Management Maths Ii Question Paper
Management Maths Ii
Course:Bachelor Of Commerce
Institution: Kca University question papers
Exam Year:2010
UNIVERSITY EXAMINATIONS: 2009/2010
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
COMMERCE
CMS 102: MANAGEMENT MATHS II – SUNDAY CLASS
DATE: AUGUST 2010 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE
(a) Determine the first derivatives of the following functions;
i. y = x 4x +10 [2 Marks]
ii. 2 2 12 f (x) = 15e x + [2 Marks]
iii. f (x) = ln(7x3 - 5x2 + 3x - 5) [2 Marks]
(b) A car costs Sh.75, 000 in the market and the running cost for x kilometers is given by
V.C(x) = x + 30x(x -1). Determine when is the total average cost minimum. [5 Marks]
(c) Joe Fundi, the caretaker manager of sky towers is in the process of deciding which of two
elevator maintenance companies to select for providing maintenance and repair services for the
buildings elevators. The guarantees for promptness and quality of service from one week to the
next for the two companies are as follows.
Quick response service Ltd JIT Elevator Service Ltd
To
From
operation
Breakdown
To
From
operation
breakdown
Operation
Breakdown
0.85
0.95
0.15
0.05
Operation
Breakdown
0.95
0.85
0.05
0.15
Advise Joe Fundi on which of the two elevator companies to select. [6 Marks]
2
(d) ABC Ltd supply tool kits for the home handyman. Each tool kit comprises a standard plastic box
which contains a variable number of tools depending on the type of tools, the market and the
wholesalers requirements. The firm has derived a profit function which shows that their profits
are dependent both on the number of tool kits supplied and the number of tools in each kit. The
profit function is as follows: -
P = 8K - 0.0001K 2 + 0.05KT - 77.5T 2 -10,000
Where P = Profit in KSh. K = Number of Kits and T = Number of tools in each kit. How many
tool kits containing how many tools should be sold to maximize profit? [6 Marks]
(e) A manufacturer has found that if he wants to increase his output, he must lower his price. His
total revenue (TR) from an output x is given by the expression TR = x(148 - x). His production
costs are Shs. 1,000 fixed and Shs. 36 per unit variable. You are required to find:
(i) The output that would maximize revenue. [2 Marks]
(ii) The maximum total revenue. [1 Mark]
(iii) The output x , that would maximize profit. [4 Marks]
QUESTION TWO
Beauty products Ltd has launched a new mouth wash branded “Aloe” into the market. The product is
competing with brands of the two company’s competitor’s: “Fluoride” produced by Cee Ltd and
“Vera” produced by Dee Ltd. In order to determine the market share of “Aloe”, the marketing
department of beauty products limited conducted a market survey in the Month of April 2008. The
results of the survey were such that:-
1. Out of 1,000 consumers who used “Aloe” at the beginning of the month of April, 100 had shifted
to “Fluoride” while 100 had shifted to “Vera” at the end of the month
2. Out of 1,000 consumers who used “Fluoride” at the beginning of the month of April, 70 and 30 of
them had shifted to “Aloe” and “Vera” respectively at the end of the month
3. Out of 1,000 consumers who used “Vera” at the beginning of the Month of April, 100 and 200 of
them had shifted to “Fluoride” and “Aloe” respectively at the end of the month
The management of beauty products Ltd are also considering undertaking a promotional campaign of
the “Aloe” brand. The promotional strategies of the company are as follows:-
Strategy 1: Offer a 50 ml packet for every 100 ml packet of “Aloe” purchased. The results of the
strategy are expected to be such that:-
3
1. Out of the total consumers using “Aloe” at the beginning of the month, 90% would be repeat
consumers, 5% would shift to “Fluoride” and 5% would shift to “Vera” at the end of the month
2. Out of the total consumers of “Fluoride” at the beginning of the month, 90% would be repeat
consumers, 7% and 3% would shift to “Aloe” and “Vera” respectively at the end of the month
3. Out of the total consumers of “Vera” at the beginning of the Month, 70% would be repeat
consumers, 20% and 10% would shift to “Aloe’ and “Fluoride” respectively at the end of the
month
Strategy 2: Advertise in the television network
The results of the strategy are expected to be such that:-
1. Out of the total consumers of “Flouride” at the beginning of the month, 87% of the would be
repeat consumers, 10% of them would shift to “Aloe” and 3% would shift to “Vera” at the end of
the month
2. Out of the total consumers of “Vera” at the beginning of the month, 25% and 10% of them would
shift to “Aloe” and “Fluoride” respectively at the end of the month
3. Out of the total consumers of “Aloe” at the beginning of the month, 8% of them would shift to
“Fluoride” and 8% would shift to “Vera” at the end of the month.
The total market for mouth wash is estimated to be worth Sh. 200 million each month.
Required:-
a) Assuming that no promotion campaign is undertaken, determine the long-run market share of each
of the brands [6 Marks]
b) Advise the management of beauty products limited on the promotional strategy (if any) that the
company should adopt in order to improve its market share in the long run. [14 Marks]
QUESTION THREE
a) A retailer of motorized bicycles has examined cost data and has determined a cost function which
expresses the annual cost of purchasing, owning, and maintaining inventory as a function of the
size (number of units) of each order it places for the bicycles. The total cost function is given by,
= ( ) = 4,860 +15q + 750,000
q
C f q
4
Where C equals annual inventory cost, stated in dollars and q equals the number of cycles
ordered each time the retailer replenishes the supply.
i. Determine the order size, which minimizes annual inventory cost. [4 Marks]
ii. What is minimum annual inventory cost expected to equal? [2 Marks]
b) An economy consists of three interdependent sectors; agriculture, mining and manufacturing. The
flow of inputs and outputs between the industries is represented in the table below:-
Inputs (Tonnes)
Output(Tonnes) Agriculture Mining Manufacturing
Final
demand
Agriculture
Mining
Manufacturing
20,000
30,000
40,000
32,500
65,000
32,500
37,500
37,500
12,500
10,000
30,000
40,000
Required:-
i. The technical coefficient matrix [2 Marks]
ii. The leontiff inverse matrix [7 Marks]
iii. The output level for each sector of the final demand for the agricultural sector increases by
1000 tons and that of the manufacturing sector falls by 500 tons and the final demand for
the mining sector remains unchanged. [3 Marks]
iv. State two assumptions of input – output models [2 Marks]
QUESTION FOUR
a) Habib’s food liner, Ngugi’s Supermarket and Quick Stop groceries are the only three grocery stores
in sand town, a small town in Northern Kenya. A market research involving shoppers over a ten
week period indicated that Habib retains 85% of its shoppers while 10% and 5% shift to Ngugi’s
and Quick Stop respectively each subsequent week. Ngugi’s lose 20% to Habib and 5% to Quick
stop. Of those who shop in Quick stop, 15% and 10% shift to habib and Ngugi respectively. The
total grocery store customers in sand town are estimated at 10,000. Each shopper generates revenue
of about KSh. 500 per week. What are the projected weekly revenues for each grocery store at
equilibrium? [10 Marks]
b) A firm has the capacity to produce three types of cakes X, Y and Z using ingredients A, B and C.
The recipe for each unit of cake is given as follows:
X – 20, 15 and 40 grams of A, B and C respectively
Y - 12, 36 and 16 grams of A, B and C respectively
5
Z - 30, 15 and 40 grams of A, B and C respectively
If 130, 225 and 200 units of A, B and C respectively must be used. Using the matrix method,
determine number of units of each Cake to produce in order to utilize completely the available
ingredients. [10 Marks]
QUESTION FIVE
(a) Evaluate the following integrals
i. ? ?(xy - x y)dxdy
1
0
2
1
2 2 [3 Marks]
ii. dx
x ? (5 +1)5
1 [3 Marks]
iii. dx
x
x ?
ln 2 2 [3 Marks]
iv. ? x exdx [3 Marks]
(b) Given the Marginal Cost (MC) = 36 + (q – 8)2 and Marginal Revenue (MR) = 100 – 2q where q is
quantity of units for a given firm. Find: -
i. The profit maximizing output of the firm. [3 Marks]
ii. The total cost and total revenue function assuming that fixed cost and fixed revenue is
equal to zero. [2 Marks]
iii. The value of maximum profit. [3 Marks]
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