| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Standard +0.3 This is a multi-part question covering completing the square (routine), finding a turning point (direct read-off), finding where a normal meets the curve (standard differentiation), and writing inequalities for a region (straightforward). All parts are standard P1 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x\pm\ldots)^2 \pm \ldots\) or states \(a = \frac{1}{2}\) | B1 | If contradiction between embedded values of \(a,b,c\) and stated values, embedded values take precedence |
| \(\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x\pm10)^2 \pm \ldots\) or states \(a=\frac{1}{2}\) and \(b=\pm10\) | M1 | Deals correctly with first two terms of \(\frac{1}{2}x^2 - 10x + 22\) |
| \(\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x-10)^2 - 28\) | A1 | Cannot just be stated values; do not isw if they multiply by 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\text{"10"},\ \text{"}-28\text{"})\) | B1ftB1ft | B1ft: one coordinate of \((10,-28)\) or follow through one of \((-b,c)\) from (a); B1ft: \((10,-28)\) or ft \((-b,c)\); accept \(x=\text{"10"}\), \(y=\text{"}-28\text{"}\); condone lack of bracketing if intention clear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Gradient of tangent \(= 8\) | B1 | Stated or implied |
| \(\frac{dy}{dx} = x - 10 = 8 \Rightarrow x = \ldots\) | M1 | Differentiates \(\frac{1}{2}x^2 - 10x + 22\) to form \(px+q\), sets equal to 8, proceeds to find \(x\); alternatively sets \(8x+\alpha = \frac{1}{2}x^2-10x+22\), discriminant \(=0\), solves for \(\alpha\) then finds \(x\) |
| \(x = 18,\ y = 4\) | A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k - \frac{1}{8}\times\text{"18"} = \text{"4"} \Rightarrow k = \frac{25}{4}\) | dM1A1 | dM1: sets \(k - \frac{1}{8}\times\text{"18"} = \text{"4"}\), proceeds to find \(k\); dependent on method mark in (i); A1: \(k=\frac{25}{4}\) oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| One of \(x\ \ldots\text{"10"}\) or \(y\ \ldots\text{"}\frac{25}{4}\text{"} - \frac{1}{8}x\) or \(y\ \ldots\frac{1}{2}x^2 - 10x + 22\) | B1ft | Follow through on their minimum point or their \(k\); allow if \(l\) still in terms of \(k\) |
| Two of \(x\ \ldots\text{"10"}\) or \(y\ \ldots\text{"}\frac{25}{4}\text{"} - \frac{1}{8}x\) or \(y\ \ldots\frac{1}{2}x^2 - 10x + 22\) | B1ft | As above |
| All three of \(x\ \ldots 10\), \(y\ \ldots\frac{25}{4} - \frac{1}{8}x\) and \(y\ \ldots\frac{1}{2}x^2 - 10x + 22\) | B1 | Ignore references to OR/AND or equivalent set notation; use of strict or inclusive inequalities must be consistent for all inequalities on last mark only; condone additional minimum interval (or greater) \(-28\ \ldots y\ \ldots 5\) |
## Question 9:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x\pm\ldots)^2 \pm \ldots$ or states $a = \frac{1}{2}$ | B1 | If contradiction between embedded values of $a,b,c$ and stated values, embedded values take precedence |
| $\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x\pm10)^2 \pm \ldots$ or states $a=\frac{1}{2}$ and $b=\pm10$ | M1 | Deals correctly with first two terms of $\frac{1}{2}x^2 - 10x + 22$ |
| $\frac{1}{2}x^2 - 10x + 22 = \frac{1}{2}(x-10)^2 - 28$ | A1 | Cannot just be stated values; do not isw if they multiply by 2 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\text{"10"},\ \text{"}-28\text{"})$ | B1ftB1ft | B1ft: one coordinate of $(10,-28)$ or follow through one of $(-b,c)$ from (a); B1ft: $(10,-28)$ or ft $(-b,c)$; accept $x=\text{"10"}$, $y=\text{"}-28\text{"}$; condone lack of bracketing if intention clear |
### Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Gradient of tangent $= 8$ | B1 | Stated or implied |
| $\frac{dy}{dx} = x - 10 = 8 \Rightarrow x = \ldots$ | M1 | Differentiates $\frac{1}{2}x^2 - 10x + 22$ to form $px+q$, sets equal to 8, proceeds to find $x$; alternatively sets $8x+\alpha = \frac{1}{2}x^2-10x+22$, discriminant $=0$, solves for $\alpha$ then finds $x$ |
| $x = 18,\ y = 4$ | A1A1 | |
### Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k - \frac{1}{8}\times\text{"18"} = \text{"4"} \Rightarrow k = \frac{25}{4}$ | dM1A1 | dM1: sets $k - \frac{1}{8}\times\text{"18"} = \text{"4"}$, proceeds to find $k$; dependent on method mark in (i); A1: $k=\frac{25}{4}$ oe |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| One of $x\ \ldots\text{"10"}$ or $y\ \ldots\text{"}\frac{25}{4}\text{"} - \frac{1}{8}x$ or $y\ \ldots\frac{1}{2}x^2 - 10x + 22$ | B1ft | Follow through on their minimum point or their $k$; allow if $l$ still in terms of $k$ |
| Two of $x\ \ldots\text{"10"}$ or $y\ \ldots\text{"}\frac{25}{4}\text{"} - \frac{1}{8}x$ or $y\ \ldots\frac{1}{2}x^2 - 10x + 22$ | B1ft | As above |
| All three of $x\ \ldots 10$, $y\ \ldots\frac{25}{4} - \frac{1}{8}x$ and $y\ \ldots\frac{1}{2}x^2 - 10x + 22$ | B1 | Ignore references to OR/AND or equivalent set notation; use of strict or inclusive inequalities must be consistent for all inequalities on last mark only; condone additional minimum interval (or greater) $-28\ \ldots y\ \ldots 5$ |
The image appears to be a blank page from a Pearson Education document, showing only the company registration information at the bottom and "PMT" in the top right corner. There is no mark scheme content visible on this page to extract.9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-24_889_666_258_703}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the curve $C$ with equation
$$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$
\begin{enumerate}[label=(\alph*)]
\item Write $\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$ in the form
$$a ( x + b ) ^ { 2 } + c$$
where $a , b$ and $c$ are constants to be found.
The point $M$ is the minimum turning point of $C$, as shown in Figure 3.
\item Deduce the coordinates of $M$
The line $l$ is the normal to $C$ at the point $P$, as shown in Figure 3.\\
Given that $l$ has equation $y = k - \frac { 1 } { 8 } x$, where $k$ is a constant,
\item \begin{enumerate}[label=(\roman*)]
\item find the coordinates of $P$
\item find the value of $k$
Question 9 continues on the next page
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 is a copy of Figure 3. The finite region $R$, shown shaded in Figure 4, is bounded by $l , C$ and the line through $M$ parallel to the $y$-axis.
\end{enumerate}\item Identify the inequalities that define $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2022 Q9 [14]}}