| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Moderate -0.3 This is a straightforward C1 question testing standard surd manipulation (expanding brackets, rationalizing denominators) and basic substitution. Part (a) involves routine algebraic techniques, while part (b) requires recognizing a simple substitution (y = x^{1/2}), but the connection is heavily signposted by 'hence'. All techniques are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(= 16 - 24\sqrt{2} + 18 = 34 - 24\sqrt{2}\) | M1 A1 | |
| (iii) \(= \frac{1}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}}\) | M1 | |
| \(= \frac{2 - \sqrt{2}}{4 - 2} = 1 - \frac{1}{2}\sqrt{2}\) | M1 A1 | |
| (b) (i) \(y^2 - 9y + 8 = 0\) | ||
| \((y - 1)(y - 8) = 0\) | M1 | |
| \(y = 1, 8\) | A1 | |
| (iii) let \(y = x^2 \Rightarrow y^2 + 8 = 9y\) | B1 | |
| \(\therefore x^2 = 1, 8\) | M1 | |
| \(x = 1\) or \((\sqrt{8})^2\) | ||
| \(x = 1\) or \(4\) | A1 | (10 marks) |
**(a)** (i) $= 16 - 24\sqrt{2} + 18 = 34 - 24\sqrt{2}$ | M1 A1 |
(iii) $= \frac{1}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ | M1 |
$= \frac{2 - \sqrt{2}}{4 - 2} = 1 - \frac{1}{2}\sqrt{2}$ | M1 A1 |
**(b)** (i) $y^2 - 9y + 8 = 0$ | |
$(y - 1)(y - 8) = 0$ | M1 |
$y = 1, 8$ | A1 |
(iii) let $y = x^2 \Rightarrow y^2 + 8 = 9y$ | B1 |
$\therefore x^2 = 1, 8$ | M1 |
$x = 1$ or $(\sqrt{8})^2$ | |
$x = 1$ or $4$ | A1 | (10 marks)\begin{enumerate}[label=(\alph*)]
\item Express each of the following in the form $p + q\sqrt{2}$ where $p$ and $q$ are rational.
\begin{enumerate}[label=(\roman*)]
\item $(4 - 3\sqrt{2})^2$
\item $\frac{1}{2 + \sqrt{2}}$ [5]
\end{enumerate}
\item
\begin{enumerate}[label=(\roman*)]
\item Solve the equation
$$y^2 + 8 = 9y.$$
\item Hence solve the equation
$$x^3 + 8 = 9x^{\frac{1}{2}}.$$ [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [10]}}