| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Solve equations with surds |
| Difficulty | Moderate -0.8 This is a straightforward rationalizing the denominator question followed by a simple linear equation with surds. Part (a) requires the standard technique of multiplying by the conjugate, and part (b) is basic algebraic manipulation. Both are routine exercises requiring only recall of standard methods with no problem-solving insight needed. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02c Simultaneous equations: two variables by elimination and substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{4-2\sqrt{2}} = \frac{1}{4-2\sqrt{2}} \times \frac{4+2\sqrt{2}}{4+2\sqrt{2}}\) | M1 | For sight of \(\frac{1}{4-2\sqrt{2}} \times \frac{4+2\sqrt{2}}{4+2\sqrt{2}}\) oe |
| \(= \frac{4+2\sqrt{2}}{16-8} = \frac{1}{2} + \frac{1}{4}\sqrt{2}\) | A1 | For achieving \(\frac{1}{2}+\frac{1}{4}\sqrt{2}\) or exact equivalent such as \(0.5+\frac{\sqrt{2}}{4}\), \(\frac{2}{4}+\frac{2}{8}\sqrt{2}\) or correct \(a\) and \(b\). ISW following correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4x = 2\sqrt{2}x + 20\sqrt{2} \Rightarrow (4-2\sqrt{2})x = 20\sqrt{2}\) | M1 | Collecting \(x\) terms on one side and constant on other. Must attempt to collect terms with a bracket or implied bracket |
| \(x = \frac{20\sqrt{2}}{(4-2\sqrt{2})} = 20\sqrt{2} \times \text{(a)}\) | dM1 | Using part (a) and attempting to find \(k\sqrt{2} \times \text{(a)}\) |
| \(x = 20\sqrt{2}\times\left(\frac{1}{2}+\frac{1}{4}\sqrt{2}\right) = 10 + 10\sqrt{2}\) | A1 | \(10\sqrt{2}+10\) or \(10+10\sqrt{2}\) but NOT \(10(\sqrt{2}+1)\). Cannot be awarded without sight of \(k\sqrt{2}\times\text{(a)}\) |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{4-2\sqrt{2}} = \frac{1}{4-2\sqrt{2}} \times \frac{4+2\sqrt{2}}{4+2\sqrt{2}}$ | M1 | For sight of $\frac{1}{4-2\sqrt{2}} \times \frac{4+2\sqrt{2}}{4+2\sqrt{2}}$ oe |
| $= \frac{4+2\sqrt{2}}{16-8} = \frac{1}{2} + \frac{1}{4}\sqrt{2}$ | A1 | For achieving $\frac{1}{2}+\frac{1}{4}\sqrt{2}$ or exact equivalent such as $0.5+\frac{\sqrt{2}}{4}$, $\frac{2}{4}+\frac{2}{8}\sqrt{2}$ or correct $a$ and $b$. ISW following correct answer |
**(2 marks)**
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4x = 2\sqrt{2}x + 20\sqrt{2} \Rightarrow (4-2\sqrt{2})x = 20\sqrt{2}$ | M1 | Collecting $x$ terms on one side and constant on other. Must attempt to collect terms with a bracket or implied bracket |
| $x = \frac{20\sqrt{2}}{(4-2\sqrt{2})} = 20\sqrt{2} \times \text{(a)}$ | dM1 | Using part (a) and attempting to find $k\sqrt{2} \times \text{(a)}$ |
| $x = 20\sqrt{2}\times\left(\frac{1}{2}+\frac{1}{4}\sqrt{2}\right) = 10 + 10\sqrt{2}$ | A1 | $10\sqrt{2}+10$ or $10+10\sqrt{2}$ but NOT $10(\sqrt{2}+1)$. Cannot be awarded without sight of $k\sqrt{2}\times\text{(a)}$ |
**(3 marks)**
---\begin{enumerate}
\item Answer this question showing each stage of your working.\\
(a) Simplify $\frac { 1 } { 4 - 2 \sqrt { 2 } }$\\
giving your answer in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are rational numbers.\\
(b) Hence, or otherwise, solve the equation
\end{enumerate}
$$4 x = 2 \sqrt { 2 } x + 20 \sqrt { 2 }$$
giving your answer in the form $p + q \sqrt { 2 }$ where $p$ and $q$ are rational numbers.
\hfill \mbox{\textit{Edexcel P1 2019 Q2 [5]}}