Sub Maths Pre Registration

MT ST HENRY’S HIGH SCHOOL MUKONO PRE -REGISTRATION EXAMINATION UGANDA ADVANCED CERTIFICATE OF EDUCATION SUBSIDIARY MATHEMATICS TIME 2:45HOURS ATTEMPT FOUR ITEMS IN THIS PAPER ITEM ONE a) The profit P in thousands of Ugandan Shillings for a small business selling handmade crafts is modeled by the equation 𝑷=−𝟐𝒏𝟐+𝟒𝟎𝒏 −𝟕𝟐, where n is the number of items sold in hundreds. The business ow ner needs to know the maximum profits to set realistic sales t argets and to determine the number of items that must be sold to maximize profits . Task: Find t he maximum profit and the number of items to be sold to maximize profit. b) A company is designing a new type of battery that discharges energy exponentially over time. The amount of energy remaining in the battery after 𝒙 hours is modelled by the function 𝑬(𝒙)=𝟑𝟐𝒙−𝟏𝟐(𝟑𝒙)+𝟐𝟕. The engineers want to find the time 𝑥 (in hours) when the energy level reaches zero, meaning the battery is fully discharged. Task: At what time 𝒙 will the battery be fully discharged? ITEM TWO a) A construction worker stacks bricks in rows following a specific pattern: in Row 1, there are 50 bricks. Each subsequent row contains 3 more bricks than the previous row. Task: i) Calculate the number of bricks in row 15. ii) What is the total number of bricks in a 20-row stack? b) The same construction company ’s start up revenue shows exponential growth : year 3 revenue: $200,000. Year 6 revenue : $1.6 million. Task: i) Determine the annual growth rate (r) ii) Using this rate, project the revenue for year 10. ITEM THREE A team of engineers is laying down two underground fiber -optic cables in Kampala . Cable A follows a path represented by the vector 𝒂=𝛽𝒊+(2+𝛽)𝒋 while Cable B follows the …