| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Completing square from standard form |
| Difficulty | Easy -1.2 This is a routine P1 completing the square question with standard follow-up parts. Part (a)(i) is a textbook exercise in completing the square, (a)(ii) requires reading the maximum from completed square form, and (b) involves finding a tangent parallel to a given line using basic differentiation. All techniques are standard with no problem-solving insight required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4+12x-3x^2 = a\pm3(x+c)^2\) or \(a+b(x\pm2)^2\) | M1 | Attempt to complete the square; look for \(b=\pm3\), \(c=\pm2\) |
| Two of: \(16-3(x-2)^2\), or \(a=16\), \(b=-3\), \(c=-2\) | A1 | |
| \(16-3(x-2)^2\) | A1 | \((16-3(2-x)^2\) scores M1A1A0) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Coordinates \(M=(2,16)\) | B1ft B1ft | ft on \((-c,a)\) from \(a+b(x+c)^2\) where \(b\neq\pm1\); wrong order e.g. \((16,2)\) scores SC B1 B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \(l_2\) has equation \(y = "8"x + k\) | M1 | ft on their gradient from \(M\) |
| Sets \(4+12x-3x^2 = "8x"+k\) and proceeds to 3TQ | dM1 | |
| Correct 3TQ: \(3x^2-4x+k-4=0\) | A1 | "=0" may be implied |
| Attempts \(b^2-4ac=0\) to find \(k\) | ddM1 | |
| \(k=\frac{16}{3} \Rightarrow y=8x+\frac{16}{3}\) | A1 | Condone just \(k=\frac{16}{3}\) if \(y=8x+k\) was stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiates \(4+12x-3x^2\) and sets equal to 8 | M1 | |
| Solves for \(x\), finds coordinates of point of contact | dM1 | Tangent meets curve at \(\left(\frac{2}{3},\frac{32}{3}\right)\) |
| Substitutes \(\left(\frac{2}{3},\frac{32}{3}\right)\) into \(y="8"x+k\) | ddM1 | |
| \(y=8x+\frac{16}{3}\) | A1 |
## Question 8 (Completing the Square/Tangent):
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4+12x-3x^2 = a\pm3(x+c)^2$ or $a+b(x\pm2)^2$ | M1 | Attempt to complete the square; look for $b=\pm3$, $c=\pm2$ |
| Two of: $16-3(x-2)^2$, or $a=16$, $b=-3$, $c=-2$ | A1 | |
| $16-3(x-2)^2$ | A1 | $(16-3(2-x)^2$ scores M1A1A0) |
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Coordinates $M=(2,16)$ | B1ft B1ft | ft on $(-c,a)$ from $a+b(x+c)^2$ where $b\neq\pm1$; wrong order e.g. $(16,2)$ scores SC B1 B0 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $l_2$ has equation $y = "8"x + k$ | M1 | ft on their gradient from $M$ |
| Sets $4+12x-3x^2 = "8x"+k$ and proceeds to 3TQ | dM1 | |
| Correct 3TQ: $3x^2-4x+k-4=0$ | A1 | "=0" may be implied |
| Attempts $b^2-4ac=0$ to find $k$ | ddM1 | |
| $k=\frac{16}{3} \Rightarrow y=8x+\frac{16}{3}$ | A1 | Condone just $k=\frac{16}{3}$ if $y=8x+k$ was stated |
**Alternative for part (b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiates $4+12x-3x^2$ and sets equal to 8 | M1 | |
| Solves for $x$, finds coordinates of point of contact | dM1 | Tangent meets curve at $\left(\frac{2}{3},\frac{32}{3}\right)$ |
| Substitutes $\left(\frac{2}{3},\frac{32}{3}\right)$ into $y="8"x+k$ | ddM1 | |
| $y=8x+\frac{16}{3}$ | A1 | |8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-22_657_659_214_646}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of the curve $C$ with equation
$$y = 4 + 12 x - 3 x ^ { 2 }$$
The point $M$ is the maximum turning point on $C$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write $4 + 12 x - 3 x ^ { 2 }$ in the form
$$a + b ( x + c ) ^ { 2 }$$
where $a , b$ and $c$ are constants to be found.
\item Hence, or otherwise, state the coordinates of $M$.
The line $l _ { 1 }$ passes through $O$ and $M$, as shown in Figure 4.\\
A line $l _ { 2 }$ touches $C$ and is parallel to $l _ { 1 }$
\end{enumerate}\item Find an equation for $l _ { 2 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q8 [10]}}