| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Easy -1.2 This is a straightforward C1 completing the square question with standard follow-up parts. Part (a) is routine algebraic manipulation, part (b) requires basic sketching skills, and part (c) involves solving a simple quadratic equation. All techniques are standard textbook exercises requiring minimal problem-solving insight. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(x^2 - 6x + 18 = (x-3)^2 + 9\) | B1, M1 A1 | (3 marks) |
| (b) "U"-shaped parabola | M1 | |
| Vertex in correct quadrant | A1ft | |
| \(P\): \((0, 18)\) (or 18 on \(y\)-axis) | B1 | |
| \(Q\): \((3, 9)\) | B1ft | (4 marks) |
| (c) \(x^2 - 6x + 18 = 41\) or \((x-3)^2 + 9 = 41\) | M1 | |
| Attempt to solve 3 term quadratic \(x = \ldots\) | M1 | |
| \(x = \frac{6 \pm \sqrt{36 - (4 \times -23)}}{2}\) (or equiv.) | A1 | |
| \(\sqrt{128} = \sqrt{64 \times 2}\) (or equiv. surd manipulation) | M1 | |
| \(3 + 4\sqrt{2}\) (Ignore other value) | A1 | (5 marks) |
**(a)** $x^2 - 6x + 18 = (x-3)^2 + 9$ | B1, M1 A1 | (3 marks)
**(b)** "U"-shaped parabola | M1 |
Vertex in correct quadrant | A1ft |
$P$: $(0, 18)$ (or 18 on $y$-axis) | B1 |
$Q$: $(3, 9)$ | B1ft | (4 marks)
**(c)** $x^2 - 6x + 18 = 41$ or $(x-3)^2 + 9 = 41$ | M1 |
Attempt to solve 3 term quadratic $x = \ldots$ | M1 |
$x = \frac{6 \pm \sqrt{36 - (4 \times -23)}}{2}$ (or equiv.) | A1 |
$\sqrt{128} = \sqrt{64 \times 2}$ (or equiv. surd manipulation) | M1 |
$3 + 4\sqrt{2}$ (Ignore other value) | A1 | (5 marks)
**Total: 12 marks**
**Notes:**
(a) M1 requires $(x \pm a)^2 \pm b \pm 18$, $a \neq 0$, $b \neq 0$. Answer only: full marks.10. Given that
$$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$
\begin{enumerate}[label=(\alph*)]
\item express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$, where $a$ and $b$ are integers.
The curve $C$ with equation $y = \mathrm { f } ( x ) , x \geqslant 0$, meets the $y$-axis at $P$ and has a minimum point at $Q$.
\item In the space provided on page 19, sketch the graph of $C$, showing the coordinates of $P$ and $Q$.
The line $y = 41$ meets $C$ at the point $R$.
\item Find the $x$-coordinate of $R$, giving your answer in the form $p + q \sqrt { } 2$, where $p$ and $q$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q10 [12]}}