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Q1.

1. a) Simplify the following
2. 5X+3XY-X-XY
3.   X(1-Y) – 8XY +5(X+Y)                                                 

iii.   4X2-4X+7Y2-3Y+2X2-Y2+XY            

iv.4(x/(2xy)) +(6y/2y)-((x2 y2 /(xy))

b). Write down an expression for the floor area of two rooms flat when the first room has one and a half times the width and twice the length of the second.

c). Solve the following simultaneous equations algebraically.

1. i) y=2x +5

y=3x + 1

1. ii) 2x-6= -3y

15-6y=4x

iii) 2x-8=-4y

4x+8y=16

1. d) Solve and factorize the following quadratic equations:
2. i)  x2− 3x +2 = 0
3. ii) x2+10x+25=0

iii) x2+2x+5=0

Write down all your calculations.

1. e) Solve log(x)+log(x−1) =1

(25marks)

Q2.

220. a) Suppose we know that when the output is 10 units, the cost of production is £ 120, and when the output is 60 units, the cost is £220. Find the equation relating the cost (as the dependent variable) to the number of products produced (as the independent variable).

b) The following equations give the daily demand for and supply of calculators:

Demand P = 100 – 5Q

Supply P = 10 + 4Q

For values of Q (the quantity of calculators per day) ranging from 0 to 20

calculate values for P (the price per calculator) and plot them in one 

diagram. Use the horizontal axis for Q and the vertical axis for P. Label each equation and the axes. Do these equations make economic sense?

Find the intersection of the two lines and confirm your answer algebraically. What is the significance of this point?

1. c) A company manufactures two products: Product A and Product B. The company has limited resources for labor hours and materials.

• Each unit of Product A requires 5 hours of labor and 3 units of material.
• Each unit of Product B requires 1 hour of labor and 4 units of material.
• The company has a maximum of 100 labor hours available.
• The company can use a maximum of 120 units of material.

Each unit of Product A generates a profit of $30, and each unit of Product B generates a profit of $40.

Problem: The company wants to maximize its profit while staying within the labour and material constraints.

Write the above problem using mathematical expressions.

Produce a graph with the two constraints and evaluate the profit where the two constraints intersect. 

d). A new machine is purchased for £80,000, with delivery charges £1000, and provision of foundations by the company’s own labour £4000. The machinery is estimated to have a useful economic life of five years, and a residual value of £5000. 

How would this machine appear on the balance sheet at the end of the second year of its life and at the end of year 5, using a straight-line method of depreciation? Plot a graph that shows the book value of the machine for all the five years.

(25marks)

Q3

1. a)    Find the mean and median for the following annual earning fingers (£) sample of 10 people. Sort and plot the data. What measure do you think best represents the set and why?

10500, 60000, 17000, 10500, 15000, 12000, 12500, 13500, 14000, 14500.

2017. b) The following are the earnings per share for a sample of 15 software companies for 2017.

Earnings per share £: 16.40, 12.99, 0.09, 3.18, 0.13, 0.41, 1.49, 1.20, 1.12, 0.51,10.13, 8.92, 3.5, 6.36, 7.83

1.   Find the mean and median. Compute algebraically the variance and the standard deviation. Show the formulas you used.
2.   Sort and plot the data and comment on the shape of the graph.
3. c)   Consider the following samples of earning data (000’s):

 Set1: 25, 12, 20.5, 12.50, 15, 20, 16.5, 18, 20, 15.5

Set2:  29, 15, 10.5, 21.5, 12, 13.5,  21.5, 18.5,17.5,16

 Required:

17.   The sample means in each case are £17.5 thousand per annum. Does this suggest that the data sets are similar? In what respects do the data differ?
18.   Find the two data sets’ min, max, ranges and standard deviations.

d) A retail company wants to analyze the sales revenue it generates from its different product categories. The company records daily sales revenue (in thousands of dollars) from five product categories over a month (30 days). The data below shows the frequency of daily sales revenue for one of the product categories:

Problem:

1. Draw a histogram for the data, representing the frequency of each sales revenue range.
2. Calculate the mean sales revenue using the midpoints of the intervals.

(25 marks)

Q4.

1. a) Accountant firm submit 3 tenders for contracts with three different multinational companies. The accounting firm consider their chances of getting the contracts are 1/2, 1/3, 1/4 respectively. What is the probability that the accounting firm will obtain,
2. One and only one contract.
3. No contracts?
4. Two contracts.

b) Each morning, a mechanic sets up a machine to make a part that must be made with precision. Experience shows that 3 days out of 4 he sets the machine up correctly. If the machine is set up correctly 90% of the parts are of the required precision. If the machine is set up incorrectly 30% of the parts are of the required precision.

Find the probability that the machine is set up correctly, given that the parts are of required precision.

8. c) When two dice are rolled, find the probability of getting a greater number on the first die than the one on the second, given that the sum should equal 8.

(25marks)

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