| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Show surd expression equals value |
| Difficulty | Moderate -0.8 This is a straightforward surds manipulation question requiring standard techniques (simplifying surds by factoring, rationalizing denominators). Part (i) involves basic simplification of surds with no problem-solving insight needed. Part (ii) is a routine rationalization exercise. Both parts are textbook-standard with clear methods, making this easier than average for A-level. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Writes \(\sqrt{180}\) as \(6\sqrt{5}\) or \(\sqrt{80}\) as \(4\sqrt{5}\) | M1 | Additional working may be seen but not required |
| \(\frac{6\sqrt{5} - 4\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2\) | A1 | Correct work leading to answer 2. M mark must have been awarded |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{4\sqrt{5}-5}{7-3\sqrt{5}} = \frac{4\sqrt{5}-5}{7-3\sqrt{5}} \times \frac{7+3\sqrt{5}}{7+3\sqrt{5}} = \ldots\) | M1 | Correct attempt to rationalise by multiplying numerator and denominator by \(7+3\sqrt{5}\). Multiplications need not be carried out |
| \(= \frac{28\sqrt{5} - 35 + 12(\sqrt{5})^2 - 15\sqrt{5}}{49 - 9 \times 5}\) | dM1 | Dependent on previous M1. Correct expression for denominator (without surds), accept \(7^2 - 3^2 \times 5\) or just 4. Attempts to multiply out numerator with sight of 4 terms |
| \(= \frac{25}{4} + \frac{13}{4}\sqrt{5}\) | A1 | Correct answer any way around. Allow \(6.25 + 3.25\sqrt{5}\). Do NOT allow \(\frac{25+13\sqrt{5}}{4}\) unless followed by "hence \(a=\frac{25}{4}, b=\frac{13}{4}\)" |
# Question 3(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Writes $\sqrt{180}$ as $6\sqrt{5}$ or $\sqrt{80}$ as $4\sqrt{5}$ | M1 | Additional working may be seen but not required |
| $\frac{6\sqrt{5} - 4\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2$ | A1 | Correct work leading to answer 2. M mark must have been awarded |
# Question 3(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{4\sqrt{5}-5}{7-3\sqrt{5}} = \frac{4\sqrt{5}-5}{7-3\sqrt{5}} \times \frac{7+3\sqrt{5}}{7+3\sqrt{5}} = \ldots$ | M1 | Correct attempt to rationalise by multiplying numerator and denominator by $7+3\sqrt{5}$. Multiplications need not be carried out |
| $= \frac{28\sqrt{5} - 35 + 12(\sqrt{5})^2 - 15\sqrt{5}}{49 - 9 \times 5}$ | dM1 | Dependent on previous M1. Correct expression for denominator (without surds), accept $7^2 - 3^2 \times 5$ or just 4. Attempts to multiply out numerator with sight of 4 terms |
| $= \frac{25}{4} + \frac{13}{4}\sqrt{5}$ | A1 | Correct answer any way around. Allow $6.25 + 3.25\sqrt{5}$. Do NOT allow $\frac{25+13\sqrt{5}}{4}$ unless followed by "hence $a=\frac{25}{4}, b=\frac{13}{4}$" |
---3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.\\
(i) Show that $\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }$ is an integer and find its value.\\
(ii) Simplify
$$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$
giving your answer in the form $a + b \sqrt { 5 }$ where $a$ and $b$ are rational numbers.\\
\hfill \mbox{\textit{Edexcel P1 2022 Q3 [5]}}