2, and w3) and the three eigenvectors of the system using the values \begin{array}{l} \mathbf{k}_{\mathbf{1}}=\mathbf{k}_{\mathbf{4}}=15 \mathbf{N} \mathbf{m} \\ \mathbf{k}_{\mathbf{2}}=\mathbf{k}_{\mathbf{3}}=\mathbf{3 5} \mathbf{N}_{\mathbf{m}} \\ \mathbf{m}_{\mathbf{1}}=\mathbf{m}_{\mathbf{2}}=\mathbf{m}_{\mathbf{3}}=\mathbf{m}_{\mathbf{4}}=\mathbf{2} \mathbf{k g} \end{array} You may use the "eig" function in MATLAB. Explain what the eigenvectors tell you about the directions the three masses will move when oscillating at the three different natural frequencies.See Answer**Q17:**Problem 2: Using the initial guesses of x1,0 = x2,0 = 1.2, perform two iterations of the generalized form of the Newton-Raphson method to solve for x1 and x2 in the simultaneous system of nonlinear equations below: \begin{array}{l} x_{2}=-x_{1}^{2}+x_{1}+0.5 \\ x_{2}+5 x_{1} x_{2}=x_{1}^{2} \end{array} Calculate the approximate relative percent error, Ea, for each unknown during each iteration. Place boxes around root estimates and ɛa's. Show all your steps.See Answer**Q18:**Problem 3: Use the power method to determine the highest eigenvalue and corresponding eigenvector for the matrix Show four iterations of your hand calculations, starting with an initial guess for the eigenvector {1 1 1}^T.Also, use the powereig.m M-file function discussed in class (posted on Canvas under Lecture 16) to obtain the eigenvalue within 0.01% accuracy using MATLAB.See Answer

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