| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 Part (a) is a standard rationalizing the denominator exercise requiring multiplication by the conjugate and simplification—pure routine technique. Part (b) is straightforward Pythagoras' theorem application with surds. Both parts are textbook exercises with no problem-solving required, making this easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{5+\sqrt{7}}{3-\sqrt{7}} \times \frac{3+\sqrt{7}}{3+\sqrt{7}}\) | M1 | |
| Numerator \(= 15 + 5\sqrt{7} + 3\sqrt{7} + 7\) | m1 | Condone one error or omission |
| Denominator \(= 9 - 7\ (= 2)\) | B1 | Must be seen as the denominator |
| Answer \(= 11 + 4\sqrt{7}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((2\sqrt{5})^2 = 20\) or \((3\sqrt{2})^2 = 18\) | B1 | Either correct |
| their \((2\sqrt{5})^2 - (3\sqrt{2})^2\) | M1 | Condone missing brackets and \(x^2\) |
| \((x^2 = 20 - 18)\) \(\Rightarrow x = \sqrt{2}\) | A1 | \(\pm\sqrt{2}\) scores A0; Answer only of 2 scores B0, M0; Answer only of \(\sqrt{2}\) scores 3 marks |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{5+\sqrt{7}}{3-\sqrt{7}} \times \frac{3+\sqrt{7}}{3+\sqrt{7}}$ | M1 | |
| Numerator $= 15 + 5\sqrt{7} + 3\sqrt{7} + 7$ | m1 | Condone one error or omission |
| Denominator $= 9 - 7\ (= 2)$ | B1 | Must be seen as the denominator |
| Answer $= 11 + 4\sqrt{7}$ | A1 | |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(2\sqrt{5})^2 = 20$ or $(3\sqrt{2})^2 = 18$ | B1 | Either correct |
| their $(2\sqrt{5})^2 - (3\sqrt{2})^2$ | M1 | Condone missing brackets and $x^2$ |
| $(x^2 = 20 - 18)$ $\Rightarrow x = \sqrt{2}$ | A1 | $\pm\sqrt{2}$ scores A0; Answer only of 2 scores B0, M0; Answer only of $\sqrt{2}$ scores 3 marks |
---2
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 5 + \sqrt { 7 } } { 3 - \sqrt { 7 } }$ in the form $m + n \sqrt { 7 }$, where $m$ and $n$ are integers.
\item The diagram shows a right-angled triangle.
The hypotenuse has length $2 \sqrt { 5 } \mathrm {~cm}$. The other two sides have lengths $3 \sqrt { 2 } \mathrm {~cm}$ and $x \mathrm {~cm}$. Find the value of $x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2009 Q2 [7]}}