June 2018 Paper 2 Q24
24 Work out the value of \(p\) when
\(9^{0.5p} \times 81 = 27^{2p - 1}\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Powers of 3 | ||
| \((3^2)^{0.5p}\) or \((3^3)^{2p - 1}\) or \(3^{2 \times 0.5p + 4}\) | M1 | oe powers of 3 eg \(3^p\) or \(3^{6p - 3}\) or \(3^{p + 4}\) brackets not needed if intention clear eg \(3^{2^{0.5p}}\) |
| \((3^2)^{0.5p}\) and \(3^4\) and \((3^3)^{2p - 1}\) or \(3^{2 \times 0.5p + 4}\) and \((3^3)^{2p - 1}\) | M1dep | oe powers of 3 eg \(3^p\) and \(3^4\) and \(3^{6p - 3}\) or \(3^{p + 4}\) and \(3^{6p - 3}\) |
| \(2 \times 0.5p + 4 = 3(2p - 1)\) or \(p + 4 = 6p - 3\) | M1dep | oe equation dep on M2 |
| 1.4 or \(\dfrac{7}{5}\) | A1 | oe |
| Alternative method 2 Powers of 9 | ||
| \(9^{0.5p + 2}\) or \((9^{1.5})^{2p - 1}\) | M1 | oe power of 9 eg \(9^{3p - 1.5}\) brackets not needed if intention clear eg \(9^{1.5^{2p - 1}}\) |
| \(9^2\) and \((9^{1.5})^{2p - 1}\) or \(9^{0.5p + 2}\) and \((9^{1.5})^{2p - 1}\) | M1dep | oe powers of 9 eg \(9^2\) and \(9^{3p - 1.5}\) or \(9^{0.5p + 2}\) and \(9^{3p - 1.5}\) |
| \(0.5p + 2 = 1.5(2p - 1)\) or \(0.5p + 2 = 3p - 1.5\) | M1dep | oe equation dep on M2 |
| 1.4 or \(\dfrac{7}{5}\) | A1 | oe |
| Alternative method 3 Powers of 27 | ||
| \(\left(27^{\frac{2}{3}}\right)^{0.5p}\) | M1 | oe power of 27 eg \(27^{\frac{2}{3} \times 0.5p}\) or \(27^{\frac{1}{3}p}\) brackets not needed if intention clear eg \(27^{\frac{2}{3}^{0.5p}}\) |
| \(\left(27^{\frac{2}{3}}\right)^{0.5p}\) and \(27^{\frac{4}{3}}\) | M1dep | oe powers of 27 eg \(27^{\frac{2}{3} \times 0.5p}\) and \(27^{\frac{4}{3}}\) or \(27^{\frac{1}{3}p}\) and \(27^{\frac{4}{3}}\) M2 \(\ 27^{\frac{2}{3} \times 0.5p + \frac{4}{3}}\) or \(27^{\frac{1}{3}p + \frac{4}{3}}\) |
| \(\dfrac{2}{3} \times 0.5p + \dfrac{4}{3} = 2p - 1\) or \(\dfrac{1}{3}p + \dfrac{4}{3} = 2p - 1\) | M1dep | oe equation dep on M2 |
| 1.4 or \(\dfrac{7}{5}\) | A1 | oe |
| Alternative method 4 Powers of 81 | ||
| \((81^{0.5})^{0.5p}\) or \((81^{0.75})^{2p - 1}\) or \(81^{0.5 \times 0.5p + 1}\) | M1 | oe powers of 81 eg \(81^{0.25p}\) or \(81^{1.5p - 0.75}\) or \(81^{0.25p + 1}\) brackets not needed if intention clear eg \(81^{0.5^{0.5p}}\) |
| \((81^{0.5})^{0.5p}\) and \((81^{0.75})^{2p - 1}\) or \(81^{0.5 \times 0.5p + 1}\) and \((81^{0.75})^{2p - 1}\) | M1dep | oe powers of 81 eg \(81^{0.25p}\) and \(81^{1.5p - 0.75}\) or \(81^{0.25p + 1}\) and \(81^{1.5p - 0.75}\) |
| \(0.5 \times 0.5p + 1 = 0.75(2p - 1)\) or \(0.25p + 1 = 1.5p - 0.75\) | M1dep | oe equation dep on M2 |
| 1.4 or \(\dfrac{7}{5}\) | A1 | oe |
Additional guidance
| Mark positively if potentially more than one scheme used | |
| Answer 1.4 | M3A1 |
| Correct equation implies M3 | |
| Just seeing expressions not in an equation and not as powers scores zero eg Alt 1 \(\ 6p - 3\) and \(p + 4\) not in an equation and not as powers of 3 | M0M0M0 |
| Allow recovery of missing brackets | |
| Use of logs with answer not 1.4 - escalate |