| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a routine C1 completing-the-square question with standard follow-up parts. Part (a) requires mechanical completion of the square, part (b) straightforward factorisation and sketching. All techniques are textbook exercises with no problem-solving or novel insight required, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(16 - 6x - x^2 = -(x^2 + 6x - 16)\) | M1 | Attempt completing the square |
| \(= 25 - (x+3)^2\) | A1 | \(p=25\), \(q=3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Maximum value \(= 25\) | B1 | Follow through from (a)(i) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(16 - 6x - x^2 = (2-x)(8+x)\) | B1 | Accept equivalent forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Curve crosses \(x\)-axis at \(x=2\) and \(x=-8\) | B1 | |
| \(y\)-intercept at \((0, 16)\) | B1 | |
| Correct inverted parabola shape through stated points | B1 |
# Question 4:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $16 - 6x - x^2 = -(x^2 + 6x - 16)$ | M1 | Attempt completing the square |
| $= 25 - (x+3)^2$ | A1 | $p=25$, $q=3$ |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Maximum value $= 25$ | B1 | Follow through from (a)(i) |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $16 - 6x - x^2 = (2-x)(8+x)$ | B1 | Accept equivalent forms |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Curve crosses $x$-axis at $x=2$ and $x=-8$ | B1 | |
| $y$-intercept at $(0, 16)$ | B1 | |
| Correct inverted parabola shape through stated points | B1 | |
---4
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $16 - 6 x - x ^ { 2 }$ in the form $p - ( x + q ) ^ { 2 }$ where $p$ and $q$ are integers.
\item Hence write down the maximum value of $16 - 6 x - x ^ { 2 }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Factorise $16 - 6 x - x ^ { 2 }$.
\item Sketch the curve with equation $y = 16 - 6 x - x ^ { 2 }$, stating the values of $x$ where the curve crosses the $x$-axis and the value of the $y$-intercept.\\[0pt]
[3 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2014 Q4 [7]}}