| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Completing the square, form and properties |
| Difficulty | Moderate -0.3 This is a structured, multi-part question that guides students through completing the square and applying it to a distance minimization problem. While it requires connecting algebra to geometry, the scaffolding (explicit hints like 'expand', 'show that', 'use your results') makes it easier than average. The techniques are standard C1 content with no novel insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-4)^2\) or \(p = 4\) | B1 | ISW for \(p = -4\) if \((x-4)^2\) seen |
| \(+1\) or \(q = 1\) | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| (Minimum value is) \(1\) | B1\(\checkmark\) [1] | Correct or FT "their \(q\)" (NOT coords) |
| Answer | Marks | Guidance |
|---|---|---|
| (Minimum occurs when \(x =\) ) \(4\) | B1\(\checkmark\) [1] | Correct or FT "their \(p\)"; may use calculus; condone \((p, **)\) for this B1 |
| Answer | Marks |
|---|---|
| \((x-5)^2 = x^2 - 10x + 25\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-5)^2 + (7-x-4)^2\) | M1 | Condone one slip in one bracket; may be seen under \(\sqrt{}\) sign |
| Answer | Marks | Guidance |
|---|---|---|
| \(= x^2 - 10x + 25 + 9 - 6x + x^2\) | A1 | From a fully correct expression |
| \(AB^2 = 2x^2 - 16x + 34 = 2(x^2 - 8x + 17)\) | A1 [3] | AG CSO |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum \(AB^2 = 2 \times \text{"their (a)(ii)"}\) | M1 | Or use of their \(x=4\) in expression; or use of \(B(4,3)\) and \(A(5,4)\) in distance formula; M0 if calculus used |
| Minimum \(AB = \sqrt{2}\) | A1 [2] | Answer only of \(2\times\) "their (a)(ii)" scores M1, A0 |
## Question 6:
**Part (a)(i)**
$(x-4)^2$ or $p = 4$ | B1 | ISW for $p = -4$ if $(x-4)^2$ seen
$+1$ or $q = 1$ | B1 [2] |
**Part (a)(ii)**
(Minimum value is) $1$ | B1$\checkmark$ [1] | Correct or FT "their $q$" (NOT coords)
**Part (a)(iii)**
(Minimum occurs when $x =$ ) $4$ | B1$\checkmark$ [1] | Correct or FT "their $p$"; may use calculus; condone $(p, **)$ for this B1
**Part (b)(i)**
$(x-5)^2 = x^2 - 10x + 25$ | B1 [1] |
**Part (b)(ii)**
$(x-5)^2 + (7-x-4)^2$ | M1 | Condone one slip in one bracket; may be seen under $\sqrt{}$ sign
$= (x-5)^2 + (3-x)^2$
$= x^2 - 10x + 25 + 9 - 6x + x^2$ | A1 | From a fully correct expression
$AB^2 = 2x^2 - 16x + 34 = 2(x^2 - 8x + 17)$ | A1 [3] | AG CSO
**Part (b)(iii)**
Minimum $AB^2 = 2 \times \text{"their (a)(ii)"}$ | M1 | Or use of their $x=4$ in expression; or use of $B(4,3)$ and $A(5,4)$ in distance formula; M0 if calculus used
Minimum $AB = \sqrt{2}$ | A1 [2] | Answer only of $2\times$ "their (a)(ii)" scores M1, A0
---6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $x ^ { 2 } - 8 x + 17$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
\item Hence write down the minimum value of $x ^ { 2 } - 8 x + 17$.
\item State the value of $x$ for which the minimum value of $x ^ { 2 } - 8 x + 17$ occurs.\\
(1 mark)
\end{enumerate}\item The point $A$ has coordinates (5,4) and the point $B$ has coordinates ( $x , 7 - x$ ).
\begin{enumerate}[label=(\roman*)]
\item Expand $( x - 5 ) ^ { 2 }$.
\item Show that $A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)$.
\item Use your results from part (a) to find the minimum value of the distance $A B$ as $x$ varies.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2009 Q6 [10]}}