| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then solve equation |
| Difficulty | Moderate -0.8 This is a straightforward C1 completing the square question with standard steps: factor out the coefficient, complete the square, then solve using the completed square form. Both parts are routine textbook exercises requiring only mechanical application of a well-practiced technique with no problem-solving or insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x^2 + 12x + 13 = 2(x^2 + 6x) + 13\) | B1 | |
| \(= 2[(x+3)^2 - 9] + 13\) | B1 | |
| \(= 2(x+3)^2 - 5\) | M1 | \(a = 2\), \(b = 3\); 13 – 2b² or 13 – b² or \(\frac{13}{2} - b^2\) (their b) |
| \(c = -5\) | A1 | 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(2(x+3)^2 - 5 = 0\) | M1 | |
| \((x+3)^2 = \frac{5}{2}\) | A1 | |
| \(x = -3 \pm \sqrt{\frac{5}{2}}\) | A1 | 3 marks; Uses correct quadratic formula or completing square method; \(x = \frac{-12 \pm \sqrt{40}}{4}\) or \((x+3)^2 = \frac{5}{2}\); \(x = -3 \pm \frac{\sqrt{10}}{2}\) or \(-3 \pm \frac{1}{2}\sqrt{10}\) |
## (i)
$2x^2 + 12x + 13 = 2(x^2 + 6x) + 13$ | B1 |
$= 2[(x+3)^2 - 9] + 13$ | B1 |
$= 2(x+3)^2 - 5$ | M1 | $a = 2$, $b = 3$; 13 – 2b² or 13 – b² or $\frac{13}{2} - b^2$ (their b)
$c = -5$ | A1 | 4 marks total
## (ii)
$2(x+3)^2 - 5 = 0$ | M1 |
$(x+3)^2 = \frac{5}{2}$ | A1 |
$x = -3 \pm \sqrt{\frac{5}{2}}$ | A1 | 3 marks; Uses correct quadratic formula or completing square method; $x = \frac{-12 \pm \sqrt{40}}{4}$ or $(x+3)^2 = \frac{5}{2}$; $x = -3 \pm \frac{\sqrt{10}}{2}$ or $-3 \pm \frac{1}{2}\sqrt{10}$
---\begin{enumerate}[label=(\roman*)]
\item Express $2x^2 + 12x + 13$ in the form $a(x + b)^2 + c$. [4]
\item Solve $2x^2 + 12x + 13 = 0$, giving your answers in simplified surd form. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2006 Q3 [7]}}