| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Show surd expression equals value |
| Difficulty | Easy -1.2 This is a routine Core 1/2 surd manipulation question requiring standard techniques: simplifying surds (√45 = 3√5), rationalizing denominators, and multiplying surds. Both parts follow textbook procedures with no problem-solving or insight needed—purely mechanical application of surd rules, making it easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(= \sqrt{9}\sqrt{5} - \frac{20\sqrt{5}}{\sqrt{5}\sqrt{5}} + \sqrt{6}\sqrt{6}\sqrt{5} = 3\sqrt{5} - 4\sqrt{5} + 6\sqrt{5}\) | M1 | Shows at least one term on LHS as multiple of \(\sqrt{5}\) with correct intermediate step |
| \(= 5\sqrt{5}\) | A1* | All three terms must show intermediate step \(3\sqrt{5} - 4\sqrt{5} + 6\sqrt{5}\) followed by \(5\sqrt{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{LHS} = \frac{17\sqrt{2}(\sqrt{2}-6)}{(\sqrt{2}+6)(\sqrt{2}-6)}\) | M1 | Multiply numerator and denominator by \(\sqrt{2}-6\) or \(6-\sqrt{2}\) |
| \(= \frac{17\times2 - 17\times6\sqrt{2}}{2-36}\) | A1 | Multiplies out to correct unsimplified answer |
| \(= \frac{34 - 102\sqrt{2}}{-34} = 3\sqrt{2} - 1\)* | A1 | Denominator must be simplified; \(\frac{34-17\times6\sqrt{2}}{-34}\) seen before reaching given answer |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $= \sqrt{9}\sqrt{5} - \frac{20\sqrt{5}}{\sqrt{5}\sqrt{5}} + \sqrt{6}\sqrt{6}\sqrt{5} = 3\sqrt{5} - 4\sqrt{5} + 6\sqrt{5}$ | M1 | Shows at least one term on LHS as multiple of $\sqrt{5}$ with correct intermediate step |
| $= 5\sqrt{5}$ | A1* | All three terms must show intermediate step $3\sqrt{5} - 4\sqrt{5} + 6\sqrt{5}$ followed by $5\sqrt{5}$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{LHS} = \frac{17\sqrt{2}(\sqrt{2}-6)}{(\sqrt{2}+6)(\sqrt{2}-6)}$ | M1 | Multiply numerator and denominator by $\sqrt{2}-6$ or $6-\sqrt{2}$ |
| $= \frac{17\times2 - 17\times6\sqrt{2}}{2-36}$ | A1 | Multiplies out to correct unsimplified answer |
| $= \frac{34 - 102\sqrt{2}}{-34} = 3\sqrt{2} - 1$* | A1 | Denominator must be simplified; $\frac{34-17\times6\sqrt{2}}{-34}$ seen before reaching given answer |
---3. Answer this question without the use of a calculator and show your method clearly.\\
(i) Show that
$$\sqrt { 45 } - \frac { 20 } { \sqrt { 5 } } + \sqrt { 6 } \sqrt { 30 } = 5 \sqrt { 5 }$$
(ii) Show that
$$\frac { 17 \sqrt { 2 } } { \sqrt { 2 } + 6 } = 3 \sqrt { 2 } - 1$$
\hfill \mbox{\textit{Edexcel C12 2016 Q3 [5]}}