| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 Part (a) is straightforward simplification of a surd (√80 = 4√5). Part (b) requires rationalizing the denominator of √80/(1+√5), which is a standard textbook exercise involving multiplying by the conjugate. Both parts test routine manipulation skills with no problem-solving or novel insight required. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks |
|---|---|
| (a) \(80 = 5 \times 16\) and \(\sqrt{80} = 4\sqrt{5}\) | B1 |
| (1 mark) | |
| (b) | Method 1: |
| \(\frac{\sqrt{80}}{\sqrt{5}+1}\) or \(\frac{c\sqrt{5}}{\sqrt{5}+1}\) | B1ft |
| \(\frac{\sqrt{80}}{\sqrt{5}+1} \cdot \frac{\sqrt{5}-1}{\sqrt{5}-1}\) or \(\frac{\sqrt{80}}{1+\sqrt{5}} \cdot \frac{1-\sqrt{5}}{1-\sqrt{5}}\) | M1 |
| \(\frac{20-4\sqrt{5}}{4}\) or \(\frac{4\sqrt{5}-20}{-4}\) | A1 |
| \(= 5 - \sqrt{5}\) | A1cao |
| Method 2: | |
| \((p+q\sqrt{5})(\sqrt{5}+1) = \sqrt{80}\) | B1ft |
| \(p\sqrt{5} + q\sqrt{5} + p + 5q = 4\sqrt{5}\) | M1 |
| \(p + 5q = 0\) and \(p + q = 4\) | A1 |
| \(p = 5, q = -1\) | A1cao |
| (4 marks) | |
| Notes: | |
| (a) | B1: Accept \(4\sqrt{5}\) or \(c = 4\) – no working necessary |
| (b) (Method 1) | |
| B1ft: Only ft on \(c\). See \(\frac{\sqrt{80}}{\sqrt{5}+1}\) or \(\frac{c\sqrt{5}}{\sqrt{5}+1}\) | |
| M1: State intention to multiply by \(\sqrt{5} - 1\) or \(1 - \sqrt{5}\) in the numerator and the denominator. | |
| A1: Obtain denominator of \(4\) for \((\sqrt{5} - 1)\) or correct simplified numerator of \(20 - 4\sqrt{5}\) or \(4(5 - \sqrt{5})\) So either numerator or denominator must be correct. | |
| A1: Correct answer only. Both numerator and denominator must have been correct and division of numerator and denominator by 4 has been performed. Accept \(p = 5, q = -1\) or accept \(5 - \sqrt{5}\) or \(-\sqrt{5} + 5\). Also accept \(5 - 1\sqrt{5}\). | |
| Common error: \(\frac{\sqrt{80}}{1+\sqrt{5}} \cdot \frac{1-\sqrt{5}}{1-\sqrt{5}} = \frac{4\sqrt{5}-20}{4} = \sqrt{5} - 5\) gets B1 M1 A1 (for correct numerator – denominator is wrong for their product) then A0. | |
| Correct answer with no working – send to review – have they used a calculator? | |
| Correct answer after trial and improvement with evidence that \((5- \sqrt{5})(\sqrt{5}+1) = \sqrt{80}\) could earn all four marks. | |
| (Method 2) | |
| B1ft: Only ft on \(c\). \((p+q\sqrt{5})(\sqrt{5}+1) = \sqrt{80}\) or \(c\sqrt{5}\) | |
| M1: Multiply out the lhs and replace \(\sqrt{80}\) by \(c\sqrt{5}\) | |
| A1: Compare rational and irrational parts to give \(p + q = 4\), and \(p + 5q = 0\) | |
| A1: Solve equations to give \(p = 5, q = -1\) |
**(a)** $80 = 5 \times 16$ and $\sqrt{80} = 4\sqrt{5}$ | B1 |
| | **(1 mark)** |
**(b)** | **Method 1:** | |
| | $\frac{\sqrt{80}}{\sqrt{5}+1}$ or $\frac{c\sqrt{5}}{\sqrt{5}+1}$ | B1ft |
| | $\frac{\sqrt{80}}{\sqrt{5}+1} \cdot \frac{\sqrt{5}-1}{\sqrt{5}-1}$ or $\frac{\sqrt{80}}{1+\sqrt{5}} \cdot \frac{1-\sqrt{5}}{1-\sqrt{5}}$ | M1 |
| | $\frac{20-4\sqrt{5}}{4}$ or $\frac{4\sqrt{5}-20}{-4}$ | A1 |
| | $= 5 - \sqrt{5}$ | A1cao |
| | | |
| **Method 2:** | |
| | $(p+q\sqrt{5})(\sqrt{5}+1) = \sqrt{80}$ | B1ft |
| | $p\sqrt{5} + q\sqrt{5} + p + 5q = 4\sqrt{5}$ | M1 |
| | $p + 5q = 0$ and $p + q = 4$ | A1 |
| | $p = 5, q = -1$ | A1cao |
| | **(4 marks)** |
| **Notes:** | |
|---|---|
| **(a)** | B1: Accept $4\sqrt{5}$ or $c = 4$ – no working necessary |
| **(b) (Method 1)** | |
| | B1ft: Only ft on $c$. See $\frac{\sqrt{80}}{\sqrt{5}+1}$ or $\frac{c\sqrt{5}}{\sqrt{5}+1}$ |
| | M1: State intention to multiply by $\sqrt{5} - 1$ or $1 - \sqrt{5}$ in the numerator and the denominator. |
| | A1: Obtain denominator of $4$ for $(\sqrt{5} - 1)$ **or** correct simplified numerator of $20 - 4\sqrt{5}$ or $4(5 - \sqrt{5})$ **So either numerator or denominator must be correct**. |
| | A1: Correct answer only. Both **numerator and denominator must have been correct and division of numerator and denominator by 4 has been performed.** Accept $p = 5, q = -1$ or accept $5 - \sqrt{5}$ or $-\sqrt{5} + 5$. Also accept $5 - 1\sqrt{5}$. |
| | Common error: $\frac{\sqrt{80}}{1+\sqrt{5}} \cdot \frac{1-\sqrt{5}}{1-\sqrt{5}} = \frac{4\sqrt{5}-20}{4} = \sqrt{5} - 5$ gets B1 M1 A1 (for correct numerator – denominator is wrong for their product) then A0. |
| | Correct answer with no working – send to review – have they used a calculator? |
| | Correct answer after trial and improvement with evidence that $(5- \sqrt{5})(\sqrt{5}+1) = \sqrt{80}$ could earn all four marks. |
| **(Method 2)** | |
| | B1ft: Only ft on $c$. $(p+q\sqrt{5})(\sqrt{5}+1) = \sqrt{80}$ or $c\sqrt{5}$ |
| | M1: Multiply out the lhs and replace $\sqrt{80}$ by $c\sqrt{5}$ |
| | A1: Compare rational and irrational parts to give $p + q = 4$, and $p + 5q = 0$ |
| | A1: Solve equations to give $p = 5, q = -1$ |
---6
\begin{enumerate}[label=(\alph*)]
\item Write $\sqrt { } 80$ in the form $c \sqrt { } 5$, where $c$ is a positive constant.
A rectangle $R$ has a length of ( $1 + \sqrt { } 5$ ) cm and an area of $\sqrt { 80 } \mathrm {~cm} ^ { 2 }$.
\item Calculate the width of $R$ in cm . Express your answer in the form $p + q \sqrt { 5 }$, where $p$ and $q$ are integers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2014 Q6 [5]}}