| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing standard surd manipulation techniques: rationalizing denominators and simplifying surds. Both parts require only direct application of well-practiced procedures with no problem-solving or insight needed, making it easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | \(\frac{7+\sqrt{5}}{3+\sqrt{5}} \cdot \frac{3-\sqrt{5}}{3-\sqrt{5}}\) | M1 |
| Numerator \(= 21 + 3\sqrt{5} - 7\sqrt{5} - (\sqrt{5})^2\) | m1 | Condone one slip: \(16 - 4\sqrt{5}\) |
| Denominator \(= 9 - 5 = 4\) | B1 | (Or \(5 - 9 = -4\) from other conjugate) |
| Answer \(= 4 - \sqrt{5}\) | A1 | CSO |
| 3(b) | \(\sqrt{45} = 3\sqrt{5}\) | B1 |
| \(\frac{20}{\sqrt{5}} = \frac{20\sqrt{5}}{5}\) | M1 | May score if combined as one expression. Must have 5 in denominator |
| Sum \(= 7\sqrt{5}\) | A1 |
**3(a)** | $\frac{7+\sqrt{5}}{3+\sqrt{5}} \cdot \frac{3-\sqrt{5}}{3-\sqrt{5}}$ | M1 | Multiply by $\frac{3-\sqrt{5}}{3-\sqrt{5}}$ or $\frac{\sqrt{5}-3}{\sqrt{5}-3}$
| Numerator $= 21 + 3\sqrt{5} - 7\sqrt{5} - (\sqrt{5})^2$ | m1 | Condone one slip: $16 - 4\sqrt{5}$
| Denominator $= 9 - 5 = 4$ | B1 | (Or $5 - 9 = -4$ from other conjugate)
| Answer $= 4 - \sqrt{5}$ | A1 | CSO
**3(b)** | $\sqrt{45} = 3\sqrt{5}$ | B1 |
| $\frac{20}{\sqrt{5}} = \frac{20\sqrt{5}}{5}$ | M1 | May score if combined as one expression. Must have 5 in denominator
| Sum $= 7\sqrt{5}$ | A1 |
**Total: 7 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
\item Express $\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }$ in the form $k \sqrt { 5 }$, where $k$ is an integer.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2009 Q3 [7]}}