| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then find vertex/turning point |
| Difficulty | Easy -1.8 This is a routine completing-the-square exercise with direct reading of vertex coordinates—purely procedural with no problem-solving required. The completing the square is straightforward (coefficient of x² is 1), and finding P and Q requires only substitution and reading from completed square form. This is easier than typical A-level questions, being foundational GCSE/AS-level material. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) = (x-2)^2 \pm \ldots\) | M1 | Achieves \((x-2)^2 \pm \ldots\) or states \(a = -2\) |
| \(f(x) = (x-2)^2 + 1\) | A1 | Correct expression; ISW after sight of this. Condone \(a=-2, b=1\). Condone \((x-2)^2+1=0\) |
| (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P = (0, 5)\) | B1 | Correct coordinates for \(P\). Allow \(x=0, y=5\) |
| (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Q = (2, 1)\) | B1ft | Correct coordinates for \(Q\). Allow \(x=2, y=1\). Follow through from part (a): allow \((-a, b)\) where \(a, b\) numeric |
| (1 mark) |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = (x-2)^2 \pm \ldots$ | M1 | Achieves $(x-2)^2 \pm \ldots$ or states $a = -2$ |
| $f(x) = (x-2)^2 + 1$ | A1 | Correct expression; ISW after sight of this. Condone $a=-2, b=1$. Condone $(x-2)^2+1=0$ |
| **(2 marks)** | | |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P = (0, 5)$ | B1 | Correct coordinates for $P$. Allow $x=0, y=5$ |
| **(1 mark)** | | |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Q = (2, 1)$ | B1ft | Correct coordinates for $Q$. Allow $x=2, y=1$. Follow through from part (a): allow $(-a, b)$ where $a, b$ numeric |
| **(1 mark)** | | |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$
(a) express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$ where $a$ and $b$ are integers to be found.
The curve with equation $y = \mathrm { f } ( x )$
\begin{itemize}
\item meets the $y$-axis at the point $P$
\item has a minimum turning point at the point $Q$\\
(b) Write down\\
(i) the coordinates of $P$\\
(ii) the coordinates of $Q$
\end{itemize}
\hfill \mbox{\textit{Edexcel Paper 1 2021 Q2 [4]}}