| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a straightforward completing-the-square exercise followed by a routine sketch. Part (a) requires standard algebraic manipulation with integer coefficients that factor nicely. Part (b) is direct application: the turning point comes immediately from the completed square form, and the y-intercept requires only substituting x=0. No problem-solving or insight needed—pure procedural recall. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 3\) | B1 | Correct value for \(a\) stated or shown by working |
| \(b = \pm 2\) | M1 | Obtains \(b = \pm 2\). Allow unsimplified e.g. \(b = \pm\frac{4}{2}, \pm\frac{12}{6}\) |
| \(3x^2 + 12x + 13 = 3(x+2)^2 + 1\) or \(a=3, b=2, c=1\) | A1 | Fully correct expression or correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct U shape with minimum in second quadrant | B1 | Must be U shape, minimum not on \(x\)-axis, not a clear V shape |
| Intercept 13 on \(y\)-axis | B1 | Allow as just 13 or \((0,13)\) or \((13,0)\) if in correct position |
| Vertex at \((-2, 1)\) | B1ft | Follow through on their \(b\) and \(c\), i.e. \(x=-b\), \(y=c\). Must correspond to sketch. |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 3$ | **B1** | Correct value for $a$ stated or shown by working |
| $b = \pm 2$ | **M1** | Obtains $b = \pm 2$. Allow unsimplified e.g. $b = \pm\frac{4}{2}, \pm\frac{12}{6}$ |
| $3x^2 + 12x + 13 = 3(x+2)^2 + 1$ or $a=3, b=2, c=1$ | **A1** | Fully correct expression or correct values |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct U shape with minimum in second quadrant | **B1** | Must be U shape, minimum not on $x$-axis, not a clear V shape |
| Intercept 13 on $y$-axis | **B1** | Allow as just 13 or $(0,13)$ or $(13,0)$ if in correct position |
| Vertex at $(-2, 1)$ | **B1ft** | Follow through on their $b$ and $c$, i.e. $x=-b$, $y=c$. Must correspond to sketch. |
---\begin{enumerate}
\item (a) Express $3 x ^ { 2 } + 12 x + 13$ in the form
\end{enumerate}
$$a ( x + b ) ^ { 2 } + c$$
where $a$, $b$ and $c$ are integers to be found.\\
(b) Hence sketch the curve with equation $y = 3 x ^ { 2 } + 12 x + 13$
On your sketch show clearly
\begin{itemize}
\item the coordinates of the $y$ intercept
\item the coordinates of the turning point of the curve
\end{itemize}
\hfill \mbox{\textit{Edexcel P1 2023 Q3 [6]}}