| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Expand polynomial with surds |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation question testing basic surd operations. Part (i) requires expanding two squared brackets and simplifying surds (recognizing √8 = 2√2), while part (ii) involves routine rearrangement of a linear equation with surds. Both parts are standard textbook exercises with clear procedures and no problem-solving insight required, making this easier than average for A-level. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02c Simultaneous equations: two variables by elimination and substitution1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(4y - 3\sqrt{3} = \frac{54+3}{{\sqrt{3}}}\), leading to isolating \(y\) terms on one side | M1 | Isolating \(y\) terms on one side of the equation |
| \(y(4 - \frac{5\sqrt{3}}{3}) = 1 + 3\sqrt{3}\), making \(y\) the subject | dM1 | Making \(y\) the subject; dependent on previous M mark |
| \(y = \frac{4 + 12\sqrt{3} + \frac{8}{3}\sqrt{3} + 15}{16 - 25}\) proceeding to rational denominator | ddM1 | Proceeds to \(y = \ldots\) with a rational denominator |
| \(y = \frac{57}{23} + \frac{41}{23}\sqrt{3}\) | A1 | Correct answer with full working shown |
## Question 3 (Alt ii):
**Part (ii) - Dividing by $\sqrt{3}$, rationalising denominator and collecting terms:**
| Working | Mark | Guidance |
|---------|------|----------|
| $4y - 3\sqrt{3} = \frac{54+3}{{\sqrt{3}}}$, leading to isolating $y$ terms on one side | M1 | Isolating $y$ terms on one side of the equation |
| $y(4 - \frac{5\sqrt{3}}{3}) = 1 + 3\sqrt{3}$, making $y$ the subject | dM1 | Making $y$ the subject; dependent on previous M mark |
| $y = \frac{4 + 12\sqrt{3} + \frac{8}{3}\sqrt{3} + 15}{16 - 25}$ proceeding to rational denominator | ddM1 | Proceeds to $y = \ldots$ with a rational denominator |
| $y = \frac{57}{23} + \frac{41}{23}\sqrt{3}$ | A1 | Correct answer with full working shown |
---\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying on calculator technology are not acceptable.\\
(i)
$$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$
Express $\mathrm { f } ( x )$ in the form $a x ^ { 2 } + b x \sqrt { 2 } + c$ where $a , b$ and $c$ are integers to be found.\\
(ii) Solve the equation
$$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$
giving your answer in the form $p + q \sqrt { 3 }$ where $p$ and $q$ are simplified fractions to be found.
\hfill \mbox{\textit{Edexcel P1 2022 Q3 [7]}}