| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Multiple Region or Composite Area |
| Difficulty | Standard +0.3 This is a standard A-level integration question requiring finding intersection points, setting up two definite integrals for areas between curves, and computing a ratio. While it involves multiple regions and several steps, all techniques are routine P2 content with no novel insights required. The algebraic manipulation is straightforward, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((0,\ 5)\) | B1 | Correct coordinates. Allow \(x=0,\ y=5\). 5 on its own is B0. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(x^2-4x+5=2 \Rightarrow x^2-4x+3=0 \Rightarrow x=\ldots\) | M1 | Sets \(C=l\) and solves for \(x\). Expect 3TQ \(x^2-4x+3(=0)\) before solving by factorising, formula, completing the square or calculator. |
| \(x(E)=1,\ x(F)=3\) | A1 | Correct values from correct method. Condone if \(E\) and \(F\) labelled wrong way round. Answers only scores M0A0. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Area \(R_1=\int_0^1(x^2-4x+5-2)\,dx=\left[\frac{x^3}{3}-2x^2+3x\right]_0^1=\frac{1}{3}-2+3=\frac{4}{3}\) | M1A1 | M1: Fully correct strategy for area of \(R_1\), attempting to integrate \(C-l\) or integrate \(C\) and subtract rectangle area. Limits must be based on part (b). A1: \(\frac{4}{3}\) or exact equivalent. |
| Area \(R_2=\frac{1}{2}\times3\times3-\text{"}\frac{4}{3}\text{"}=\frac{19}{6}\) | M1A1 | M1: Fully correct strategy for area of \(R_2\). Many strategies possible (e.g. \(GDF-\)"\(R_1\)", \(EDH+EHF\), \(DJFG-\)"\(R_1\)"\(-DJF\)). A1: \(\frac{19}{6}\) or exact equivalent. |
| \(\frac{\text{area of }R_1}{\text{area of }R_2}=\frac{\frac{4}{3}}{\frac{19}{6}}=\frac{8}{19}\) | A1 | cao, all previous marks must have been scored. |
## Question 9(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(0,\ 5)$ | B1 | Correct coordinates. Allow $x=0,\ y=5$. 5 on its own is B0. |
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## Question 9(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $x^2-4x+5=2 \Rightarrow x^2-4x+3=0 \Rightarrow x=\ldots$ | M1 | Sets $C=l$ and solves for $x$. Expect 3TQ $x^2-4x+3(=0)$ before solving by factorising, formula, completing the square or calculator. |
| $x(E)=1,\ x(F)=3$ | A1 | Correct values from correct method. Condone if $E$ and $F$ labelled wrong way round. Answers only scores M0A0. |
---
## Question 9(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Area $R_1=\int_0^1(x^2-4x+5-2)\,dx=\left[\frac{x^3}{3}-2x^2+3x\right]_0^1=\frac{1}{3}-2+3=\frac{4}{3}$ | M1A1 | M1: Fully correct strategy for area of $R_1$, attempting to integrate $C-l$ or integrate $C$ and subtract rectangle area. Limits must be based on part (b). A1: $\frac{4}{3}$ or exact equivalent. |
| Area $R_2=\frac{1}{2}\times3\times3-\text{"}\frac{4}{3}\text{"}=\frac{19}{6}$ | M1A1 | M1: Fully correct strategy for area of $R_2$. Many strategies possible (e.g. $GDF-$"$R_1$", $EDH+EHF$, $DJFG-$"$R_1$"$-DJF$). A1: $\frac{19}{6}$ or exact equivalent. |
| $\frac{\text{area of }R_1}{\text{area of }R_2}=\frac{\frac{4}{3}}{\frac{19}{6}}=\frac{8}{19}$ | A1 | cao, all previous marks must have been scored. |\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
\section*{Solutions based entirely on calculator technology are not acceptable.}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f6af51c1-5f85-4952-b3c4-9dca42b2a309-26_761_940_411_566}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows
\begin{itemize}
\item the curve $C$ with equation $y = x ^ { 2 } - 4 x + 5$
\item the line $l$ with equation $y = 2$
\end{itemize}
The curve $C$ intersects the $y$-axis at the point $D$.\\
(a) Write down the coordinates of $D$.
The curve $C$ intersects the line $l$ at the points $E$ and $F$, as shown in Figure 3.\\
(b) Find the $x$ coordinate of $E$ and the $x$ coordinate of $F$.
Shown shaded in Figure 3 is
\begin{itemize}
\item the region $R _ { 1 }$ which is bounded by $C , l$ and the $y$-axis
\item the region $R _ { 2 }$ which is bounded by $C$ and the line segments $E F$ and $D F$
\end{itemize}
Given that $\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k$, where $k$ is a constant,\\
(c) use algebraic integration to find the exact value of $k$, giving your answer as a simplified fraction.
\hfill \mbox{\textit{Edexcel P2 2023 Q9 [8]}}