Part 1 (Functions & Graphs)

The research department in Gycoland established the following price-demand, cost, and revenue functions for their AI-based drone hardware:

Price-demand function

Cost function

Revenue function

where x is in thousands of units, and C(x) and R(x) are in thousands of dollars. All three functions have domain .

(a) Graph the cost function and the revenue function simultaneously in the same coordinate system.

(b) Write a profit function .

(c) Determine algebraically when Q(x) = 0, Q(x) < 0 and Q(x) > 0, to the nearest unit.

(d) “The forecast manager suggested that the company should produce 18800 units of the AI-based drone to maximise profit.

i. Determine algebraically the maximum profit (to the nearest thousand dollars) and the output (to the nearest unit) that produces the maximum profit.

ii. Calculate the cost price of the AI-based drone (to the nearest dollar) at this output?

iii. Hence, comment on whether the company should proceed on with the amount of output that the forecast manager has suggested?

Part 2 (Data Description & Probability Distribution)

The average number of AI-based drones that Gycoland can manufacture weekly in a random sample of 30 weeks (within a year) is listed below:

(a) Construct a frequency and relative frequency table using class intervals of width 20, starting at 19.5.

(b) Construct a histogram and frequency polygon.

(c) Find the mean and standard deviation for the grouped data.

(d) What is the probability that a randomly chosen week (outside of the sample), Gycoland will manufacture more than 70 AI-based drones?

(e) The plant manager in charge of the production of the AI-based drones discover that the data collected incorrectly recorded 7 less AI-base drones production per week to all the data listed above. Comment how this will affect the mean and the standard deviation of the grouped data in part (a). (Detailed workings need to be shown.)

Part 3 (Logic, Sets and Counting)

Due to an increase in the purchasing demand of luxury housing by Singaporean, Gycoland decided to develop more projects and build more residential housing to fulfil the rapid demand. Hence, the HR department decided to hire more employees to fill up the position in the Property Development Department and Construction Management Department respectively.

(a) Assuming all applicants fulfilled all the job requirements for any position posted by Gycoland, how many ways can the HR department choose 7 applicants among 20 applicants who applied for positions in the Property Development department and 4 applicants among 12 applicants who applied for positions in the Construction Management department?

(b) Among the 4 successful applicants that has applied for the positions in the Construction Management Department in part (a), how many ways can the department select a Project Manager, a General Contractor, a Design Phase Manager and an Integrated Project Leader?

(c) Among the 7 successful applicants that has applied for the positions in the Property Development department in part (a), how many ways can the department select a Financial Analyst, 3 Construction Operations Manager, a Land Acquisition Manager, a Purchasing Agent?

(d) There are currently a total of 25 employees in both the Property Development Department and Construction Management Department. With the inclusion of all the successful applicants in part (a), how many ways can the company select 2 teams of 2 employees to work on a particular mini-integrated project?

Part 4 (Probability)

Gary, a project manager from the Construction Management Department, is given a task to handle a new development of a luxury residence condominium. However, he has a dilemma in the exact location where the new condominium is to be built. As a result, he intends to throw a die twice to help him determine where is the final location of the new condominium.

· If the outcome is 1, the condo will be built 100m up from the current position.

· Else, if it is 2, the condo will be built 100m to the right from the current position.

· Else, if it is 3, the condo will be built 100m down from the current position.

· Else, if it is 4, the condo will be built 100m to the left from the current position.

· Otherwise, if it is 5 or 6, the condo stays at the current position.

What is the probability that after the two rolls, the condo will be built,

(a) at the original position that Gary intends to develop on?

(b) exactly 200 metres away from the original position that Gary intends to develop on?

(c) more than 100 metre but less than 200 metres away from the original position that Gary intends to develop on?

(d) The probability that the new luxury residence condominium will compete its development in 2 years’ time is 0.45. The probability that the condo is built exactly 200 metres away from the original position that Gary intends to develop on and will compete its development in 2 years’ time is 0.05.

i. State, with supported working, whether the event of the completion of condominium in 2 years’ time is independent to the event of the condominium built exactly 200 metres away from the original position that Gary intends to develop on.

ii. Hence, find the probability that the condo will complete in development in 2 years’ time given that the condo is built exactly 200 metres away from the original position that Gary intends to develop on?