| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 8 |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Area using coordinate formula |
| Difficulty | Moderate -0.8 This is a straightforward coordinate geometry question requiring standard techniques: finding a perpendicular line equation (gradient = -5/3), then finding intersection points and calculating triangle area. All steps are routine AS-level procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step calculation and fraction manipulation required. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_5}
Figure 4
The line $l_1$ has equation $y = \frac{3}{5}x + 6$
The line $l_2$ is perpendicular to $l_1$ and passes through the point $B(8, 0)$, as shown in the sketch in Figure 4.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation for line $l_2$ is
$$5x + 3y = 40$$ [3]
\end{enumerate}
Given that
\begin{itemize}
\item lines $l_1$ and $l_2$ intersect at the point C
\item line $l_1$ crosses the $x$-axis at the point A
\end{itemize}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the exact area of triangle $ABC$, giving your answer as a fully simplified fraction in the form $\frac{p}{q}$ [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2024 Q5 [8]}}