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: READING TIME: |
2 hours writing 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 CALCULATORS |
YES YES |
NO NO |
Only HP 10bII+ or Casio FX82 (any suffix) calculator permitted
| SPECIFICALLY PERMITTED ITEMS if yes, items permitted are: |
YES | NO |
Candidates must complete this section if required to write answers within this paper
STUDENT ID: __ __ __ __ __ __ __ __ DESK NUMBER: __ __ __ __ __
SAMPLE EXAM
Important: There are eights questions. Please attempt all the questions, show
all the steps of your calculations, and provide explanations to justify your answers. To obtain full marks, it is important to provide complete answers supported 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 ” PPXY # = ” 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) = 13Q3 – 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 2 [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 = Z0Q0 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 and K, 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 ! 1 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 Food B |
10 mg 100 mg 40 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 + pb2 – 4ac
2a
; x2 =
-b – pb2 – 4ac
2a ; if b2 – 4ac ≥ 0
• Inverse of a 2 × 2 matrix:
A = ” a b c d # ; A-1 = ad -1 bc ” -dc a -b # (if ad – bc 6= 0)
• Arithmetic sequences (a; a + d; a + 2d; : : :):
sum of the first n(n ≥ 1) terms is Sn =
nX i
=1
Ti = n
2
[2a + (n – 1)d]
• Geometric sequences (a; ar; ar2; : : :):
sum of the first n(n ≥ 1) terms is Sn =
nX i
=1
Ti = a(1 – rn)
1 – r
• Annuities, Loans (m-payments per year)
Value at End of Year n: Vn = P0 1 + mr n + r=m A0 h1 + mr n – 1i
| 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) | f0(x) |
| Power Rule Constant Natural Exponential Exponential Logarithm Product Rule Quotient Rule Chain Rule |
xp; p 6= 0 K ex ax ln(x) u(x) · v(x) u(x) v(x) u(v(x)) |
pxp-1 0 ex ax ln(a) 1=x u0(x) · v(x) + u(x) · v0(x) u0(x)·v(x)-u(x)·v0(x) (v(x))2 u0(v(x)) v0(x) |
· • Integration Rules
| Rule | f(x) | f(x)dx |
| Power Rule One exception to power rule Integral of a constant Natural Exponential |
xp, p 6= -1 x-1 K ex |
xp+1 p+1 + C ln(x) + C Kx + C ex + C |
R • Integration by substitution
Zab f(’(t))’0(t)dt = Z’’(a()b) f(x)dx
• Integration by parts
Z u(x) · v0(x)dx = u(x)v(x) – Z v(x) · u0(x)dx
• First-order differential equation: for any constant k,
dy
dx = ky; y = Aekx; A is real
The post Mathematics for Business appeared first on My Assignment Online.