June 2022 Paper 2 Q9
9 Given that
\[\log_2 x^3 - \log_2 y^2 = 9\]show that
\[x = Ay^p\]where \(A\) is an integer and \(p\) is a rational number. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Uses a log (or index) law correctly on an algebraic term \(\log A \pm \log B\) \(n\log A\) | B1 | 1.1b |
| Raises 2 to the power of both sides (removal of \(\log_2\)) Or writes 9 as \(9\log_2 2\) or \(\log_2 512\) OE | M1 | 1.1a |
| Obtains correct equation without logs Or obtains \(\log_2(x) = \log_2\left(8y^{\frac{2}{3}}\right)\) | A1 | 1.1b |
| Completes a reasoned argument to obtain \(x = 8y^{\frac{2}{3}}\) | R1 | 2.1 |
| (4 marks) |