Higher June 2024 Paper 6 Q19
19 The diagram shows a circle with centre (0, 0) and a tangent at (12, \(-5\)).
The tangent at (12, \(-5\)) crosses the \(x\)-axis at (\(p\), 0).
Not to scale
Find the exact value of \(p\).
You must show your working. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\frac{169}{12}\) or \(14\frac{1}{12}\) or \(14.08\dot{3}\) with correct working | 5 | ‘Correct working’ requires evidence of at least M2M1 or M1M1M1 SC3 applies to all methods | |
| Gradients and equation of straight line M2 for gradient of tangent = \(\frac{12}{5}\) oe soi or for gradient of tangent = \(\frac{5}{p - 12}\) or M1 for gradient of radius = \(-\frac{5}{12}\) soi or gradient of tangent = \(\frac{-1}{\textit{their gradient of radius}}\) AND | Mark to candidate’s advantage if gradients not labelled Do not mix marks across two methods (ie do not give gradient marks and another M1 for 13 from Pythagoras) | ||
| M2 for \(0 = \frac{12}{5}p - \frac{169}{5}\) or better or for \(5 = \frac{12}{5}(p - 12)\) or better or for \(\frac{5}{p - 12} = \frac{12}{5}\) or better or M1 for \(y = \textit{their } \frac{12}{5}x + c\) or better or for \(y = \frac{5}{p - 12}x + c\) or better or for \(y - {}^{-}5 = \textit{their } \frac{12}{5}(x - 12)\) or better or \(\frac{-1}{\textit{their gradient of radius}} = \frac{5}{p - 12}\) or better | M2 must be a correct equation solely in terms of \(p\) or solely in terms of \(x\) e.g. \(0 = \frac{12}{5}x - \frac{169}{5}\) or better For M1 allow FT their \(\frac{12}{5}\) if from \(m_1 = \frac{-1}{m_2}\) | ||
| If 0, 1 or 2 scored, instead award SC3 for answer \(\frac{169}{12}\) or \(14\frac{1}{12}\) or \(14.08\dot{3}\) with no or insufficient working | PTO for alternative methods | ||
Alternative methods
| Alternative method (Pythagoras and trig): | Alternative method (Pythagoras and equations) |
|---|---|
| M1 for [O to (12, -5)] \(\sqrt{12^2 + 5^2}\) may be implied by 13 in working or on diagram M1 for [angle at O] \(\sin\theta = \frac{5}{13}\) or \(\cos\theta = \frac{12}{13}\) or \(\tan\theta = \frac{5}{12}\) or better (do not imply from just an angle) AND M2 for [\(p =\)] \(\dfrac{13}{\frac{12}{13}}\) or M1 for \(\cos \textit{their}\theta = \frac{13}{p}\) or \(p = \frac{13}{\cos \textit{their}\theta}\) (Their \(\theta\) from earlier trig work) | M2 for \(13^2 + (p - 12)^2 + 5^2 = p^2\) oe or M1 for \((p - 12)^2 + 5^2\) oe or for \(\sqrt{12^2 + 5^2}\) may be implied by 13 in working or on diagram AND M2 for \(338 - 24p = 0\) oe or M1 for \(169 + p^2 - 24p + 144 + 25 = p^2\) or better |