Advanced Math

Question: Solve the following LPP using Big M Method. Each carries 15 marks. Problem -1 Minimize Z = 7×1 + 15×2 + 20×3 Subject to 2×1+4×2+6×3 >=24 3×1 + 9×2 + 6×3 >=30 X1, X2, X3 >=0 Problem 2: Maximize Z = 6×1 + 4×2 Subject to: 2×1+3×2<=30 3x1+2x2<=24 X1+x2>=3 x1, x2>=0 Marking Scheme: 15 Marks for each question: (5 marks for converting LPP to standard form; 2 marks for initial simplex table; 6 marks@ 3 marks for each iteration process and 1 mark for the optimal value and 1 mark for final solution)

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(20 points) You are given an ordered sequence of n cities, as well as the distances between each pair of cities. Your goal is partition the cities into two subsequences (not necessarily contiguous) such that person A visits all cities in the first subsequence (in order), person B visits al cities in the second subsequence (in order), and such that the sum of the total distances travelled by A and B is minimized. Assume that person A and person B start initially at the first city in their respective subsequences

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Consider the following closed-loop system R(s) Y(s) Gc(s) G(s) H(s) K Ge(s) = K G(s) = 0.5 s(s + 2) H(s) = K2s +1 a) [2 Marks] Determine the transfer function and the characteristic equation of the closed-loop system in terms of the gains Kį and K2. Show your work. = 0.05 and the peak time b) [5 Marks] Determine the gains Kį and K, so that the unit-step response has a maximum overshoot of Mp = 0.5sec. Show your work and justify your answer. of tp c) [3 Marks] Determine the rise-time (first-order approximation) (tr) and the settling-time (2% criteria) (ts) of the unit-step response. Show your work. d) [5 Marks] Assume that K1 = 32, determine the gain K2 to achieve the fastest unit-step response without overshoot. Find the characteristic equation and poles location of the closed-loop system for these values of Kį and K2. Show your work and justify your answer.

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Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (x + y)y’ = x – y 2xyy’ = x^2 + 2y^2 xy’ = y + 2 squareroot xy (x – y)y’ = x + y x(x + y)y’ = y(x – y) (x + 2y)y’ = y xy^2y’ = x^3 + y^3 x^2y’ = xy + x^2 e^y/x x^2y’ = xy + y^2 xyy’ = x^2 + 3y^2 (x^2 – y^2)y’ = 2xy xyy’ = y^2 + x squareroot 4x^2 + y^2 xy’ = y + squareroot x^2 + y^2 yy’ + x = squareroot x^2 + y^2 x(x + y)y’ + y(3x + y) = 0 y’ = squareroot x + y + 1 y’ = (4x + y)^2 (x + y)y’ = 1 x^2y’ + 2xy = 5y^3 y^2y’ + 2xy^3 = 6x y’ = y + y^3 x^2y’ + 2xy = 5y^4 xy’ + 6y = 3xy^4/3 2xy’ + y^3e^-2x = 2xy y^2(xy’ + y)(1 + x^4)^1/2 = x 3y^2y’ + y^3 = e^-x 3xy^2y’ = 3x^4 + y^3 xe^yy’ = 2(e^y + x^3e^2x) (2x sin y cos y)y’ = 4x^2 + sin^2 y (x + e^y)y’ = xe^-y – 1

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1.6 Problems Find general solutions of the differential equations in Prob. lems 1 through 30. Primes denote derivatives with respect to x throughout. 1. (x + y) = x – y 2. 2xyy’ = x2 + 2y2 3. xy = y + 2xy 4. (x – y)y’ = x + y 5. x(x + y)y’ = y(x – y) 6. (x + 2y)y’ = y 7. xy2y’ = x + y 3 8. x2y = xy+x2e9x 9. x2y = xy+y? 10. xyy’ = x2 + 3y2 11. (x2 – y2)y’ = 2xy 12. xyy’ = y2 + x4x2 + y2 13. xy’ = y + x2 + y2 14. yy’ + x = x2 + y2 15. x(x + y)y’ + y(3x + y) = 0 16. y’ = (x + y +1 17. y’ = (4x + y)2 18. (x + y)y’ = 1 19. xºy’ + 2xy = 5y3 20. y2y + 2xy3 = 6x 21. y’= y + y3 22. xy’ + 2xy = 5y4 23. xy’ + 6y = 3x y 4/3 24. 2xy + ye-2x = 2xy 25. y (xy+y)(1+x) 1/2 = x 26. 3y2y’ + y3 = e-* 27. 3.xy?y’ = 3×4 + y3 28. xe) y = 2(e) +x22x) 29. (2x sin y cos y)y’ = 4×2 + sin? y 30. (x + 2) y = xe-y – 1

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(10 marks) Betty is considering two mobile phone plans. Plan 1 would cost $18 per month and would give her 1000 minutes per month of free local calls. Extra local calls would cost $0.15 per minute, and any long-distance calls within the country would cost $0.30 per minute. Plan 2 would cost $75 per month, and would allow a month of unlimited calling within the country. Suppose that she wishes to make a minutes of local calls and b minutes of in-country long-distance calls per month. (a) Using a max function write an expression for the total cost of Plan 1 in terms of a and b . (b) Find the values for a and b that would make Plan 1 better (i.e. cheaper) than Plan 2. (c) Based on the solution for (b), show the region on the graph below where Plan 1 is better than Plan 2

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