a) Differentiate with respect to x by using the first principle. (6marks)
b) If y = v2
+ 1 and v = x3
+ x + 4, find the derivative of y with respect to x using the chain rule (4marks)
c) Given demand function p = , where p is price and q is quantity, find the
maximum revenue. (8marks)
d) Evaluate (4marks)
e) Ali buys a flat for which he agrees to make 10 annual instalments of Shs. 20,000 each, the first being made at the end of 3 years. Find the equivalent cash price of this property, if money is worth 5% per annum effective. (4marks)
f) If +
X
= , find matrix
X. (2marks)
g) Find the inverse of . (2marks)
QUESTION TWO (20 MARKS)
a) By an
‘Economic Order Quantity’
we mean quantity Q, which when produced in each order, minimizes the total cost C incurredin producing and storing material for certain time period to fulfill a given rate of demand for the material during the period. The material demanded for a firm is 10,000 units per year, the cost price of material is sh. 1 per unit and its cost sh. 25 to make the factory ready for production run of the items, regardless of the number of units ’x’ produced in a run and the cost of storing the material is 12.5 percent per year of the shilling value of average inventory x/2 in hand.
i) Show that C = 10,000 + 250,000/x + x/16. (5marks)
ii) Find the ‘Economic Order Quantity’ and the total cost C corresponding to that. (6marks)
iii) Find the total cost when each order is placed for 2,500 units. (2marks)
b) Differentiatethe functions with respect to
x
ii)
2
(3marks)
a) If A =, show that A3
– 3A – 2
I
3
= 0 and hence find A-1. (12 marks)
b) Find the area of the region included between the curve y2
=
x
and the straight line
x
+ y = 2. (8marks)
QUESTION FOUR (20MARKS)
a) Magnate Ventures an advertising agency is preparing an advertising campaign for a group of agencies. These agencies have decided that their target customers should have the following characteristics with importance (weightage) as given below.
|
characteristics
|
Weightage(%)
|
Age
Annual income
female
|
25 – 45 years
Above Sh. 80,000
Married
|
20
40
40
|
The agency has made a careful analysis of three media and has compiled the following data:
Data Item
|
Media
|
|
|
|
Magazine
|
Radio
|
Television
|
Audience size (‘000)
Cost per advertisement (Sh.)
Min. no. of advertisements allowed
|
800
10,000
15
|
1000
30,000
10
|
1500
100,000
10
|
Max. no. of advertisements allowed
Reader Characteristics
Age 25 – 45 years
Annual income above Sh. 80,000
Females married
|
25
70%
60%
45%
|
15
60%
50%
40%
|
15
50%
40%
30
|
The advertising agency has earmarked a budget of Sh. 1000,000 for this purpose.
Formulate the Linear programming model for the problem to maximize the total expected
effective exposure. (12marks)
b) Find the local maximum and local minimum values, if any, of the function:
y
= (8marks)
QUESTION FIVE (20MARKS)
a) The manufacturing cost of an item consists of Sh. 10,000 as overheads; material cost Sh. 4 per item and the labour cost Sh. x2/90 for x items produced. Find how many items will be
produced to have the average cost minimum. (6marks)
b) For a monopolist market, the demand law is p = (6 – x)2
and marginal cost is x + 14. Find the consumers’ surplus. (7marks)
c) Jean purchases 3 pens, 2 exercise books and 1 pencil and pays Sh. 41. From the same shop, Joan purchases 2 pens, 1 exercise book and 2 pencils and pays Sh. 29, while Jaedon purchases 2 pens, 2 exercise books and 2 pencils and pays Sh. 44. Translate the problem into a system of equations. Solve the system of equations by matrix method and hence find the
cost of 1 pen, 1 exercise book and 1 pencil.