| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Region definition with inequalities |
| Difficulty | Standard +0.3 Part (a) requires standard substitution to solve simultaneous equations (line and quadratic), yielding a quadratic to solve. Part (b) asks students to write inequalities defining a shaded region, which is slightly less routine but still a standard P1 skill. Overall slightly easier than average due to straightforward algebraic manipulation with no conceptual surprises. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y - 5x = 75\), \(y = 2x^2 + x - 21 \Rightarrow 2x^2 + x - 21 = 5x + 75 \Rightarrow 2x^2 - 4x - 96 = 0\) or \(x^2 - 2x - 48 = 0\) | M1 | Equates line and curve and rearranges to 3TQ in \(x\); alternatively eliminates \(x\) to obtain 3TQ in \(y\) |
| \(x^2 - 2x - 48 = 0 \Rightarrow (x-8)(x+6) = 0 \Rightarrow x = -6, 8\) | dM1 | Solves 3TQ by factorisation, formula or completing the square; cannot just state roots without correct intermediate working |
| \(x = -6 \Rightarrow y = 45\) or \(x = 8 \Rightarrow y = 115\) | dM1 | Uses at least one \(x\) value to find at least one \(y\) value; depends on first M1 and having solved 3TQ |
| \(P(-6, 45)\) and \(Q(8, 115)\) | A1 | All correct; must be correctly paired; condone coordinates written without brackets; depends on all previous marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. \(y \leqslant 2x^2 + x - 21\), \(y - 5x \leqslant 75\), \(y \geqslant 0\), \(x \leqslant a\) where \(-3.5 \leqslant a < 3\) | M1 | Obtains at least 2 of the required inequalities; allow equivalents e.g. \(y < 5x + 75\) |
| \(x \leqslant a\) where \(-3.5 \leqslant a < 3\) (or \(a \leqslant x \leqslant b\) where \(a \leqslant -15\), \(-3.5 \leqslant b \leqslant 3\)) | A1 | Any 3 of the 4 required inequalities |
| All 4 correct and consistent: \(y \leqslant 2x^2+x-21\), \(y \leqslant 5x+75\), \(y \geqslant 0\), \(x \leqslant a\) | A1 | Inequalities for \(y\) must be either all strict or all non-strict; extra incorrect inequalities score A0 |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y - 5x = 75$, $y = 2x^2 + x - 21 \Rightarrow 2x^2 + x - 21 = 5x + 75 \Rightarrow 2x^2 - 4x - 96 = 0$ or $x^2 - 2x - 48 = 0$ | M1 | Equates line and curve and rearranges to 3TQ in $x$; alternatively eliminates $x$ to obtain 3TQ in $y$ |
| $x^2 - 2x - 48 = 0 \Rightarrow (x-8)(x+6) = 0 \Rightarrow x = -6, 8$ | dM1 | Solves 3TQ by factorisation, formula or completing the square; cannot just state roots without correct intermediate working |
| $x = -6 \Rightarrow y = 45$ or $x = 8 \Rightarrow y = 115$ | dM1 | Uses at least one $x$ value to find at least one $y$ value; depends on first M1 and having solved 3TQ |
| $P(-6, 45)$ and $Q(8, 115)$ | A1 | All correct; must be correctly paired; condone coordinates written without brackets; depends on all previous marks |
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## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. $y \leqslant 2x^2 + x - 21$, $y - 5x \leqslant 75$, $y \geqslant 0$, $x \leqslant a$ where $-3.5 \leqslant a < 3$ | M1 | Obtains at least 2 of the required inequalities; allow equivalents e.g. $y < 5x + 75$ |
| $x \leqslant a$ where $-3.5 \leqslant a < 3$ (or $a \leqslant x \leqslant b$ where $a \leqslant -15$, $-3.5 \leqslant b \leqslant 3$) | A1 | Any 3 of the 4 required inequalities |
| All 4 correct and consistent: $y \leqslant 2x^2+x-21$, $y \leqslant 5x+75$, $y \geqslant 0$, $x \leqslant a$ | A1 | Inequalities for $y$ must be either all strict or all non-strict; extra incorrect inequalities score A0 |6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-14_899_901_251_584}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
\section*{In this question you must show all stages of your working. \\
Solutions relying on calculator technology are not acceptable.}
Figure 3 shows
\begin{itemize}
\item the line $l$ with equation $y - 5 x = 75$
\item the curve $C$ with equation $y = 2 x ^ { 2 } + x - 21$
\end{itemize}
The line $l$ intersects the curve $C$ at the points $P$ and $Q$, as shown in Figure 3 .
\begin{enumerate}[label=(\alph*)]
\item Find, using algebra, the coordinates of $P$ and the coordinates of $Q$.
The region $R$, shown shaded in Figure 3, is bounded by $C , l$ and the $x$-axis.
\item Use inequalities to define the region $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2024 Q6 [7]}}