| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| 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 requiring standard techniques: simplifying surds and rationalizing denominators. Part (a) involves basic simplification and rationalization, while part (b) uses the standard method of multiplying by the conjugate. Both are textbook exercises with no problem-solving or insight required, making this easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{\sqrt{63}}{3} = \sqrt{7}\) or \(\frac{3\sqrt{7}}{3}\) | B1 | or \(\frac{(\sqrt{7}\sqrt{63}+14\times3)}{3\sqrt{7}}\) |
| \(\frac{14}{\sqrt{7}} = 2\sqrt{7}\) or \(\frac{14\sqrt{7}}{7}\) | B1 | or \(\frac{\sqrt{7}}{\sqrt{7}}(\quad)\) M1 |
| \(\Rightarrow\) sum \(= 3\sqrt{7}\) | B1 | \(\Rightarrow\) correct answer with all working correct A2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Multiply by \(\frac{\sqrt{7}+2}{\sqrt{7}+2}\) | M1 | |
| Denominator \(= 7 - 4 = 3\) | A1 | |
| Numerator \(= (\sqrt{7})^2 + \sqrt{7} + 2\sqrt{7} + 2\) | m1 | multiplied out (allow one slip) \(9 + 3\sqrt{7}\) |
| Answer \(= \sqrt{7} + 3\) | A1 |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{\sqrt{63}}{3} = \sqrt{7}$ or $\frac{3\sqrt{7}}{3}$ | B1 | or $\frac{(\sqrt{7}\sqrt{63}+14\times3)}{3\sqrt{7}}$ |
| $\frac{14}{\sqrt{7}} = 2\sqrt{7}$ or $\frac{14\sqrt{7}}{7}$ | B1 | or $\frac{\sqrt{7}}{\sqrt{7}}(\quad)$ M1 |
| $\Rightarrow$ sum $= 3\sqrt{7}$ | B1 | $\Rightarrow$ correct answer with all working correct A2 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Multiply by $\frac{\sqrt{7}+2}{\sqrt{7}+2}$ | M1 | |
| Denominator $= 7 - 4 = 3$ | A1 | |
| Numerator $= (\sqrt{7})^2 + \sqrt{7} + 2\sqrt{7} + 2$ | m1 | multiplied out (allow one slip) $9 + 3\sqrt{7}$ |
| Answer $= \sqrt{7} + 3$ | A1 | |
---2
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { \sqrt { 63 } } { 3 } + \frac { 14 } { \sqrt { 7 } }$ in the form $n \sqrt { 7 }$, where $n$ is an integer.
\item Express $\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 2 }$ in the form $p \sqrt { 7 } + q$, where $p$ and $q$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q2 [7]}}