Standard +0.3 This is a straightforward Further Maths question involving logarithms and area calculation. Parts (a)-(c) require basic logarithm manipulation and solving log equations—standard techniques. Part (d) involves comparing curved area to trapezoid area, which is computational but conceptually simple. The multi-step nature and FM context place it slightly above average, but no novel insight is required.
\includegraphics{figure_9}
The shape ABC shown in the diagram is a student's design for the sail of a small boat.
The curve AC has equation \(y = 2 \log_2 x\) and the curve BC has equation \(y = \log_2\left(x - \frac{3}{2}\right) + 3\).
State the x-coordinate of point A. [1]
Determine the x-coordinate of point B. [3]
By solving an equation involving logarithms, show that the x-coordinate of point C is 2. [4]
It is given that, correct to 3 significant figures, the area of the sail is 0.656 units\(^2\).
Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines. [4]
\begin{enumerate}[label=(\alph*)]
\item \includegraphics{figure_9}
The shape ABC shown in the diagram is a student's design for the sail of a small boat.
The curve AC has equation $y = 2 \log_2 x$ and the curve BC has equation $y = \log_2\left(x - \frac{3}{2}\right) + 3$.
State the x-coordinate of point A. [1]
\item Determine the x-coordinate of point B. [3]
\item By solving an equation involving logarithms, show that the x-coordinate of point C is 2. [4]
It is given that, correct to 3 significant figures, the area of the sail is 0.656 units$^2$.
\item Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q9 [12]}}