| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 This is a straightforward C1 indices and surds question testing basic techniques: (i) is routine fractional index evaluation, (ii) is simple index law manipulation, and (iii) is standard rationalizing the denominator. All parts are textbook exercises requiring only recall of standard methods with no problem-solving or insight needed. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| \(27^{-\frac{1}{3}} = \frac{1}{27^{\frac{1}{3}}} = \frac{1}{9}\) | M1, A1 | 2 marks; \(\frac{1}{27^{\frac{1}{3}}}\) or \(27^{\frac{1}{3}} = 9\) or \(3^{-2}\) soi |
| Answer | Marks | Guidance |
|---|---|---|
| \(5\sqrt{5} = 5^{\frac{3}{2}}\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1-\sqrt{5}}{3+\sqrt{5}} = \frac{(1-\sqrt{5})(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}\) | M1 | Multiply numerator and denominator by conjugate |
| \(= \frac{8-4\sqrt{5}}{4}\) | B1 | |
| \(= 2 - \sqrt{5}\) | A1 | 3 marks; \((\sqrt{5})^2 = 5\) soi |
## (i)
$27^{-\frac{1}{3}} = \frac{1}{27^{\frac{1}{3}}} = \frac{1}{9}$ | M1, A1 | 2 marks; $\frac{1}{27^{\frac{1}{3}}}$ or $27^{\frac{1}{3}} = 9$ or $3^{-2}$ soi
## (ii)
$5\sqrt{5} = 5^{\frac{3}{2}}$ | B1 | 1 mark
## (iii)
$\frac{1-\sqrt{5}}{3+\sqrt{5}} = \frac{(1-\sqrt{5})(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}$ | M1 | Multiply numerator and denominator by conjugate
$= \frac{8-4\sqrt{5}}{4}$ | B1 |
$= 2 - \sqrt{5}$ | A1 | 3 marks; $(\sqrt{5})^2 = 5$ soi
---\begin{enumerate}[label=(\roman*)]
\item Evaluate $27^{\frac{2}{3}}$. [2]
\item Express $5\sqrt{5}$ in the form $5^n$. [1]
\item Express $\frac{1 - \sqrt{5}}{3 + \sqrt{5}}$ in the form $a + b\sqrt{5}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2006 Q2 [6]}}