I need a full solutions to page 2 linear programming questions 1 (a) and (b)
School of Mathematics Level I Semester 1 & 2RCA Week 12 Supplementary Assessment Instructions • Answer all questions for modules for which you are registered and which appear in this booklet. • You have a maximum period of time to complete and upload your solutions of 12+ 12q hours, where q is the number of full questions that you are required to complete. Any solutions submitted after your personal deadline will not be marked. • This assessment is not intended to need the full time period you have available; the time period indicates only the timescale within which you need to complete this assessment. The available time period automatically takes into account an adjustment for those stu- dents with Reasonable Adjustment Plans (RAPs) or individual personal circumstances. • Solutions must be neatly handwritten; typed solutions will not be marked. • Make sure that you read and follow the instructions for each module, in particular the page limit for your answers. Solutions will only be marked up to the specified page limit. • Solutions to each question should be uploaded to Canvas individually as a single PDF document and may be submitted at any point within the time period available. Solutions may be re-uploaded, however incorrectly submitted solutions, i.e. uploaded to the wrong question, may not be marked. It is your responsibility to ensure that your solutions have successfully uploaded to Canvas by your personal deadline. • Whilst you may use your notes and other materials available to you, submitted answers should be your own. The School of Mathematics has enhanced processes in place to identify any instances of plagiarism or collusion. There is no reduction in the threshold for plagiarism associated with this assessment. Appropriate University penalties will be applied to any individual found to be involved in any way with an act of plagiarism or collusion. • When your answers to each question are graded the marker will take into account a num- ber of factors to determine your awarded grade including: whether your answers are fully correct to all aspects of the question; the overall level of mathematical precision and rigor demonstrated in your arguments; the appropriateness of your mathematical explanation; and, your overall presentation of the mathematical material. The index uses hyperlinks to aid navigation and on each page there is a “Back to index” link. Page 2 Turn over Index 1 LI Algebra & Combinatorics 1 2 LI Linear Algebra & Linear Programming 3 LI Multivariable & Vector Analysis 4 LI Real & Complex Analysis 1AC2 06 27363 Level I LI Algebra & Combinatorics 1 Full marks may be obtained with complete answers to BOTH the following questions. Each answer must be no more than THREE sides of A4, any work in excess of this will not be marked. 1. (a) (i) Find y,z ∈ Z such that 44y+34z = 2. (ii) Solve the simultaneous congruences x≡ 7 mod 34 x≡ 19 mod 44 giving your solution in the form x≡ s mod n, where s ∈ Z and n ∈ N. You should give sufficient explanation to justify that you have found all solutions in (a)(ii). (b) Let f = (14263) ∈ S6 and g = (13562) ∈ S6. (i) Calculate f ◦g. (ii) Find h ∈ S6 such that f ◦h = g. (c) Let f ∈ S5. (i) Suppose that o( f ) is even. Prove that o( f ) | 12. (ii) Suppose that f 4 = id. Prove that there exists a ∈ {1,2,3,4,5} such that f (a) = a. 2. (a) Draw a connected graph G with degree sequence (1,2,2,2,2,3). (b) How many copies of C5 are there in K10? (c) How many copies of G are there in K10 (where G is the graph you drew in part (a))? (d) Let F be a set of subsets of N such that for every F,F ′ ∈ F with F 6= F ′ we have |F ∩F ′| ≤ 2. Prove that F is countable. End of LI Algebra & Combinatorics 1 Page 1 Back to the index 2LALP 06 25765 Level I LI Linear Algebra & Linear Programming Full marks may be obtained with a complete answer to the following question. Your answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. (a) Consider the following linear programming problem: (P) Maximise 7x1− x2 +10x4, subject to x1 +3x2 + x3 = 1, −x2 +4x3−2x4 ≥−10, 5x1−2x2− x4 ≤ 8, x1 ≥ 0, x2 ≥ 0, x4 ≥ 0, x3 ∈ R. i) Transform the problem (P) to a standard form. ii) Write down the dual problem to the problem (P). iii) Write down the optimality conditions to the problem (P). iv) Show that xT = (0,0,1,0,0,14,8) is a BFS of the standard form problem. (b) In the middle ages all commercial transactions took place at the market. There were 10 merchants with a market stall each that supply food to 8 cities. For a while they competed with each other, then they decided to work together and get organised in order to meet market requirements at minimum travel cost. Each merchant posses a certain amount of food, say mi, and each city needs an amount of food n j. Let ci j denote the cost to transport the food from the merchant i’s store to the city j. Set this up as a LP problem but do not solve. End of LI Linear Algebra & Linear Programming Page 2 Back to the index 2MVA 06 25667 Level I LI Multivariable & Vector Analysis Full marks may be obtained with a complete answer to the following question. Your answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. Let C1 be the planar curve given by ~r(t) = (cos(2t)+2t sin(2t),sin(2t)−2t cos(2t)) , for t > 0. (1) (a) Find the unit tangent vector T̂ (t) to C1. (b) Find the unit normal vector n̂(t) to C1. (c) Sketch the portion of the curve C1 given by the parametrisation (1) when t ∈ (0,5π]. Now consider the vector field ~F : R2→ R2, ~F(x,y) = (x+ y,y), and let C2 be the closed curve in the xy-plane formed by the triangle with vertices at the origin and the points (1,0) and (0,1). (d) Give a sketch of the vector field ~F in the first quadrant, and use it to predict whether the net flux out of the region R := the interior of C2 will be positive or negative. (e) Compute the flux integral ∮ C2 ~F · n̂ds using the appropriate double integral. (Specify which orientation you are using for C2.) End of LI Multivariable & Vector Analysis Page 3 Back to the index 2RCA 06 25666 Level I LI Real & Complex Analysis Full marks may be obtained with a complete answer to the following question. Your answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. (a) Suppose that f is an analytic function on the unit disk D = {z ∈ C : |z| < 1} such that re f (z) = 2021 for all z in d. using the cauchy–riemann equations, show that f is constant in d. (b) compute the integral ∫ γ f (z)dz, where f (z) = ez 2 z2−6z , and (b1) γ = {z ∈ c : |z−2|= 1}, (b2) γ = {z ∈ c : |z−2|= 3}, (b3) γ = {z ∈ c : |z−2|= 6}. (c1) find all complex solutions of the equation sin(z) = 2. (c2) determine and classify all singularities of the function f (z) = 1 2− sin(z) . end of li real & complex analysis page 4 back to the index 1}="" such="" that="" re="" f="" (z)="2021" for="" all="" z="" in="" d.="" using="" the="" cauchy–riemann="" equations,="" show="" that="" f="" is="" constant="" in="" d.="" (b)="" compute="" the="" integral="" ∫="" γ="" f="" (z)dz,="" where="" f="" (z)="ez" 2="" z2−6z="" ,="" and="" (b1)="" γ="{z" ∈="" c="" :="" |z−2|="1}," (b2)="" γ="{z" ∈="" c="" :="" |z−2|="3}," (b3)="" γ="{z" ∈="" c="" :="" |z−2|="6}." (c1)="" find="" all="" complex="" solutions="" of="" the="" equation="" sin(z)="2." (c2)="" determine="" and="" classify="" all="" singularities="" of="" the="" function="" f="" (z)="1" 2−="" sin(z)="" .="" end="" of="" li="" real="" &="" complex="" analysis="" page="" 4="" back="" to="" the="">