2024 YIJC H1 Q5

A company offers three varieties of pens: Pinnacle, Quillmaster, and RapidWrite. The cost of a Pinnacle pen is \(0.30\) lower than the cost of \(10\) RapidWrite pens. A combination of \(5\) Pinnacle pens, \(4\) Quillmaster pens, and \(3\) RapidWrite pens costs \(29.25\). Additionally, the cost of \(13\) Quillmaster pens is the same as the cost of a Pinnacle pen.

1. By writing down three linear equations, determine the cost of a pen for each variety.[4]
2. The company sells value packs, each consisting of a Pinnacle pen, a Quillmaster pen and a RapidWrite pen. Determine the cost of a value pack if it is sold at a discount of \(20\)%.[1]

The sales director believes that the connection between the sales revenue, \(S\) thousand dollars, and the time, \(t\) years, can be modelled by

\[S=3+0.2\ln(3t+1) \text{ for}\hspace{0.5em} 0\le t\le 6.\]

3. Sketch the graph of \(S\) against \(t\).[2]
4. Find \(\frac{\mathrm{d}S}{\mathrm{d}t}\) and explain in context what the expression for \(\frac{\mathrm{d}S}{\mathrm{d}t}\) indicates about sales revenue for the first \(6\) years.[3]
5. Using your calculator, find the value of \(\displaystyle\int_0^4 3+0.2\ln(3t+1) \, \mathrm{d}t\). In the context of the question, what does this value represent?[2]