Edexcel C1 — Question 3 6 marks

Exam BoardEdexcel
ModuleC1 (Core Mathematics 1)
Marks6
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Mark schemeDownload PDF ↗
TopicIndices and Surds
TypeExpand and simplify surd expressions
DifficultyModerate -0.5 This is a straightforward similar rectangles problem requiring calculation of a scale factor and rationalization of a surd denominator. While it involves multiple steps (finding scale factor, multiplying, rationalizing), these are all standard C1 techniques with no conceptual difficulty or novel insight required. The surd manipulation is routine for this level, making it slightly easier than average.
Spec1.02b Surds: manipulation and rationalising denominators
\includegraphics{figure_1} Figure 1 shows the rectangles \(ABCD\) and \(EFGH\) which are similar. Given that \(AB = (3 - \sqrt{5})\) cm, \(AD = \sqrt{5}\) cm and \(EF = (1 + \sqrt{5})\) cm, find the length \(EH\) in cm, giving your answer in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are integers. [6]
AnswerMarks Guidance
\(\frac{EH}{AD} = \frac{EF}{AB} \therefore \frac{EH}{\sqrt{5}} = \frac{1+\sqrt{5}}{3-\sqrt{5}}\)M1
\(\frac{1+\sqrt{5}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{3+\sqrt{5}+3\sqrt{5}+5}{9-5} = 2 + \sqrt{5}\)M2 A1
\(\therefore EH = \sqrt{5}(2 + \sqrt{5}) = 5 + 2\sqrt{5}\)M1 A1 (6) 
$\frac{EH}{AD} = \frac{EF}{AB} \therefore \frac{EH}{\sqrt{5}} = \frac{1+\sqrt{5}}{3-\sqrt{5}}$ | M1 |
$\frac{1+\sqrt{5}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{3+\sqrt{5}+3\sqrt{5}+5}{9-5} = 2 + \sqrt{5}$ | M2 A1 |
$\therefore EH = \sqrt{5}(2 + \sqrt{5}) = 5 + 2\sqrt{5}$ | M1 A1 | (6)
\includegraphics{figure_1}

Figure 1 shows the rectangles $ABCD$ and $EFGH$ which are similar.

Given that $AB = (3 - \sqrt{5})$ cm, $AD = \sqrt{5}$ cm and $EF = (1 + \sqrt{5})$ cm, find the length $EH$ in cm, giving your answer in the form $a + b\sqrt{5}$ where $a$ and $b$ are integers. [6]

\hfill \mbox{\textit{Edexcel C1  Q3 [6]}}
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