Easy -1.2 This is a straightforward rationalizing the denominator question requiring multiplication by the conjugate and simplification of surds. It's a standard C1 exercise with clear method and minimal steps, making it easier than average but not trivial due to the surd arithmetic involved.
Attempt to rationalise denominator – must attempt to multiply
\(\sqrt{20} = 2\sqrt{5}\) soi
B1
\(\frac{-1+3\sqrt{5}}{9-5}\)
A1
Either numerator or denominator correct and simplified to no more than two terms
\(-\frac{1}{4}+\frac{3}{4}\sqrt{5}\)
A1
Fully correct and fully simplified. Allow \(\frac{-1+3\sqrt{5}}{4}\), order reversed etc. Do not ISW if then multiplied by 4 etc.
Alternative: M1 correct method to solve simultaneous equations formed from equating expression to \(a\sqrt{5}+b\); B1 \(\sqrt{20}=2\sqrt{5}\) soi; A1 either \(a\) or \(b\) correct; A1 both correct
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3+\sqrt{20}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}$ | M1 | Attempt to rationalise denominator – must attempt to multiply |
| $\sqrt{20} = 2\sqrt{5}$ soi | B1 | |
| $\frac{-1+3\sqrt{5}}{9-5}$ | A1 | Either numerator or denominator correct and simplified to no more than two terms |
| $-\frac{1}{4}+\frac{3}{4}\sqrt{5}$ | A1 | Fully correct and fully simplified. Allow $\frac{-1+3\sqrt{5}}{4}$, order reversed etc. Do not ISW if then multiplied by 4 etc. |
**Alternative:** M1 correct method to solve simultaneous equations formed from equating expression to $a\sqrt{5}+b$; B1 $\sqrt{20}=2\sqrt{5}$ soi; A1 either $a$ or $b$ correct; A1 both correct
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