| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 This is a routine C1 surds question testing standard techniques: simplifying surds by factoring, basic arithmetic with surds, and rationalizing denominators. All parts follow textbook procedures with no problem-solving required, making it easier than average but not trivial due to the multi-step nature of part (b). |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{48} = 4\sqrt{3}\) | B1 | condone \(k = 4\) stated |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4\sqrt{3} + 6\sqrt{3}}{2\sqrt{3}}\) | M1 | attempt to write each term in form \(k\sqrt{3}\) with at least 2 terms correctly obtained |
| A1 | correct unsimplified in terms of \(\sqrt{3}\) only | |
| \(= 5\) | A1 also | must simplify fraction to 5 |
| Alternative 1: \(\times\frac{\sqrt{12}}{\sqrt{12}}\) (or \(\times\frac{\sqrt{3}}{\sqrt{3}}\)) | M1 | correct with integer terms \(= \frac{24 + 36}{12}\) |
| Alternative 2: \(\frac{\sqrt{48} + \sqrt{108}}{\sqrt{12}}\) | M1 | \(= \sqrt{4} + \sqrt{9}\) |
| Alternative 3: \(\sqrt{\frac{48}{12}} + 2\sqrt{\frac{27}{12}}\) | M1 | \(= 2 + 2\sqrt{\frac{9}{4}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1 - 5\sqrt{5}}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}}\) | M1 | |
| \((numerator) = 3 - \sqrt{5} - 15\sqrt{5} + 25\) | m1 | correct unsimplified but must write \(5\sqrt{5}\sqrt{5} = 25\) PI by 28 seen later |
| \((denominator = 9 - 5) = 4\) | B1 | must be seen as denominator |
| giving \(\frac{28 - 16\sqrt{5}}{4}\) | ||
| \((answer) = 7 - 4\sqrt{5}\) | A1 | \(m = 7, n = -4\) |
**2(a)(i)**
$\sqrt{48} = 4\sqrt{3}$ | B1 | condone $k = 4$ stated
**2(a)(ii)**
$\frac{4\sqrt{3} + 6\sqrt{3}}{2\sqrt{3}}$ | M1 | attempt to write each term in form $k\sqrt{3}$ with at least 2 terms correctly obtained
| A1 | correct unsimplified in terms of $\sqrt{3}$ only
$= 5$ | A1 also | must simplify fraction to 5
**Alternative 1:** $\times\frac{\sqrt{12}}{\sqrt{12}}$ (or $\times\frac{\sqrt{3}}{\sqrt{3}}$) | M1 | correct with integer terms $= \frac{24 + 36}{12}$ | A1 | $= 5$ | A1 also |
**Alternative 2:** $\frac{\sqrt{48} + \sqrt{108}}{\sqrt{12}}$ | M1 | $= \sqrt{4} + \sqrt{9}$ | A1 | $= 5$ | A1 also |
**Alternative 3:** $\sqrt{\frac{48}{12}} + 2\sqrt{\frac{27}{12}}$ | M1 | $= 2 + 2\sqrt{\frac{9}{4}}$ | A1 | $= 5$ | A1 also |
if hybrid of methods used, award M1 and most appropriate first A1
NMS (answer = 5) scores full marks
**2(b)**
$\frac{1 - 5\sqrt{5}}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$ | M1 |
$(numerator) = 3 - \sqrt{5} - 15\sqrt{5} + 25$ | m1 | correct unsimplified but must write $5\sqrt{5}\sqrt{5} = 25$ PI by 28 seen later
$(denominator = 9 - 5) = 4$ | B1 | must be seen as denominator
giving $\frac{28 - 16\sqrt{5}}{4}$ | |
$(answer) = 7 - 4\sqrt{5}$ | A1 | $m = 7, n = -4$
---2
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\sqrt { 48 }$ in the form $k \sqrt { 3 }$, where $k$ is an integer.
\item Simplify $\frac { \sqrt { 48 } + 2 \sqrt { 27 } } { \sqrt { 12 } }$, giving your answer as an integer.
\end{enumerate}\item Express $\frac { 1 - 5 \sqrt { 5 } } { 3 + \sqrt { 5 } }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2011 Q2 [8]}}