| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Region definition with inequalities |
| Difficulty | Standard +0.3 Part (a) is a standard simultaneous equations problem requiring substitution and solving a quadratic. Part (b) requires identifying which inequalities define a shaded region, which is slightly more conceptual but still routine for P1. Overall, this is slightly easier than average as it involves well-practiced techniques with no novel problem-solving required. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02i Represent inequalities: graphically on coordinate plane1.02q Use intersection points: of graphs to solve equations |
| VIIV SIHI NI IIIIM IONOO | VIIIV SIHI NI JIIIM I ON OO | VIAV SIHI NI III HM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(x + y = 6,\ y = 6x - 2x^2 + 1 \Rightarrow 6 - x = 6x - 2x^2 + 1 \Rightarrow 2x^2 - 7x + 5 = 0\) | M1 | Uses line and curve to obtain 3TQ in \(x\) or \(y\) equal to zero; condone slips in rearrangement |
| \(2x^2 - 7x + 5 = 0 \Rightarrow (2x-5)(x-1) = 0 \Rightarrow x = \frac{5}{2},\ 1\) | M1 | Solves their 3TQ by factorising, formula or completing the square; must see correct line of working |
| \(x = \frac{5}{2} \Rightarrow y = \frac{7}{2}\); or \(x = 1 \Rightarrow y = 5\) | dM1 | Solves for \(y\) for at least one value of \(x\); dependent on previous M mark |
| \((1, 5)\) and \((2.5, 3.5)\) | A1 | Both pairs correct; condone omission of brackets; cannot award if \(x\) and \(y\) reversed |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(y \geqslant 6x - 2x^2 + 1\) and \(x + y \leqslant 6\) | M1 | Any 2 of the inequalities |
| \(x \geqslant a\) where \(1 \leqslant a \leqslant 2.5\) | A1 | Any 3 of the inequalities correct; ignore references to and/or |
| \(y \geqslant 0\) (or \(0 \leqslant y \leqslant c\) where \(c \geqslant 3.5\)); allow strict or non-strict inequalities | A1 | All 4 correct; withhold final mark if "or" or \(\cup\) used between inequalities |
## Question 4:
**Part (a):**
| Working | Mark | Guidance |
|---------|------|----------|
| $x + y = 6,\ y = 6x - 2x^2 + 1 \Rightarrow 6 - x = 6x - 2x^2 + 1 \Rightarrow 2x^2 - 7x + 5 = 0$ | M1 | Uses line and curve to obtain 3TQ in $x$ or $y$ equal to zero; condone slips in rearrangement |
| $2x^2 - 7x + 5 = 0 \Rightarrow (2x-5)(x-1) = 0 \Rightarrow x = \frac{5}{2},\ 1$ | M1 | Solves their 3TQ by factorising, formula or completing the square; must see correct line of working |
| $x = \frac{5}{2} \Rightarrow y = \frac{7}{2}$; or $x = 1 \Rightarrow y = 5$ | dM1 | Solves for $y$ for at least one value of $x$; dependent on previous M mark |
| $(1, 5)$ and $(2.5, 3.5)$ | A1 | Both pairs correct; condone omission of brackets; cannot award if $x$ and $y$ reversed |
**Part (b):**
| Working | Mark | Guidance |
|---------|------|----------|
| $y \geqslant 6x - 2x^2 + 1$ and $x + y \leqslant 6$ | M1 | Any 2 of the inequalities |
| $x \geqslant a$ where $1 \leqslant a \leqslant 2.5$ | A1 | Any 3 of the inequalities correct; ignore references to and/or |
| $y \geqslant 0$ (or $0 \leqslant y \leqslant c$ where $c \geqslant 3.5$); allow strict or non-strict inequalities | A1 | All 4 correct; withhold final mark if "or" or $\cup$ used between inequalities |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-10_689_917_264_507}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\section*{In this question you must show all stages of your working.}
\section*{Solutions relying on calculator technology are not acceptable.}
Figure 1 shows a line $l$ with equation $x + y = 6$ and a curve $C$ with equation $y = 6 x - 2 x ^ { 2 } + 1$
The line $l$ intersects the curve $C$ at the points $P$ and $Q$ as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Find, using algebra, the coordinates of $P$ and the coordinates of $Q$.
The region $R$, shown shaded in Figure 1, is bounded by $C , l$ and the $x$-axis.
\item Use inequalities to define the region $R$.\\
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIIV SIHI NI IIIIM IONOO & VIIIV SIHI NI JIIIM I ON OO & VIAV SIHI NI III HM ION OC \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2022 Q4 [7]}}