| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Simplify using given results |
| Difficulty | Moderate -0.8 This is a structured multi-part question with clear signposting where parts (a) and (b) provide the results needed for part (c). Part (a) is routine algebraic expansion, part (b) is standard rationalizing the denominator, and part (c) requires recognizing how to apply the given results but the path is heavily scaffolded. Easier than average due to the explicit guidance structure. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(r - \frac{1}{r}\right)^2 = r^2 - r\times\frac{1}{r} - r\times\frac{1}{r} + \frac{1}{r^2}\) | M1 | Expands bracket to obtain 3 or 4 terms with at least 2 correct (may be unsimplified). Allow use of different variable e.g. \(x\) |
| \(= r^2 + \frac{1}{r^2} - 2\) | A1 | Correct simplified expansion in terms of \(r\). Accept terms in any order. Accept e.g. \(r^2 + r^{-2} - 2\) and \(r^2 + \left(\frac{1}{r}\right)^2 - 2\). Do not ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{1}{3+2\sqrt{2}} = \frac{1}{3+2\sqrt{2}} \times \frac{3-2\sqrt{2}}{3-2\sqrt{2}}\) | M1 | Correct process to rationalise denominator — look for multiplying numerator and denominator by \(3-2\sqrt{2}\) or any multiple. No processing required, just look for the statement |
| \(= \frac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2} = 3-2\sqrt{2}\) | A1 | Shows an intermediate line before obtaining \(3-2\sqrt{2}\). Do not allow \(\frac{3-2\sqrt{2}}{1}\) as final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{1}{3+2\sqrt{2}} = p+q\sqrt{2} \Rightarrow 1=(p+q\sqrt{2})(3+2\sqrt{2}) = 3p+4q+2p\sqrt{2}+3q\sqrt{2}\) \(\Rightarrow 3p+4q=1,\quad 2p+3q=0\) | M1 | Sets \(\frac{1}{3+2\sqrt{2}}=p+q\sqrt{2}\), multiplies up, compares rational and irrational parts to produce 2 equations in \(p\) and \(q\) |
| \(\Rightarrow q=-2,\ p=3,\quad \frac{1}{3+2\sqrt{2}} = 3-2\sqrt{2}\) | A1 | Solves simultaneously showing working and obtains \(3-2\sqrt{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}}\right)^2 = 3+2\sqrt{2} + \frac{1}{3+2\sqrt{2}} - 2\) | M1 | Applies result from (a) or correct result to \(\left(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}}\right)^2\) |
| \(= 3+2\sqrt{2} + 3-2\sqrt{2} - 2\ ...\ (=4)\) | dM1 | Depends on first mark. Applies result from (b) and cancels \(\sqrt{2}\) terms to achieve a constant |
| so \(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}} = 2\) | A1 | Reaches correct answer from correct work. No requirement to justify the positive square root |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}} = 2 \Rightarrow 3+2\sqrt{2}-1 = 2\sqrt{3+2\sqrt{2}}\) | M1 | Multiplies equation through by \(\sqrt{3+2\sqrt{2}}\) and applies \(\sqrt{3+2\sqrt{2}}\times\sqrt{3+2\sqrt{2}} = 3+2\sqrt{2}\) |
| \(\Rightarrow \left(2+2\sqrt{2}\right)^2 = 4\left(3+2\sqrt{2}\right)\) | dM1 | Depends on first mark. Squares both sides (may or may not have cancelled the 2 before squaring) |
| \(\Rightarrow 4+8\sqrt{2}+8 = 12+8\sqrt{2}\ \checkmark\) Hence true | A1 | Achieves similar expression for both sides and gives a (minimal) conclusion |
## Question 6:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(r - \frac{1}{r}\right)^2 = r^2 - r\times\frac{1}{r} - r\times\frac{1}{r} + \frac{1}{r^2}$ | M1 | Expands bracket to obtain 3 or 4 terms with at least 2 correct (may be unsimplified). Allow use of different variable e.g. $x$ |
| $= r^2 + \frac{1}{r^2} - 2$ | A1 | Correct simplified expansion in terms of $r$. Accept terms in any order. Accept e.g. $r^2 + r^{-2} - 2$ and $r^2 + \left(\frac{1}{r}\right)^2 - 2$. Do **not** ISW |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{3+2\sqrt{2}} = \frac{1}{3+2\sqrt{2}} \times \frac{3-2\sqrt{2}}{3-2\sqrt{2}}$ | M1 | Correct process to rationalise denominator — look for multiplying numerator and denominator by $3-2\sqrt{2}$ or any multiple. No processing required, just look for the statement |
| $= \frac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2} = 3-2\sqrt{2}$ | A1 | Shows an intermediate line before obtaining $3-2\sqrt{2}$. Do **not** allow $\frac{3-2\sqrt{2}}{1}$ as final answer |
### Part (b) ALT:
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{3+2\sqrt{2}} = p+q\sqrt{2} \Rightarrow 1=(p+q\sqrt{2})(3+2\sqrt{2}) = 3p+4q+2p\sqrt{2}+3q\sqrt{2}$ $\Rightarrow 3p+4q=1,\quad 2p+3q=0$ | M1 | Sets $\frac{1}{3+2\sqrt{2}}=p+q\sqrt{2}$, multiplies up, compares rational and irrational parts to produce 2 equations in $p$ and $q$ |
| $\Rightarrow q=-2,\ p=3,\quad \frac{1}{3+2\sqrt{2}} = 3-2\sqrt{2}$ | A1 | Solves simultaneously **showing working** and obtains $3-2\sqrt{2}$ |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}}\right)^2 = 3+2\sqrt{2} + \frac{1}{3+2\sqrt{2}} - 2$ | M1 | Applies result from (a) or correct result to $\left(\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}}\right)^2$ |
| $= 3+2\sqrt{2} + 3-2\sqrt{2} - 2\ ...\ (=4)$ | dM1 | Depends on first mark. Applies result from (b) and cancels $\sqrt{2}$ terms to achieve a constant |
| so $\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}} = 2$ | A1 | Reaches correct answer from correct work. **No requirement to justify the positive square root** |
### Part (c) ALT:
| Working | Mark | Guidance |
|---------|------|----------|
| $\sqrt{3+2\sqrt{2}} - \frac{1}{\sqrt{3+2\sqrt{2}}} = 2 \Rightarrow 3+2\sqrt{2}-1 = 2\sqrt{3+2\sqrt{2}}$ | M1 | Multiplies equation through by $\sqrt{3+2\sqrt{2}}$ and applies $\sqrt{3+2\sqrt{2}}\times\sqrt{3+2\sqrt{2}} = 3+2\sqrt{2}$ |
| $\Rightarrow \left(2+2\sqrt{2}\right)^2 = 4\left(3+2\sqrt{2}\right)$ | dM1 | Depends on first mark. Squares both sides (may or may not have cancelled the 2 before squaring) |
| $\Rightarrow 4+8\sqrt{2}+8 = 12+8\sqrt{2}\ \checkmark$ Hence true | A1 | Achieves similar expression for both sides and gives a (minimal) conclusion |\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.\\
(a) Expand and simplify
\end{enumerate}
$$\left( r - \frac { 1 } { r } \right) ^ { 2 }$$
(b) Express $\frac { 1 } { 3 + 2 \sqrt { 2 } }$ in the form $p + q \sqrt { 2 }$ where $p$ and $q$ are integers.\\
(c) Use the results of parts (a) and (b), or otherwise, to show that
$$\sqrt { 3 + 2 \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 + 2 \sqrt { 2 } } } = 2$$
\hfill \mbox{\textit{Edexcel P1 2023 Q6 [7]}}