| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Solve equations with surds |
| Difficulty | Moderate -0.8 This is a routine C1 surds question testing standard techniques: rationalizing denominators, simplifying surds, and solving linear equations with surds. All parts follow textbook procedures with no problem-solving required, making it easier than average but not trivial since it requires careful algebraic manipulation across multiple steps. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Numerator \(= 5 + 3\sqrt{5} + 2\sqrt{5} + 6 = \pm(\sqrt{5} + 3)(\sqrt{5} + 2)\) | M1, M1 | Multiplying top & bottom by \(\pm(\sqrt{5} + 2)\); Multiplying out (condone one slip) \(\pm(\sqrt{5} + 3)(\sqrt{5} + 2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Final answer \(= 5\sqrt{5} + 11\) | A1, A1 | With clear evidence that denominator \(= 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{45} = 3\sqrt{5}\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \([x 2\sqrt{5} = 7\sqrt{5} - 3\sqrt{5}]\) or \(x\sqrt{20} = 2\sqrt{20}\) | M1 | Both sides; or \(x = \sqrt{4}\) |
| \(x = 2\) | A1 | CSO |
## 3(a)
$\frac{\sqrt{5} + 3}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2}$
Numerator $= 5 + 3\sqrt{5} + 2\sqrt{5} + 6 = \pm(\sqrt{5} + 3)(\sqrt{5} + 2)$ | M1, M1 | Multiplying top & bottom by $\pm(\sqrt{5} + 2)$; Multiplying out (condone one slip) $\pm(\sqrt{5} + 3)(\sqrt{5} + 2)$
$= \frac{5\sqrt{5} + 11}{\text{}}$
Final answer $= 5\sqrt{5} + 11$ | A1, A1 | With clear evidence that denominator $= 1$
**Total (part a): 4**
## 3(b)(i)
$\sqrt{45} = 3\sqrt{5}$ | B1 | 1 mark
## 3(b)(ii)
$\sqrt{20} = \sqrt{4}\sqrt{5}$ or $4\sqrt{5} = \sqrt{4} \times \sqrt{20}$
or attempt to have equation with $\sqrt{5}$ or $\sqrt{20}$ only
$[x 2\sqrt{5} = 7\sqrt{5} - 3\sqrt{5}]$ or $x\sqrt{20} = 2\sqrt{20}$ | M1 | Both sides; or $x = \sqrt{4}$
$x = 2$ | A1 | CSO
**Total (part b): 3**
**Total: 8**
---3
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }$ in the form $p \sqrt { 5 } + q$, where $p$ and $q$ are integers.
\item \begin{enumerate}[label=(\roman*)]
\item Express $\sqrt { 45 }$ in the form $n \sqrt { 5 }$, where $n$ is an integer.
\item Solve the equation
$$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$
giving your answer in its simplest form.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q3 [8]}}