| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 9 |
| Topic | Modulus function |
| Type | Sketch y=|linear| and y=linear with unknown constants, then solve |
| Difficulty | Challenging +1.2 This is a multi-part modulus question requiring sketching, solving an inequality with parameters, and finding a range. Part (a) is routine, part (b) requires standard case analysis for |3x-2a| ≤ x+a yielding an interval in terms of a, and part (c) requires substituting this interval into g(x) and analyzing the V-shaped graph. While it involves parameters throughout and multiple steps, the techniques are standard A-level modulus methods without requiring novel insight—moderately above average difficulty due to the parameter manipulation and multi-stage reasoning. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02t Solve modulus equations: graphically with modulus function |
12.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph with equation
$$y = | 3 x - 2 a |$$
where $a$ is a positive constant.\\
State the coordinates of each point where the graph cuts or meets the coordinate axes.
\item Solve, in terms of $a$, the inequality
$$| 3 x - 2 a | \leqslant x + a$$
Given that $| 3 x - 2 a | \leqslant x + a$
\item find, in terms of $a$, the range of possible values of $\mathrm { g } ( x )$, where
$$\mathrm { g } ( x ) = 5 a - \left| \frac { 1 } { 2 } a - x \right|$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q12 [9]}}