| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| 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 testing standard techniques: squaring a surd, expanding brackets with surds, and rationalizing a denominator. All parts follow textbook procedures with no problem-solving required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((3\sqrt{2})^2 = 18\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((3\sqrt{2}-1)^2 = \text{'their }18\text{'} - 3\sqrt{2} - 3\sqrt{2} + 1\) | M1 | FT their \((3\sqrt{2})^2\) |
| \(= 18 - 3\sqrt{2} - 3\sqrt{2} + 1\) | A1 | \((= 19 - 6\sqrt{2})\) |
| \((3+\sqrt{2})^2 = 9 + 3\sqrt{2} + 3\sqrt{2} + 2\) | B1 | \((= 11 + 6\sqrt{2})\) |
| \(\Rightarrow \text{Sum} = 30\) | A1cso |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{4\sqrt{5}-7\sqrt{2}}{2\sqrt{5}+\sqrt{2}} \times \frac{2\sqrt{5}-\sqrt{2}}{2\sqrt{5}-\sqrt{2}}\) | M1 | |
| Numerator \(= 8(\sqrt{5})^2 - 4\sqrt{5}\sqrt{2} - 14\sqrt{5}\sqrt{2} + 7(\sqrt{2})^2\) | m1 | Correct unsimplified \((= 54 - 18\sqrt{10})\) |
| Denominator \(= (2\sqrt{5})^2 - (\sqrt{2})^2 = 18\) | B1 | Must be seen as denominator |
| \(\Rightarrow \text{Answer} = 3 - \sqrt{10}\) | A1cso |
# Question 3:
## Part (a)(i)
| Working | Mark | Guidance |
|---------|------|----------|
| $(3\sqrt{2})^2 = 18$ | B1 | |
## Part (a)(ii)
| Working | Mark | Guidance |
|---------|------|----------|
| $(3\sqrt{2}-1)^2 = \text{'their }18\text{'} - 3\sqrt{2} - 3\sqrt{2} + 1$ | M1 | FT their $(3\sqrt{2})^2$ |
| $= 18 - 3\sqrt{2} - 3\sqrt{2} + 1$ | A1 | $(= 19 - 6\sqrt{2})$ |
| $(3+\sqrt{2})^2 = 9 + 3\sqrt{2} + 3\sqrt{2} + 2$ | B1 | $(= 11 + 6\sqrt{2})$ |
| $\Rightarrow \text{Sum} = 30$ | A1cso | |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{4\sqrt{5}-7\sqrt{2}}{2\sqrt{5}+\sqrt{2}} \times \frac{2\sqrt{5}-\sqrt{2}}{2\sqrt{5}-\sqrt{2}}$ | M1 | |
| Numerator $= 8(\sqrt{5})^2 - 4\sqrt{5}\sqrt{2} - 14\sqrt{5}\sqrt{2} + 7(\sqrt{2})^2$ | m1 | Correct unsimplified $(= 54 - 18\sqrt{10})$ |
| Denominator $= (2\sqrt{5})^2 - (\sqrt{2})^2 = 18$ | B1 | Must be seen as denominator |
| $\Rightarrow \text{Answer} = 3 - \sqrt{10}$ | A1cso | |3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Simplify $( 3 \sqrt { 2 } ) ^ { 2 }$.
\item Show that $( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }$ is an integer and find its value.
\end{enumerate}\item Express $\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }$ in the form $m - \sqrt { n }$, where $m$ and $n$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2012 Q3 [9]}}