| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Expand and simplify surd expressions |
| Difficulty | Moderate -0.8 This is a straightforward surd manipulation question requiring standard techniques: expanding brackets using (a-b)² in part (a), then rationalizing the denominator in part (b). Both are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the two-part structure and potential for arithmetic errors. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| \((3-\sqrt{5})^2 = 9 - 6\sqrt{5} + 5 = 14 - 6\sqrt{5}\) | M1 A1 | M1 for expanding; A1 for \(m=14, n=-6\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{14-6\sqrt{5}}{1+\sqrt{5}} \times \frac{1-\sqrt{5}}{1-\sqrt{5}}\); numerator: \((14-6\sqrt{5})(1-\sqrt{5}) = 14 - 14\sqrt{5} - 6\sqrt{5} + 30 = 44 - 20\sqrt{5}\); denominator: \(1-5 = -4\); \(= \frac{44-20\sqrt{5}}{-4} = -11 + 5\sqrt{5}\) | M1 A1 M1 A1 | First M1 for multiplying by conjugate; second M1 for expanding numerator correctly |
# Question 2:
## Part (a)
| $(3-\sqrt{5})^2 = 9 - 6\sqrt{5} + 5 = 14 - 6\sqrt{5}$ | M1 A1 | M1 for expanding; A1 for $m=14, n=-6$ |
## Part (b)
| $\frac{14-6\sqrt{5}}{1+\sqrt{5}} \times \frac{1-\sqrt{5}}{1-\sqrt{5}}$; numerator: $(14-6\sqrt{5})(1-\sqrt{5}) = 14 - 14\sqrt{5} - 6\sqrt{5} + 30 = 44 - 20\sqrt{5}$; denominator: $1-5 = -4$; $= \frac{44-20\sqrt{5}}{-4} = -11 + 5\sqrt{5}$ | M1 A1 M1 A1 | First M1 for multiplying by conjugate; second M1 for expanding numerator correctly |
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\begin{enumerate}[label=(\alph*)]
\item Express $( 3 - \sqrt { 5 } ) ^ { 2 }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
\item Hence express $\frac { ( 3 - \sqrt { 5 } ) ^ { 2 } } { 1 + \sqrt { 5 } }$ in the form $p + q \sqrt { 5 }$, where $p$ and $q$ are integers.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2010 Q2 [6]}}