| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Expand and simplify surd expressions |
| Difficulty | Easy -1.2 This is a routine C1 surds question requiring standard techniques: (a) expanding a binomial with surds, and (b) rationalizing a denominator by multiplying by the conjugate. Both are textbook exercises with straightforward algebraic manipulation and no problem-solving insight required, making it easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(3 + 1 + 2\sqrt{3}\) | M1 | Multiplied out; at least 3 terms with \(\sqrt{3}\) term |
| \(= 4 + 2\sqrt{3}\) | A1 | \(m = 4\), \(n = 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Multiplying top and bottom by \(\sqrt{3} + 1\) | M1 | |
| Denominator \(= 3 - 1 = 2\) | B1 | |
| Expression \(= \frac{4 + 2\sqrt{3}}{2}\) | ||
| \(= 2 + \sqrt{3}\) | A1 | CSO \(m = 2\), \(n = 1\) |
## Question 5:
**Part (a)**
| Working | Marks | Guidance |
|---------|-------|----------|
| $3 + 1 + 2\sqrt{3}$ | M1 | Multiplied out; at least 3 terms with $\sqrt{3}$ term |
| $= 4 + 2\sqrt{3}$ | A1 | $m = 4$, $n = 2$ |
**Part (b)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Multiplying top and bottom by $\sqrt{3} + 1$ | M1 | |
| Denominator $= 3 - 1 = 2$ | B1 | |
| Expression $= \frac{4 + 2\sqrt{3}}{2}$ | | |
| $= 2 + \sqrt{3}$ | A1 | **CSO** $m = 2$, $n = 1$ |
**Total: 5 marks**
---5 Express each of the following in the form $m + n \sqrt { 3 }$, where $m$ and $n$ are integers:
\begin{enumerate}[label=(\alph*)]
\item $( \sqrt { 3 } + 1 ) ^ { 2 }$;
\item $\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q5 [5]}}