2018 AJC Promo Q11

1. For all real numbers, the least integer function denoted by \(y=\mathrm{f}\left( x \right)\), returns the least integer greater than or equal to \(x\). For example, \(\mathrm{f}\left( 1 \right)=1\), \(\mathrm{f}\left( 1.5 \right)=2\) and \(\mathrm{f}\left( -2.3 \right)=-2\). Show that when \(\mathrm{f}\left( 2x+5 \right)=12\), then \(3 < x\le \frac{7}{2}\).[2]

Taxi fares are determined by the flag-down fare, the distance and time taken for your journey, and any applicable time-based and location surcharges. A taxi company has their non-peak period unit charges tabulated in the following table,

A woman takes a taxi from her home in a journey covering \(x\)km.

2. Denoting \(\mathrm{f}\left( x \right)\) to be the taxi fare for the journey, write down \(\mathrm{f}\left( x \right)\) in the form
   \(\mathrm{f}\left( x \right)=8.7+0.22\,\mathrm{f}\left( \frac{ax+b}{7} \right)\) for \(x>11\),
   where \(\mathrm{f}\) is the least integer function and, \(a\) and \(b\) are constants to be determined.[3]
3. Explain with reasons if the woman is able to deduce the actual distance travelled for her taxi ride based on the amount that she paid.[3]

To further entice users, the company rolls out a royalty programme that credits loyalty points based on distance travelled using the following function,

Points awarded, \(\mathrm{g}\left( x \right)\left\{ \begin{matrix} 30\ln \left( x+1 \right)\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,x>5, \\ 0\,\,\,\,\,\,\,\,\,\mathrm{otherwise}\mathrm{.} \\ \end{matrix} \right.\)

A man travels from his office to meet his clients using the taxi company. The distance he travels varies according to \(t\), a variable relating to the traffic conditions and is given by the function,

\(\mathrm{h}\left( t \right)=1.25\sin t+6\) for \(0<t<d\).

By considering the function \(\mathrm{gh}\),

4. Explain why \(d\) cannot exceed \(4.069\), correct to \(3\) decimal places, for him to earn points for each trip he makes.[2]
5. Find exactly the range of points awarded for the man when he takes a taxi ride under the loyalty programme for \(d=\frac{5\pi }{6}\).[3]