| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.8 This is a routine C1 question testing basic surd manipulation: simplifying √45 requires factorizing under the root, and rationalizing the denominator is a standard textbook exercise using conjugate multiplication. Both parts are direct application of memorized techniques with no problem-solving required. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(3\sqrt{5}\) (or \(a = 3\)) | B1 | |
| (b) \(\frac{2(3 + \sqrt{5})}{(3-\sqrt{5})} \times \frac{(3+\sqrt{5})}{(3+\sqrt{5})}\) | M1 | |
| \((3-\sqrt{5})(3+\sqrt{5}) = 9 - 5 = 4\) (Used as or intended as denominator) | B1 | |
| \((3+\sqrt{5})(p \pm q\sqrt{5}) = \ldots 4\) terms (\(p \neq 0, q \neq 0\)) (Independent) | M1 | |
| or \((6 + 2\sqrt{5})(p \pm q\sqrt{5}) = \ldots 4\) terms (\(p \neq 0, q \neq 0\)) | ||
| [Correct version: \((3+\sqrt{5})(3+\sqrt{5}) = 9 + 3\sqrt{5} + 3\sqrt{5} + 5\), or double this.] | ||
| \(\frac{2(14 + 6\sqrt{5})}{4} = 7 + 3\sqrt{5}\) | A1 A1 | 1st A1: \(b = 7\), 2nd A1: \(c = 3\) |
| Answer | Marks |
|---|---|
| Guidance: (b) 2nd M mark for attempting \((3+\sqrt{5})(p + q\sqrt{5})\) is generous. Condone errors. |
**(a)** $3\sqrt{5}$ (or $a = 3$) | B1 |
**(b)** $\frac{2(3 + \sqrt{5})}{(3-\sqrt{5})} \times \frac{(3+\sqrt{5})}{(3+\sqrt{5})}$ | M1 |
$(3-\sqrt{5})(3+\sqrt{5}) = 9 - 5 = 4$ (Used as or intended as denominator) | B1 |
$(3+\sqrt{5})(p \pm q\sqrt{5}) = \ldots 4$ terms ($p \neq 0, q \neq 0$) (Independent) | M1 |
or $(6 + 2\sqrt{5})(p \pm q\sqrt{5}) = \ldots 4$ terms ($p \neq 0, q \neq 0$) | |
[Correct version: $(3+\sqrt{5})(3+\sqrt{5}) = 9 + 3\sqrt{5} + 3\sqrt{5} + 5$, or double this.] | |
$\frac{2(14 + 6\sqrt{5})}{4} = 7 + 3\sqrt{5}$ | A1 A1 | 1st A1: $b = 7$, 2nd A1: $c = 3$
**Total: 6 marks**
| Guidance: (b) 2nd M mark for attempting $(3+\sqrt{5})(p + q\sqrt{5})$ is generous. Condone errors.
---5. (a) Write $\sqrt { 45 }$ in the form $a \sqrt { 5 }$, where $a$ is an integer.\\
(b) Express $\frac { 2 ( 3 + \sqrt { 5 } ) } { ( 3 - \sqrt { 5 } ) }$ in the form $b + c \sqrt { 5 }$, where $b$ and $c$ are integers.\\
\section*{6.}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-07_607_844_310_555}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve passes through the points $( 0,3 )$ and $( 4,0 )$ and touches the $x$-axis at the point $( 1,0 )$.
On separate diagrams sketch the curve with equation\\
\hfill \mbox{\textit{Edexcel C1 2006 Q5 [6]}}