If y(x) satisfies y(x) ex + xy(x) = 1, find y(x). If the position r of a particle P is given by. r = cos(2t)i + sin(2t)j + 4j, where i and j are two unit vectors along the x-axis and y-axis, respectively. What is the velocity (vector) of this particle? What is its acceleration at t = /2? In the preceding question, show that the path (or orbit) of the particle (P) is a circle. Where is the centre of the orbit? In the 3D coordinates, what is the rotation matrix for rotating an angle o about the y- axis? Express it as a 3 by 3 matrix and explain briefly. 50. Using the Taylor expansions, show that In(1 + x) = x where x > -1. 51. The exponential distribution has a probability density function. p(x) = de-2x, x 0. Find its expectation or mean 23 3 E(X)= fxp(x) dx 52. What is the total impedance of the AC circuit shown in Fig. 2? [Hint: Use complex number representations with the AC frequency w.] Vo 13 R C2 C1 Figure 2: A linear circuit model for one resistor and two capacitors. 53. A student tries to generate 1000 pseudo-random numbers using a recurrence relation Xn+1 = (axa + b) mod m, where abm are integers. The 'mod' is to find the reminder after division. For example, 5 mod 3 = 2. The student's code is given below, but it does run correctly. % Initial value % a % b % m for n=1:1000; d=a*x[n]+b; x(1)=1; a=8121 b=28411, m=134456; x[n+1]=d-floor(d/m)*m; end % Plotting plot(x) First correct the mistakes and try to add a histogram to see what probability distribution this method actually generates. Does it matter if x = 0 is used (in stead of xo = 1)? Can you generate 10 000 random numbers in the range of [0, 4] using this method?