Easy -1.2 This is a straightforward algebraic manipulation requiring rationalizing the denominator by multiplying by the conjugate (3 + √5). It's a single-step technique with no conceptual difficulty beyond basic surd manipulation, making it easier than average but not trivial since students must recognize and execute the rationalization method correctly.
1 In this question you must show detailed reasoning.
Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
Multiplying numerator and denominator by \(3+\sqrt{5}\) or \(-3-\sqrt{5}\); Alternative M1: correct method to solve simultaneous equations formed from equating expressions to \(a+b\sqrt{5}\)
Correct simplified denominator; A1 either \(a\) or \(b\) correct
\(= 18+6\sqrt{5}\)
A1 (AO 1.1)
Final answer cao, therefore final answer of only \(6(3+\sqrt{5})\) is A0; A1 both correct
**Question 1:**
$x = \frac{24}{3-\sqrt{5}} = \frac{24(3+\sqrt{5})}{(3-\sqrt{5})(3+\sqrt{5})}$ | M1 (AO 1.1) | Multiplying numerator and denominator by $3+\sqrt{5}$ or $-3-\sqrt{5}$; Alternative M1: correct method to solve simultaneous equations formed from equating expressions to $a+b\sqrt{5}$
$= \frac{24(3+\sqrt{5})}{9-3\sqrt{5}+3\sqrt{5}-5} = \frac{24(3+\sqrt{5})}{4}$ | A1 (AO 1.1) | Correct simplified denominator; A1 either $a$ or $b$ correct
$= 18+6\sqrt{5}$ | A1 (AO 1.1) | Final answer cao, therefore final answer of only $6(3+\sqrt{5})$ is A0; A1 both correct
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1 In this question you must show detailed reasoning.
Solve the equation $x ( 3 - \sqrt { 5 } ) = 24$, giving your answer in the form $a + b \sqrt { 5 }$, where $a$ and $b$ are positive integers.
\hfill \mbox{\textit{OCR PURE Q1 [3]}}