Download A selection of past exam questions

Sample Exam Questions
A manufacturer has observed last year that the annual sales of a particular
product were 3000 units when it was retailed at a unit price of £300. In a previous
year annual sales were only 1000 units when the unit retail price was £600.
i)
Find a linear function for annual sales volume in terms of price
and forecast the annual sales volume for a unit price of £300.
ii)
Write down an expression for annual revenue in terms of price.
What is the mathematical name for this type of function?
iii)
What range of prices generates revenue, what price maximises
revenue and what is the maximum revenue that can obtained for
this product?
iv)
Assuming that the manufacturer can gear production to match
demand what production level maximises annual revenue?
v)
Sketch a graph of annual revenue against production and indicate
the features.
Pineworld PLC manufacture and sell handcrafted pine furniture direct to the
public through numerous retail outlets throughout the UK. The firm's weekly
revenue function for one particular product, 'the standard dining chair' is given
by:
R = -0.25q2 + 150q
where R is revenue (£'s) and q is output (units)
Weekly fixed costs for this product are £4096 whilst unit production costs are
£70.
For this particular product:
i)
Write down the firm's weekly profit function in terms q.
What is the mathematical name for this type of function?
ii)
What range of output generates profits for the firm?
What output level maximises profits for the firm?
iii)
Find an expression for the firm's average costs per unit.
What is the mathematical name for this type of function?
iv)
What happens to average costs when: output is very small
and when output is very large?
v)
Sketch graphs of revenue, cost, profit and average costs against
output on the same set of axes and indicate their features. From
your graph or otherwise find the output level when average cost
per unit equals total cost.
(i)
Find the INVERSE of the matrix:
2
1
2
-1
3
3
1
2
3
(ii)
Check that your inverse is correct
(iii)
Use your inverse from part (i) to solve the following system of
equations:
2x
x
2x
- y + z=4
+ 3y + 2z = 5
+ 3y + 3z = 8
An internet advertising agency wants to model memory recall of adverts (M)
in terms of time elapsed since viewing the advert (t). The agency has
conducted an experiment in which a random sample of the population were
asked to watch internet adverts and then were asked to recall them after a
certain time. The agency discovered that after 30 minutes of viewing adverts
the subjects only recalled 60% of the adverts whilst after 60 minutes only 40%
of adverts were recalled.
i)
Find a hyperbolic model for memory recall (M) in terms
time elapsed since viewing the advert (t)
ii)
What does your model predict for recall after 2 hours?
iii)
What percentage of adverts, according to your model,
are held in long term memory?
iv)
Sketch a graph of your model and indicate the features.
Market research suggests that the potential market for a particular product is
800,000 units. Market penetration had reached 50% by the end of the first
year and reached 75 % at the end of the second year.
i)
Find a model for sales (S) over time (t) in years of the form:
S = a + bect
ii)
What is the mathematical name for this type of function?
iii)
Use your model in part (i) to forecast sales after 3 years
iv)
Sketch a graph of sales over time and indicate the features
Weekly sales of 350 units of a particular product have been observed when the
product is priced at £40 but a unit price of £70 weekly sales are 200 units.
(i)
Find a linear model for weekly sales in terms of price and
forecast weekly sales at a unit price of £60.
(ii)
Find an expression for weekly sales revenue in terms of
price for a manufacturer of this product. What is the
mathematical name for this revenue function?
(iii)
What range of prices generates revenue, what price
maximises revenue and what is the maximum revenue
that the manufacturer in part (ii) can achieve?
(iv)
Assuming that the manufacturer can gear production
to meet, what level of production maximises revenue?
(v)
Sketch a graph of weekly revenue against production
level and indicate the features.
A company manufacture a product having fixed costs of £1600 and unit
production costs of £50 and have a revenue function given by:
R = -q(q – 150) where R is revenue (£’s) and q is production output in units
(i) Write down the company’s profit function in terms of q
What is the mathematical name for this type of function?
(ii) What range of production output generates profits and
What level of output maximises profits?
(iii) Write down an expression for average costs per unit.
What is the mathematical name for this type of function?
(iv) What happens to average costs when output is very:
(a) small and (b) large?
(v) Sketch graphs of: revenue, cost, profit and average cost and indicate
their features. When does average cost equal total cost?
Find the inverse of the matrix:
3
2
3
0
4
4
2
3
4
(i)
Check that your inverse is correct
(ii)
Use the inverse matrix to solve the system of equations:
3x +
2x +
3x +
2y
4y
4y
+
+
3z
4z
=
=
=
12
17
22
When a product is priced at £10 per unit demand has been observed to be 900
units whilst at a unit price of £20 demand of 500 units has been observed.
(i)
Find a hyperbolic model for demand in terms of price.
(ii)
What does you model in part (i) predict for a unit price of £40?
(iii)
What is the minimum level of demand predicted by your model?
(iv)
Sketch a graph of the demand function and indicate the features.
Market research for a particular product suggests that the potential market has a
ceiling of 20 million units and that no sales will be achieved without marketing.
After spending £½ million on marketing 10 million were sold.
(i)
(ii)
Find a model for sales (S) in terms of advertising expenditure (x)
of the form:
S = a + be-cx
What is the mathematical name of the function in part (i)?
(iii)
Use you model in part (i) to forecast sales when £1million is
spent on marketing.
(iv)
Sketch a graph of the model found in part (i).
A firm sells 400 units per week when it prices a particular product at £25 but
sells only 300 units per week when the price is £30.
(a)
Find a linear function for quantity sold in terms of price.
(b)
Algebraically rearrange your expression in part (a) to give price in
terms of quantity sold.
(c)
Assume that the firm can gear production to meet demand. Write
down an expression for the revenue of the firm in terms of quantity
produced. What range of production generates revenue for the firm ?
What production level maximises the revenue of the firm ?
(d)
Assume that for this product the firm has fixed costs of production
of £660 and unit production costs of £20. Write down a linear cost
function in terms of quantity produced.
(e)
From parts (c) and (d) write down an expression for profit in terms
of production. What range of production generates profit ? What
production level maximises profit ?
(f)
Sketch graphs of revenue, cost and profit on the same set of axes and
indicate the features. What are the mathematical names of the
revenue
and profit functions ?
(a)
Use ‘pivoting’ to find the INVERSE of the matrix:
1
2
3
(b)
3
3
1
Three products X, Y and Z are to be processed through a production
system having three machines: A, B and C. The unit processing times
in hours of each product on each machine and the total production
time
available this week are given below. Using your result from part (a)
find the quantities of the three products X, Y and Z that can be
produced this week utilising all the available machine time.
Machine A
Machine B
Machine C
(c)
2
1
3
Unit production time Total available
X
Y
Z
time (hours)
1
2
3
15
2
1
3
15
3
3
1
20
What would be your answer to part (b) if this week the machines had
the following hours available:
Machine A
15
Machine B
15
Machine C
30
At the end of each of the last three years a firm has recorded the total output
and total production costs for a particular product. These data are:
Year
1999
2000
2001
Output
(1,000’s units)
3
4
5
Total Costs
(£1,000’s)
700
500
400
(a)
Find an exact fitting polynomial model for Total Costs in terms of
output. What is the mathematical name of this function?
(b)
What are the firm’s fixed costs of production ? If output in 1995
was 6,00 units what Total Costs would your model predict ?
(c)
Use your result from part (b) to find the output level that minimises
Total Costs.
(d)
Write down an expression for AVERAGE COSTS per year in terms
of output. What is the mathematical name of this function ?
(e)
Sketch graphs of Total Costs and Average Costs on the same set of
axes and indicate their important features.
(f)
What output level minimises Average costs?
At what output level are Total and Average costs the same ?
A European defence consortium has orders for 100 new fighter planes
guaranteed by their host governments before production commences and
never expects to have total sales in excess of 500 planes. The companies
in the consortium operate 50 weeks per year. After 2 years (100 weeks)
production and marketing the consortium have made a total of 200 sales.
(a)
Find a hyperbolic model for Total Sales over time in weeks.
(b)
What level of Total Sales does your model in part (a) predict
after 6 years of operation ?
(c)
How many years would your model in (a) predict for Total
Sales to reach 400 planes ?
(d)
Sketch a graph of Total Sales over time and indicate its features.
(e)
What other mathematical function might you suggest as a
suitable model for Total Sales over time ?
A firm manufactures and sells q units of a product at price = £(575 – ½ q)
which has unit costs of £(q2 – 25q) and fixed costs of £45,000.
(a)
Write down expressions for: revenue, profit and average cost
in terms of output(q) of the firm.
(b)
Find expressions for: marginal revenue, marginal cost, marginal
profit and marginal average costs in terms of output (q).
(c)
Find the output levels of the firm that:
(i)
(ii)
(iii)
(iv)
maximise revenue
minimise costs
maximise profits
minimise average costs
and confirm that the output levels found do indeed maximise
or minimise these functions
The demand function for a product is given by: Demand = 50 + 200
(p – 5)
(a)
Calculate the ARC elasticity of demand with respect to price between:
(i)
(ii)
(iii)
p = 6 and p = 15
p = 6 and p = 25
p = 15 and p = 25
and in each case comment on whether demand is elastic or inelastic
(b)
Find an expression for POINT elasticity of demand with respect
to price in terms of price.
(c)
Calculate POINT elasticity of demand at p = 6 and p = 25 and
comment on their values and on the relationship between ARC
and POINT elasticity
(d)
What price(s) give unit POINT elasticity?
The end-of-month sales volume of a product that had been on the market for
one year can be modelled by:
Sales = -2t3 + 90t2 + 20t + 50
where t is time in months.
(a)
Calculate the total annual sales during last year
(i.e. at the end of the first year on the market in 2001).
(b)
What total annual sales does the model predict during
this year (i.e. 2002)?
(c)
Forecast the expected total annual sales during
of the each of the next two years (i.e. 2003 and 2004).
(d)
From your answers to parts (a), (b) and (c) or otherwise,
explain what is happening to expected future sales and
comment on the validity of the model above.
The demand for product 1 is related to its price and the price of a substitute
(product 2) by:
q1 = 150 - 2p1 - p2
(a)
Find expressions for the partial elasticities of demand with
respect to the prices of the two products.
(b)
Using your result from (a) comment on the elasticities
at prices p1 = £25 and p2 = £30
(c)
What prices make both elasticities unity?
(d)
Calculate the approximate change in demand for product 1
resulting from a 5% increase in the present price of product 1
and a 10% decrease in the present price of product 2.
The production output quantities and associated costs of a particular process
at the end of each month for last year were recorded as follows:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Cost (£’s)
1100
1300
900
1900
2000
2200
1700
1900
1700
1800
1600
1000
Quantity
20
40
20
60
80
70
40
60
40
50
40
30
(a)
Draw a scatter-gram to show cost against output.
(b)
Using least squares regression find the equation of the best fit
linear model for cost as a function of output. What according to
your model are the fixed costs and unit costs of the process?
(c)
Show the line of best fit on your scatter-gram and discuss how
good your model is in explaining the relationship between cost
and output.
(d)
How could the formula below substantiate your answer to part (c)?
n
n
n
i 1
i 1
i 1
(n  x i y i   x i  y i ) 2
R2 
n
n
n
 n 2
2 
2
2
n  x i  (  x i )  n  y i  (  y i ) 
i 1
i 1
 i 1
  i 1

(a)
Interest on a loan is charged at a compound rate of 1% per month.
(i)
What is the Nominal Annual rate of interest?
(ii)
What is the APR?
(b)
A saver takes out 5 year Endowment scheme offering a fixed
annual growth rate of 4% p.a. What is the maturity value of
the Endowment scheme if payments of £1000 are made into
the scheme at the start of each year?
(c)
A firm applies reducing balance depreciation of 20% p.a. to a
machine purchased 3 years ago for £100,000. What is the current
book value of the machine? Calculate the book value at the end
of each of the next 5 years and plot a graph of book value over
time. How would you describe this graph?
A student takes out a loan, which provides £4000 at the start of each of the
three years of study. An interest rate of 5% p.a. is charged on the outstanding
debt at the end of each year and no repayments need to be made until after
graduation. After graduation the loan reverts to a standard repayment loan
over 5 years with equal end of year instalments with interest charged at the
bank base rate.
(a)
Assuming no repayments are made, what will the
total debt be at the end of the three years?
(b)
If the bank base rate on graduation is 6% p.a.,
what will be the annual instalments?
(c)
After two instalments are made, the bank base rate falls
to 4% p.a. and the student borrows a further £5000 on
the same loan arrangement to purchase a car. What will
the instalments be for the remaining years?
A firm must purchase machinery to do a specific job. It has a choice between two
alternative machines whose operating costs fall at the BEGINNING of each year
as shown below:Year
1
2
3
4
5
6
Machine 1
2000
400
800
-
Machine 2
3200
200
400
600
800
1000
At the end of three years the first machine will have no scrap value and will have to
be replaced by another machine of the same type with identical costs. In each case
the first year cost includes the purchase cost of the machine and installation and
running costs. In subsequent years the cost includes maintenance and running costs.
(a)
Show that, over a six-year period at a 10% p.a. discounting
Rate that machine 1 is preferred to machine 2.
(b)
Investigate, by the use of a graph, how the discounting rate
affects the decision between the two machines.
.
Consider the following constraints of a Linear Programme:
x1  4
x2  8
x1 + x2  10
2x1 + 6x2  6
2x1 + x2  8
x2  4
x1  0
x2  0
(a)
Draw an accurate graph showing the constraints, feasible
region and the vertices of the feasible region.
(b)
How might you describe the 4th constraint?
(c)
Put appropriate slack/surplus variables into the constraints.
(d)
Find the optimum values of: the decision variables, objective
function and slack/surplus variables for the following objectives:
(i)
(ii)
maximise x1 + x2
minimise x2 - x1
(e)
How would the solutions in part (d) be affected if the
first constraint was x1  4?
(f)
How would the solutions in part (d) be affected if the
extra constraint 2x1 + 2x2  24 was introduced?
A local shop-keeper employs two people George and Mildred to work at least 80
hours per week between them. This week George and Mildred want to work at least
30 hours each but George is not prepared to work more than 60 hours. Being older
than George, Mildred is paid £8 per hour while George is only paid £6 per hour. The
shop-keeper only has £800 cash available to pay wages this week.
(a)
Identify the decision variables, constraints and objective
function in a linear programming model of the shop-keeper’s
staffing problem.
(b)
Solve the problem and comment on the solution and the
use of resources.
(c)
How would your solution in part (b) be affected if:
(i) Mildred wanted to work at least 40 hours
and (ii) George was given a pay rise of £1 per hour.
The marginal cost of production for a particular product has been modelled by:
MC = 30q2 - 45q - 7500
where q is the quantity produced.
(a)
If fixed costs of production are £25,500 find an expression
for the total cost function C.
(b)
If marginal revenue for this product has been modelled by:
MR = 30q2 - 75q
find an expression for the revenue function R.
(c)
Find an expression for profit and hence, or otherwise determine
the production quantity that maximises profit.
Mr Y wishes to take out a 20 year mortgage for £150,000. His local building
society offer him the following mortgage options:
(a)
A standard repayment mortgage with an annual variable interest rate
(b)
An endowment mortgage with a growth rate of 2% below the
mortgage rate
(c)
A fixed rate mortgage at 4%, which reverts to the standard variable
rate after 5 years.
If the standard variable rate is expected to remain constant at 6% p.a.
calculate
the annual repayments that Mr X would need to make for each type of
mortgage and advise Mr X, which mortgage has the least total cost.
A firm is considering the purchase of one of two possible machines. Machine 1 costs
£6000 whilst machine 2 costs £5000. The expected end of year revenues for the two
machines are given below:
Year:
Machine 1
Machine 2
1
2000
1700
2
2000
1800
3
2000
1900
4
2000
1900
5
2000
1700
(a)
If future values are discounted at 10% p.a. calculate the NPV of
the two machines and advise the firm on the choice of machine
at this discounting rate.
(b)
How would your result in part (a) differ if future values are
discounted at 20%?
(c)
If IRR was the decision criterion, which machine should be
purchased?
(d)
By sketching a graph or otherwise, show how the choice of
machine varies according to the discounting rate.
6
2000
1500
A small engineering firm manufactures nuts and bolts. One type of nut (1cm long)
and bolt (4cm long) are made from the same hexagonal bar. This size of nut and bolt
both have a retail price of £2.50. The unit manufacturing cost of a bolt is £1 whilst the
unit manufacturing cost of a nut is £2. The production manager has allocated £450 to
the production of these nuts and bolts and has 4 metres of the appropriate hexagonal
bar in stock.
(a)
Identify the decision variables, constraints and objective function in a
linear programming model of the firm’s production decision problem.
(b)
Solve the problem and comment on the use of resources and the
production levels for this type of nut and bolt.
(c)
How would your solution in part (b) be affected if 8 metres of the
appropriate
hexagonal bar were available?
A new product was launched in ten different stores last week. Each store allocated
different areas for displaying the product and the following data was recorded:
Store
Area (m2)
Sales(qty)
1
10
130
2
16
180
3
18
190
4
12
140
5
10
90
6
12
150
7
16
170
8
14
150
(a)
Draw a scattergram to show sales against display area.
(b)
Using least squares regression find the equation of the best fit
linear model for sales as a function of sales area. What is the
slope and what is the intercept of your linear model ?
(c)
Use your model to forecast sales at the given sales areas and
hence show the line of best fit on your scattergram.
(d)
Calculate the Mean Absolute Error and comment on your model
9
12
140
10
20
220