| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.3 This is a straightforward C1 surd manipulation question requiring basic simplification of square roots and rationalizing a denominator. Part (a) involves factoring out perfect squares (routine), and part (b) uses the result from (a) to simplify a fraction—standard textbook exercise with no problem-solving insight needed. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5\sqrt{2} - 3\sqrt{2}\) | M1 | \(\sqrt{50} = 5\sqrt{2}\) or \(\sqrt{18} = 3\sqrt{2}\) and the other term in the form \(k\sqrt{2}\). May be implied by correct answer \(2\sqrt{2}\) |
| \(= 2\sqrt{2}\) or \(a=2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{12\sqrt{3}}{``2"\sqrt{2}}\) | M1 | Uses part (a) by replacing denominator by their \(a\sqrt{2}\) where \(a\) is numeric |
| \(\frac{12\sqrt{3}}{``2"\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{6}}{4}\) | dM1 | Rationalises by correct method e.g. multiplies numerator and denominator by \(k\sqrt{2}\) to obtain multiple of \(\sqrt{6}\). Dependent on first M1 |
| \(= 3\sqrt{6}\) or \(b=3, c=6\) | A1 | cao and cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{12\sqrt{3}}{\sqrt{50}-\sqrt{18}} \times \frac{\sqrt{50}+\sqrt{18}}{\sqrt{50}+\sqrt{18}}\) | M1 | Rationalising denominator by multiplying by \(k(\sqrt{50}+\sqrt{18})\) |
| \(\frac{60\sqrt{6}+36\sqrt{6}}{50-18}\) | dM1 | Replacing numerator by \(\alpha\sqrt{6} + \beta\sqrt{6}\). Dependent on first M1 |
| \(= 3\sqrt{6}\) or \(b=3, c=6\) | A1 | cao and cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{12\sqrt{3}}{2\sqrt{2}} = \frac{6\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{108}}{\sqrt{2}} = \sqrt{54} = \sqrt{9 \cdot 6}\) | dM1 | Cancels to obtain multiple of \(\sqrt{6}\). Dependent on first M1 |
| \(= 3\sqrt{6}\) or \(b=3, c=6\) | A1 | cao and cso |
# Question 3(a):
$$\sqrt{50} - \sqrt{18}$$
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5\sqrt{2} - 3\sqrt{2}$ | M1 | $\sqrt{50} = 5\sqrt{2}$ or $\sqrt{18} = 3\sqrt{2}$ **and** the other term in the form $k\sqrt{2}$. May be implied by correct answer $2\sqrt{2}$ |
| $= 2\sqrt{2}$ or $a=2$ | A1 | |
**Total: [2]**
---
# Question 3(b):
$$\frac{12\sqrt{3}}{\sqrt{50}-\sqrt{18}}$$
**WAY 1:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12\sqrt{3}}{``2"\sqrt{2}}$ | M1 | Uses part (a) by replacing denominator by their $a\sqrt{2}$ where $a$ is numeric |
| $\frac{12\sqrt{3}}{``2"\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{6}}{4}$ | dM1 | Rationalises by correct method e.g. multiplies numerator and denominator by $k\sqrt{2}$ to obtain multiple of $\sqrt{6}$. **Dependent on first M1** |
| $= 3\sqrt{6}$ or $b=3, c=6$ | A1 | cao and cso |
**WAY 2:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12\sqrt{3}}{\sqrt{50}-\sqrt{18}} \times \frac{\sqrt{50}+\sqrt{18}}{\sqrt{50}+\sqrt{18}}$ | M1 | Rationalising denominator by multiplying by $k(\sqrt{50}+\sqrt{18})$ |
| $\frac{60\sqrt{6}+36\sqrt{6}}{50-18}$ | dM1 | Replacing numerator by $\alpha\sqrt{6} + \beta\sqrt{6}$. **Dependent on first M1** |
| $= 3\sqrt{6}$ or $b=3, c=6$ | A1 | cao and cso |
**WAY 3:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12\sqrt{3}}{2\sqrt{2}} = \frac{6\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{108}}{\sqrt{2}} = \sqrt{54} = \sqrt{9 \cdot 6}$ | dM1 | Cancels to obtain multiple of $\sqrt{6}$. **Dependent on first M1** |
| $= 3\sqrt{6}$ or $b=3, c=6$ | A1 | cao and cso |
**Total: 5 marks**
---\begin{enumerate}[label=(\alph*)]
\item Simplify
$$\sqrt { 50 } - \sqrt { 18 }$$
giving your answer in the form $a \sqrt { 2 }$, where $a$ is an integer.
\item Hence, or otherwise, simplify
$$\frac { 12 \sqrt { 3 } } { \sqrt { 50 } - \sqrt { 18 } }$$
giving your answer in the form $b \sqrt { c }$, where $b$ and $c$ are integers and $b \neq 1$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2016 Q3 [5]}}