Consider the system of independent components connected as in the figure below. The system works only if there is a path of functional components from left to right. Components 1 and 2 are connected in parallel, so that subsystem works if and only if either 1 or 2 works. Since 3 and 4 are connected in series, that subsystem works if and only if both 3 and 4 work. Suppose that P(1 works) = 0.73, P(2 works) = 0.76, P(3 works) = 0.83, and P(4 works) = 0.87, and these events are mutually independent. 1 2 (a) Find the probability that the system works. (b) Find the probability that component 1 works, given that the system works.

4. Use the SOR method with w = 1.2 to solve the linear systems 4x1 + x2 + X3+ Xs = 6, -x – 3x2 + x3 + x4 = 6, 2x, + x2 + 5xz – X4 - x5 = 6, -XI - x2 - X3 + 4x4 = 6, 2x2 - x3 + xX4 +4x3 = 6. with a tolerance TOL = 10-3 in the loo norm.

Solve the following system using the Gaussian algorithm: 2x1 - x2 + 3x3 = 0 -2x1 + x2 - X3 = -2 x1 - 3x2 + 6x3 = 3 Let x be the vector x = |x2, where x1, x2 and x3 are solutions of the system, and y be the vector y =|1 Evaluate the scalar product of x and y. [Please enter your answer numerically in decimal format. You will be marked correct as long as what you enter is within 0.25 of the correct answer. So for example, if the correct answer is 6.78 then any input that lies between between 6.53 and 7.03 will be marked as correct.] Answer:

Solve the following system using the Gaussian algorithm: 2x1 - x2 + 3x3 = 3 -2x1 + x2 – x3 = 0 x1 - 3x2 + 6x3 = 3 Let x be the vector x = x2l, where x1, x2 and x3 are solutions of the system, and y be the vector y = 1. Evaluate the scalar product of x and y. 0.