| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show line is tangent, verify |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing routine techniques: completing the square (standard C1 skill), reading off the minimum point, substituting to form a quadratic equation, and solving it by factorisation. All steps are mechanical with no problem-solving insight required, making it easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-2)^2 + 5\) | B1 | — |
| B1 | 2 | \(p = 2\), \(q = 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum point \((2, 5)\) or \(x = 2, y = 5\) | B2√ | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(12 - 2x = x^2 - 4x + 9 \Rightarrow x^2 - 2x - 3 = 0\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-3)(x+1) = 0\) | M1 | Attempt at factors or quadratic formula or one value spotted |
| \(x = 3, -1\) | A1 | — |
| Substitute one value of x to find y | M1 | — |
| Points are \((3, 6)\) and \((-1, 14)\) | A1 | 4 |
## 3(a)(i)
$(x-2)^2 + 5$ | B1 | —
| B1 | 2 | $p = 2$, $q = 5$
## 3(a)(ii)
Minimum point $(2, 5)$ or $x = 2, y = 5$ | B2√ | 2 | B1 for each coordinate correct or ft; **Alt method** M1, A1 sketch, differentiation
## 3(b)(i)
$12 - 2x = x^2 - 4x + 9 \Rightarrow x^2 - 2x - 3 = 0$ | B1 | 1 | AG (be convinced) (must have = 0)
Or $x^2 - 4x + 9 + 2x = 12$
## 3(b)(ii)
$(x-3)(x+1) = 0$ | M1 | Attempt at factors or quadratic formula or one value spotted
$x = 3, -1$ | A1 | — | Both values correct & simplified
Substitute one value of x to find y | M1 | —
Points are $(3, 6)$ and $(-1, 14)$ | A1 | 4 | y-coordinates correct linked to x values
---3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $x ^ { 2 } - 4 x + 9$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
\item Hence, or otherwise, state the coordinates of the minimum point of the curve with equation $y = x ^ { 2 } - 4 x + 9$.
\end{enumerate}\item The line $L$ has equation $y + 2 x = 12$ and the curve $C$ has equation $y = x ^ { 2 } - 4 x + 9$.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of the points of intersection of $L$ and $C$ satisfy the equation
$$x ^ { 2 } - 2 x - 3 = 0$$
\item Hence find the coordinates of the points of intersection of $L$ and $C$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q3 [9]}}