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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: \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
Q3: 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
Q4: 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
Q5: 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
Q6: - 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
Q7: 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
Q8: 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
Q9: 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
Q10: 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
Q11: 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
Q12: 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
Q13: Problem 1: Use the Gauss-Seidel iterative method to solve the following system until the percent relative error fallsbelow ɛ, = 5%. \begin{array}{c} 10 x_{1}+2 x_{2}-x_{3}=27 \\ -3 x_{1}-6 x_{2}+2 x_{3}=-615 \\ x_{1}+x_{2}+5 x_{3}=-215 \end{array} Show all the steps of your hand calculations. Verify if the above linear system is diagonally dominant.Also, use the GaussSeidel.m M-file function discussed in class (posted on Canvas under Lecture 13) to solve the solution of the same problem using MATLAB.See Answer
Q14: a) Find anorthogonal matrix P and a diagonal matrix D such that P D P^{T}=\left(\begin{array}{rrr} 2 & 2 & -2 \\ 2 & 5 & -4 \\ -2 & -4 & 5 \end{array}\right) b) Use your answer to part (a) to write the quadratic form Q(x,y, z) as a sum of onlysquare terms, where Q(x, y, z)=2 x^{2}+5 y^{2}+5 z^{2}+4 x y-4 x z-8 y zSee Answer
Q15: 8) Solve the following quadratic equations using the indicated method; a) Solve 3x^2 – 4x – 4 = 0, using factoring. b) Solve 3x^2 + 6x + 2 = 0, using the quadratic formulaSee Answer
Q16: Find a 2 x 2 matrix that is both Hermitian and unitary, and whose entries are not allreal numbers.See Answer
Q17: 1. a) In general, given the quadratic equation ax +bx +c = 0, find an expression (formula) for: a) the sum of the roots b) the product of the roots b) How does this rule compare to the general rule you developed in Part I?See Answer
Q18: a) Given a quadratic with{2/5} a double root, determine the equation. Show your work.See Answer
Q19: If -6 is one root of the equation 4x – 3x + c =0, find c and then find the other root.See Answer
Q20: Problem 1: As we saw in class, a linear, second order, homogeneous differential equation has the form a_{2}(x) \frac{d^{2} y}{d x^{2}}+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y=0 where y is a function of x that is at least twice differentiable and a2(x), a1(x), and ao(x) are functions of x. We will define a function L which takes as inputs (at least) twice differentiable functions y as follows: 1. Choose functions a2(x), a1(x), and ao(x). 2. Write L=a_{2}(x) \frac{d^{2}}{d x}+a_{1}(x) \frac{d}{d x}+a_{0}(x) 3. Define L(y)as L(y)=\left(a_{2}(x) \frac{d^{2}}{d x}+a_{1}(x) \frac{d}{d x}+a_{0}(x)\right) y =a_{2}(x) \frac{d^{2} y}{d x^{2}}+a_{1}(x) \frac{d y}{d x}+a_{0}(x) ySee Answer

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