Office Use Only Semester One 2019 Examination Period Faculty of Business and Economics EXAM CODES: ETF2700-ETF5970 TITLE OF PAPER: Mathematics for Business – SAMPLE EXAM EXAM DURATION: 2 hours writing time READING TIME: 10 minutes THIS PAPER IS FOR STUDENTS STUDYING AT: (tick where applicable) Caulfield Clayton Parkville Peninsula Monash Extension Off Campus Learning Malaysia Sth Africa Other (specify) During an exam, you must not have in your possession any item/material that has not been authorised for your exam. This includes books, notes, paper, electronic device/s, mobile phone, smart watch/device, calculator, pencil case, or writing on any part of your body. Any authorised items are listed below. Items/materials on your desk, chair, in your clothing or otherwise on your person will be deemed to be in your possession. No examination materials are to be removed from the room. This includes retaining, copying, memorising or noting down content of exam material for personal use or to share with any other person by any means following your exam. Failure to comply with the above instructions, or attempting to cheat or cheating in an exam is a discipline offence under Part 7 of the Monash University (Council) Regulations, or a breach of instructions under Part 3 of the Monash University (Academic Board) Regulations. AUTHORISED MATERIALS OPEN BOOK YES NO CALCULATORS YES NO Only HP 10bII+ or Casio FX82 (any suffix) calculator permitted SPECIFICALLY PERMITTED ITEMS YES NO if yes, items permitted are: Candidates must complete this section if required to write answers within this paper STUDENT ID: __ __ __ __ __ __ __ __ DESK NUMBER: __ __ __ __ __ MP LE EX AM Important: There are eights questions. Please attempt all the questions, show all the steps of your calculations, and provide explanations to justify your an- swers. To obtain full marks, it is important to provide complete answers sup- ported by logically sound explanations, unless the question explicitly states that no explanation is needed. It is not sufficient to simply provide calculator instructions. Total marks: 60 [This is a sample exam, which is NOT part of the unit assessment. The final examination is worth 60% towards the final mark of this unit]. Formulae are provided at the end of this paper. Question 1: 3+2+5=10 Marks The demand and supply functions for two goods X and Y are given below. Demand: QX = 120− 2PX + 3PY , QY = 150 + 6PX − 4PY Supply: QX = −240 + 6PX , QY = −150 + 6PY (a) Determine the matrix A in the equation of the form A [ PX PY ] = [ 360 300 ] which determine market equilibrium. (b) Show that A is invertible, by verifying that det(A) 6= 0. (c) Using A−1, obtain the prices of the two goods at market equilibrium. Show your working and provide answers correct to 2 decimal places. Question 2: 2+3+3+2=10 Marks A monopolist faces a demand function P = 173− 2Q, with P denoting the market price and Q denoting the quantity demanded. There is a fixed cost of 200, the total variable cost function is TV C(Q) = 1 3 Q3 − 10Q2 + 188Q. (a) Obtain an expression for the profit function f(Q) defined for Q > 0. (b) Determine the stationary point(s) of the profit function. (c) Assume the maximal point of the profit function exist. What is the maximal value of the profit function? (d) Compute the elasticity ElQf(Q) at point Q = 10, to 2 decimal places. Provide a brief interpretation of the computed value of the price elasticity, in the context of this example. Question 3: 5 Marks Suppose the market price P can be written as a function of the market demand quantity Q ∈ [0, 5) P (Q) = 100− 10Q. The current demand quantity is Q0 = 2. Showing all steps of your working, evaluate the consumer surplus in this market CS = ∫ Q0 0 P (Q)dQ− P0Q0 where Q0 is the current market demand quantity. Question 4: 3+1+3+1=8 Marks Energy company GreenVolt (GV) owns a property at the wind-swept, sunny location of Ocean Heads. GV is evaluating two projects: a wind farm and a solar energy plant. The wind farm requires an initial investment of $10m, and a $5m loss is expected for the first year. For the following 3 years, GV expects annual returns of $8m from electricity sales. The solar plant requires an initial investment of $15m. GV expects a loss of $2m for the first year and annual returns of $9m for the following 3 years. Assume a discount (interest) rate of 8% compounded annually. (a) Calculate, in $m to 3 decimal places, the present value of the two projects. (b) Based on the present values that you have carried out in (a), explain which one you think is preferable. (c) Calculate, in percentage to 3 decimal places, the internal rate of return of the two projects. (d) On the basis of the internal rate of return of the two projects, which project do you think is preferable? Question 5: 2+2+3+1=8 Marks Given the Cobb-Douglas production function Q = 100L0.3K, L,K > 0. (a) Write down the equation of the isoquant for Q = 800 in the form K = f(L) (b) Show by differentiation that f(L) is convex. (c) Find the values of L andK, to 2 decimal places, for which the production is maximised under the budget restriction L+ 2K = 30 using Lagrange method. (d) If the budget increases by 1 (that is, increases from 30 to 31), compute the resulting change (rounded off to 3 decimal places) in the maximal level of production, using the Lagrange multiplier method. Question 6: 1+4+1=6 Marks The following difference equation models the salary scale for part-time staff Yt = 20 + 1.2Yt−1 where Yt denotes the salary (in dollars) in year t = 0, 1, 2, . . .. (a) If Y0 = 2300, deduce Y1, Y2 and Y3 directly from the difference equation. (b) Solve the difference equation. In other words, determine Yt for all t = 0, 1, . . . . (c) Calculate the number of year until the salary first exceeds $15,000. Question 7: 4+2=6 Marks Suppose that a firm’s capital stock K(t) satisfies the differential equation dK dt = I − δK(t) where investment I is constant, and δK(t) denotes depreciation, with δ a positive constant (a) Find the solution of the equation if the capital stock at time t = 0 is K0. (b) Let δ = 0.05 and I = 10. Explain what happens as t → ∞ when: (i) K0 = 150; (ii) K0 = 250. Question 8: 2+5=7 Marks A daily diet mix requires a minimum of: 160 mg of Vitamin K and 1000 mg of Vitamin D. Two foods A and B contain these vitamins: Vitamin K Vitamin D Cost per 1 kg Food A 10 mg 100 mg 40 Food B 8 mg 40 mg 20 (a) Write down the inequality constraints for each vitamin. Denote the consumption (in kg) of food A and food B as x and y. (b) Using extreme point theorem, determine the number of units of food A and B which fulfil the daily requirements at a minimum cost. You may assume the optimal point exist. END OF EXAMINATION Formulae are provided in the next two pages. Formulae • ‘abc’ formula for quadratic equation ax2 + bx+ c = 0, a 6= 0: x1 = −b+√b2 − 4ac 2a , x2 = −b−√b2 − 4ac 2a , if b2 − 4ac ≥ 0 • Inverse of a 2× 2 matrix: A = [ a b c d ] , A−1 = 1 ad− bc [ d −b −c a ] (if ad− bc 6= 0) • Arithmetic sequences (a, a+ d, a+ 2d, . . .): sum of the first n(n ≥ 1) terms is Sn = n∑ i=1 Ti = n 2 [2a+ (n− 1)d] • Geometric sequences (a, ar, ar2, . . .): sum of the first n(n ≥ 1) terms is Sn = n∑ i=1 Ti = a(1− rn) 1− r • Annuities, Loans (m-payments per year) Value at End of Year n: Vn = P0 ( 1 + r m )n + A0 r/m [( 1 + r m )n − 1 ] Debt Repayments: A0 = L · r/m 1− (1 + r/m)−n , Net Present Value: V0 = A0 1− (1 + r/m)−n r/m • Differentiation Rules Rule f(x) f ′(x) Power Rule xp, p 6= 0 pxp−1 Constant K 0 Natural Exponential ex ex Exponential ax ax ln(a) Logarithm ln(x) 1/x Product Rule u(x) · v(x) u′(x) · v(x) + u(x) · v′(x) Quotient Rule u(x) v(x) u′(x)·v(x)−u(x)·v′(x) (v(x))2 Chain Rule u(v(x)) u′(v(x)) · v′(x) • Integration Rules Rule f(x) ∫ f(x)dx Power Rule xp, p 6= −1 xp+1 p+1 + C One exception to power rule x−1 ln(x) + C Integral of a constant K Kx+ C Natural Exponential ex ex + C • Integration by substitution∫ b a f(ϕ(t))ϕ′(t)dt = ∫ ϕ(b) ϕ(a) f(x)dx • Integration by parts∫ u(x) · v′(x)dx = u(x)v(x)− ∫ v(x) · u′(x)dx • First-order differential equation: for any constant k, dy dx = ky, y = Aekx, A is real
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