| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Topic | Modulus function |
| Type | Graph y=a|bx+c|+d given: solve equation or inequality |
| Difficulty | Standard +0.8 This is a multi-part modulus function question requiring: (a) finding the vertex (straightforward), (b) solving an equation with modulus (standard technique with case consideration), and (c) finding the range of gradients for tangency/intersection conditions, which requires geometric insight about lines through the origin intersecting a V-shaped graph. Part (c) elevates this above routine exercises as it demands understanding of critical gradients and set notation. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3dcde139-bc6b-412d-8d1f-c45543d67430-16_703_851_150_701}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of the graph with equation
$$y = 2 | x + 4 | - 5$$
The vertex of the graph is at the point $P$, shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $P$.
\item Solve the equation
$$3 x + 40 = 2 | x + 4 | - 5$$
A line $l$ has equation $y = a x$, where $a$ is a constant.\\
Given that $l$ intersects $y = 2 | x + 4 | - 5$ at least once,
\item find the range of possible values of $a$, writing your answer in set notation.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q8 [7]}}