Moderate -0.8 This is a straightforward rationalizing the denominator question requiring multiplication by the conjugate and algebraic simplification. It's a standard AS-level technique with no conceptual difficulty beyond the routine procedure, making it easier than average but not trivial since it requires careful algebraic manipulation to express the final answer in terms of n.
7 The expression
$$\frac { 3 - \sqrt { } n } { 2 + \sqrt { } n }$$
can be written in the form \(a + b \sqrt { } n\), where \(a\) and \(b\) and \(n\) are rational but \(\sqrt { } n\) is irrational.
Find expressions for \(a\) and \(b\) in terms of \(n\).
AO 1.1a — multiplies numerator and denominator by \((2-\sqrt{n})\); condone sign errors; PI by correct simplification
\(\frac{6 + n - 5\sqrt{n}}{4-n}\)
A1
AO 1.1b — correct simplified numerator and denominator (not necessarily as a fraction)
\(a = \frac{6+n}{4-n}\), \(b = \frac{-5}{4-n}\)
A1
AO 1.1b — correct expressions for \(a\) and \(b\), or expression with \(a\) and \(b\) correctly identified
## Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3-\sqrt{n}}{2+\sqrt{n}} \times \frac{2-\sqrt{n}}{2-\sqrt{n}}$ | M1 | AO 1.2 — multiplies top and bottom by $2-\sqrt{n}$; PI by subsequent work |
| $\frac{6 - 3\sqrt{n} - 2\sqrt{n} + n}{4 + 2\sqrt{n} - 2\sqrt{n} - n}$ | M1 | AO 1.1a — multiplies numerator and denominator by $(2-\sqrt{n})$; condone sign errors; PI by correct simplification |
| $\frac{6 + n - 5\sqrt{n}}{4-n}$ | A1 | AO 1.1b — correct simplified numerator and denominator (not necessarily as a fraction) |
| $a = \frac{6+n}{4-n}$, $b = \frac{-5}{4-n}$ | A1 | AO 1.1b — correct expressions for $a$ and $b$, or expression with $a$ and $b$ correctly identified |
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7 The expression
$$\frac { 3 - \sqrt { } n } { 2 + \sqrt { } n }$$
can be written in the form $a + b \sqrt { } n$, where $a$ and $b$ and $n$ are rational but $\sqrt { } n$ is irrational.
Find expressions for $a$ and $b$ in terms of $n$.\\
\hfill \mbox{\textit{AQA AS Paper 2 2022 Q7 [4]}}