Final Examination Mathematics - Toán cao cấp FTU

FOREIGN TRADE UNIVERSITY MATHEMATICS DEPARTMENT HS-Onl-Mid.21-22 FINAL EXAMINATION MATHEMATICS I Semester : I Year: 2021 – 2022 Class : Exam date: Start time: Time duration: 60 minutes Question 1: An economy depends on two basic products: wheat and oil. To produce 1 metric ton of wheat requires 0.25 metric ton of wheat and 0.33 metric ton of oil. The production of 1 metric ton of oil consumes 0.08 metric ton of wheat and 0.11 metric ton of oil. Find the production that will satisfy a demand of 500 metric tons of wheat and 1000 metric tons of oil. Question 2: a) A = ⎡ 1 -2 -5 ⎤ ; B = ⎡ 2 1 -9 ⎤ ; C = 2A – 3B; D = Aᵀ B. ⎣ -6 -10 4 ⎦ ⎣0 7 4⎦ Find C and D-1 (if the inverse matrix exists; if not, compute det(D)). Write the final results. Excel is allowed to be used. b) Provided a set of vectors S = {x₁ = (-1, 2, -3); x₂ = (0, -2, 5); x₃ = (1, 0, 2)}. Find Dim(span(S)). Question 3: Solve the following linear programming problem graphically Maximize z=x+y Subject to 2x + y ≥ 3 x + 2y ≥ 3 x+y≤6 x ≥ 0, y ≥ 0. Question 4: Given the linear programming problem that could be solved by the simplex method. Minimize z = x₁ + 2x₂ + 4x₃ Subject to x₁ + x₂ ≤ 10 x₁ + 2x₂ + x₃ = 14 x₁ ≥ 0, x₂ ≥ 0, x₃ free. Use Excel Solver to solve the problem. Write the final results. Question 5: Determine the linear independence of the set of following vectors a₁ = ⎡ 1 ⎤ ; a₂ = ⎡ 0 ⎤ ; a₃ = ⎡ 5 ⎤ ; a₄ = ⎡ 2 ⎤ . ⎢ -3 ⎥ ⎢ -3 ⎥ ⎢ -1 ⎥ ⎢ -1 ⎥ ⎣ 0⎦ ⎣ 2⎦ ⎣ 5⎦ ⎣ 5⎦