| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.3 This is a routine surds question testing standard techniques: simplifying surds, rationalizing denominators, and algebraic manipulation. Part (a)(i) is trivial recall, (a)(ii) requires simplification then basic rationalization, and (b) is a standard textbook exercise in rationalizing with conjugates. All steps are mechanical with no problem-solving or insight required, making it easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{18} = 3\sqrt{2}\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2\sqrt{2}}{3\sqrt{2} + 4\sqrt{2}}\) | M1 | attempt to write each term in form \(n\sqrt{2}\) with at least 2 terms correct |
| A1 | correct unsimplified | |
| \(= \frac{2}{7}\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{7\sqrt{2} - \sqrt{3}}{2\sqrt{2} - \sqrt{3}} \times \frac{2\sqrt{2} + \sqrt{3}}{2\sqrt{2} + \sqrt{3}}\) | M1 | |
| (numerator \(=\)) \(14 \times 2 - 2\sqrt{6} + 7\sqrt{6} - 3\) | m1 | correct unsimplified but must simplify \((\sqrt{2})^2, (\sqrt{3})^2\) and \(\sqrt{2} \times \sqrt{3}\) correctly |
| (denominator \(= 8 - 3 =\)) \(5\) | B1 | must be seen or identified as denominator giving \(\frac{25 + 5\sqrt{6}}{5}\) |
| (Answer \(=) 5 + \sqrt{6}\) | A1 cso | 4 marks |
**3(a)(i)**
$\sqrt{18} = 3\sqrt{2}$ | B1 | 1 mark | Condone $k = 3$
**3(a)(ii)**
$\frac{2\sqrt{2}}{3\sqrt{2} + 4\sqrt{2}}$ | M1 | attempt to write each term in form $n\sqrt{2}$ with at least 2 terms correct
| A1 | correct unsimplified
$= \frac{2}{7}$ | A1 | 3 marks | or $\times\frac{\sqrt{2}}{\sqrt{2}}$ with M1, integer terms $= \frac{6 + 8}{?}$ as A1, $= \frac{2}{7}$ as A1
**3(b)**
$\frac{7\sqrt{2} - \sqrt{3}}{2\sqrt{2} - \sqrt{3}} \times \frac{2\sqrt{2} + \sqrt{3}}{2\sqrt{2} + \sqrt{3}}$ | M1 |
(numerator $=$) $14 \times 2 - 2\sqrt{6} + 7\sqrt{6} - 3$ | m1 | correct unsimplified but must simplify $(\sqrt{2})^2, (\sqrt{3})^2$ and $\sqrt{2} \times \sqrt{3}$ correctly
(denominator $= 8 - 3 =$) $5$ | B1 | must be seen or identified as denominator giving $\frac{25 + 5\sqrt{6}}{5}$
(Answer $=) 5 + \sqrt{6}$ | A1 cso | 4 marks | $m = 5, n = 6$
---3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\sqrt { 18 }$ in the form $k \sqrt { 2 }$, where $k$ is an integer.
\item Simplify $\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }$.
\end{enumerate}\item Express $\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }$ in the form $m + \sqrt { n }$, where $m$ and $n$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2013 Q3 [8]}}