| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | October |
| Marks | 5 |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Equation of line through two points |
| Difficulty | Moderate -0.8 Part (a) is a routine straight-line equation problem requiring gradient calculation and substitution (standard GCSE/AS topic). Part (b) asks for inequalities defining a region, which is straightforward once the line equation is found—students just need to determine which inequality signs apply between the line and curve. Both parts are mechanical with no problem-solving insight required, making this easier than average A-level content. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02i Represent inequalities: graphically on coordinate plane1.03a Straight lines: equation forms y=mx+c, ax+by+c=0 |
The line $l$ passes through the points $A(-3, 0)$ and $B\left(\frac{5}{3}, 22\right)$
\begin{enumerate}[label=(\alph*)]
\item Find the equation of $l$ giving your answer in the form $y = mx + c$ where $m$ and $c$ are constants. [3]
\end{enumerate}
\includegraphics{figure_2}
Figure 2 shows the line $l$ and the curve $C$, which intersect at $A$ and $B$.
Given that
\begin{itemize}
\item $C$ has equation $y = 2x^2 + 5x - 3$
\item the region $R$, shown shaded in Figure 2, is bounded by $l$ and $C$
\end{itemize}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item use inequalities to define $R$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q3 [5]}}