Mathematical Techniques

​ ​​Find the area enclosed by the curve Y = 8X3 + 3X2 – 2X, the X axis and within the interval 2 ≤ X ≤ 4.

​ ​ ​​A simple economy consists of two sectors; Wool production and Hides tanning.  Each unit of output from Wool production requires 0.6 units worth of input from Wool production and 0.3 units worth of input from Hides tanning.  

​ ​​A company sells a product and market research shows that the demand function for the product is linear.  The price and quantity combinations observed in the market are as follows: 

​ ​ ​​The total revenue function of producing product “G” is quadratic in nature. The following data shows the number of units of product G sold and the corresponding sales revenue in thousands of shillings:

​ ​ ​​TOC Limited, an oil processing company purchased a machine that they intend to use in one of their production processes. The machine is expected to have a useful life of 10 years after which it can be disposed for Sh.200,000.

​ ​ ​​Clekex Cleaning Limited manufactures bar soaps for a hotel industry. The company specialises in three different types of bar soaps weighing 0.5kg, 0.75kg and 1kg respectively. The three types of bar soaps are made by two machines, machine A and machine B. The number of hours required for each type of bar soap in each machine is shown below: 

​ ​​An economy is based on 2 sectors; Agriculture and Manufacturing. Production of a shilling worth of Agriculture requires an input of 0.3 from the Agriculture sector and 0.1 from the Manufacturing sector.

Production of a shilling worth of manufacturing requires an input of 0.2 from the Agriculture sector and 0.4 from the Manufacturing sector.

Required:

Find the output from each sector that is needed to satisfy a final demand of Sh.12 billion for Agriculture and Sh.8 billion for Manufacturing.

​ ​​​​Explain FOUR limitations of quantitative analysis.

​​State FOUR applications of mathematical functions in business.

​ ​​BMM Limited produces X number of items of product “Wonder” in a month at a cost described by the equation C = 5x + 4,000. The Management Accountant of the firm estimates that at a selling price of Sh.22 per unit, 18,000 units of “Wonder” could be sold. If the firm increases the unit price to Sh.30, only 10,000 units of “Wonder” can be sold.

​ ​ ​ ​​The Sales Manager of XYZ Ltd. estimates that the revenue function of the company is linear in nature. He has observed that at a selling price of Sh.200, the company sold 6,000 units. When the company increased the selling price per unit to Sh.700, the company sales decreased to 1,000 units. 

​​State FOUR characteristics of a good measure of dispersion.

​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​​Soundex Company has received an order to supply 900 tables, 1,200 desks and 2,000 chairs. The management of Soundex Company has decided that 500 tables, 800 desks and 1,300 chairs could be supplied from their city centre branch and the balance of the order could be filled from their industrial area branch.

Production of each table requires 3 hours in the machining department, 5 hours in the assembly department and 4 hours in the finishing department. Production of a desk requires 4 hours in the machining department, 8 hours in the assembly department and 6 hours in the finishing department. Production of a chair requires 2 hours in the machining department, 3 hours in the assembly department and 5 hours in the finishing department.

The cost of an hour in the machining, assembly and finishing department is Sh.50, Sh.100 and Sh.150 respectively.

Required:

Using matrix algebra:

(i) Calculate the total number of hours required at each branch.

(ii) Calculate the total cost incurred at each branch and the total cost incurred by Soundex Company to supply the order.

​ ​ ​​The Production Manager of AMIK Company has provided the following information relating to the number of units produced per day in the last 172 days:

​ ​ ​​A curve is defined by the quadratic function 

​​Outline four qualities of a good average.

​ ​ Given that A =​​​\(\begin{bmatrix} 2 & -1 & 2 \\ 1 & 0 & 3 \\ 3 & -2 & -5 \end{bmatrix}\)​ and

​ ​ ​​ Explain three types of Kurtosis that a graphical representation of a frequency distribution can assume.

​​The average revenue function of a certain company is given by the function AR = 2,000 - 24q. The cost function is given by the function C = 6q² + 1,440q + 1,280. In both cases, q represents the quantity in units. 

​ ​ ​​XYZ Ltd. produces three products namely A, B and C. The company presents the profit per unit of the products it produces and sells as follows: Sh.2,000, Sh.3,000 and Sh.4,000 for products A, B and C respectively during the month of January 2022. Sh.7,000, Sh.9,000 and Sh.4,000 for products A, B and C respectively during the month of February 2022. Sh.1,000, Sh.4,000 and Sh.2,000 for products A, B and C respectively during the month of March 2022. The total profits in the months of January, February and March 2022 are Sh.17 million, Sh.45 million and Sh.16 million respectively. 

​ ​ ​ ​ ​ ​ ​​The following data shows the age distribution of 350 employees of a multi-national company.

​ ​​Hexadol Limited has been in operation for the last 5 years. The Company's annual revenue function and annual cost function take the form of quadratic functions. 

​​Agro manufacturers produce three products; Chat, Item and Wit (in thousands) whose demand and cost functions are given as follows:

Chat: AR = 16 – 3Q ; ATC = 4Q + 8

Item: P = 10 – Q – ​\(2Q^2\)​ ; ATC = Q + 4

Wit: P = 100 – ½Q ; ATC = 300 + 2Q – 2​\(Q^2\)​

Required:

(a) Output and price levels that will maximize profits.

(b) Maximum profit for each product.

(c) Total profit for the production of the three products at the optimal point.

​ ​ ​​The frequency distribution of after tax earnings for Applewood Ltd. for 180 months to 31 December 2020 was as follows:

​​ State five advantages of the arithmetic mean as a measure of central tendency.

​​ State four applications of matrices in business.

​ ​ ​ ​ ​​The Aviation Perspectives Agency (APA) recently conducteda market survey to determine the air travel preferences of 170 travellers in a certain country. The air travel preferences were represented by the following airlines; Excellent Airways, Safari Connections and Tumaini Services.

The following results were obtained from the survey:

100 travellers preferred Excellent Airways.

40 travellers preferred Safari Connections.

75 travellers preferred Tumaini Services.

20 travellers preferred Excellent Airways and Safari Connections.

28 travellers preferred Excellent Airways and Tumaini Services.

18 travellers preferred Safari Connections and Tumaini Services.

9 travellers did not record any preference.

Required:

(i) Represent the above information in a Venn diagram.

(ii) Determine the number of travellers who preferred all the three airlines.

(iii) The number of travellers who preferred the Safari Connections and Tumaini Services, but not the Excellent Airways. 

​ ​​The demand function for a certain product is given by:

​​Citing an example in each case, explain the difference between a continuous function and a discrete function.