June 2018 Paper 2 Q8
8 Here is a sketch of triangle \(ABC\).
\(P\) is a point on \(AB\).
\(AP : PB\) is \(3 : 1\)
Work out the length \(PC\).
Give your answer to 4 significant figures. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| (\(x\)-coordinate of \(P =\)) 5.5 and (\(y\)-coordinate of \(P =\)) 2.25 | B2 | oe may be seen on diagram B1 (\(x\)-coordinate of \(P =\)) 5.5 or (\(y\)-coordinate of \(P =\)) 2.25 or \(x\)-coordinate of \(P = 2.25\) and \(y\)-coordinate of \(P = 5.5\) |
| \((9 - \text{their } 5.5)^2 + (12 - \text{their } 2.25)^2\) or \(3.5^2 + 9.75^2\) or \(12.25 + 95.06(25)\) or \(12.25 + 95.063\) or 107.3(125) or 107.313 | M1 | oe eg \(\sqrt{3.5^2 + 9.75^2}\) or \(\dfrac{1717}{16}\) \(1 \lt\) their \(5.5 \lt 7\) \(1 \lt\) their \(2.25 \lt 6\) |
| 10.36 | A1ft | correct or ft their 5.5 and/or their 2.25 must be rounded to 4 sig figs |
| Alternative method 2 Uses \(AC = 10\), \(BC = \sqrt{125}\) or \(5\sqrt{5}\) or 11.18… and \(AB = \sqrt{61}\) or 7.81… | ||
| \(\cos^{-1} \dfrac{10^2 + 7.81\ldots^2 - 11.18\ldots^2}{2 \times 10 \times 7.81\ldots}\) or [76.67, 76.71] | M1 | oe eg \(\cos^{-1} 0.23(0\ldots)\) or \(\cos^{-1} 0.231\) may be on diagram angle \(BAC\) |
| \((0.75 \times 7.81\ldots)^2 + 10^2\) \(- 2 \times (0.75 \times 7.81\ldots) \times 10 \times \cos\) their [76.67, 76.71] | M1dep | |
| [107.3, 107.4] | A1 | |
| 10.36 | A1 | |
| Alternative method 3 Uses \(AC = 10\), \(BC = \sqrt{125}\) or \(5\sqrt{5}\) or 11.18… and \(AB = \sqrt{61}\) or 7.81… | ||
| \(\cos^{-1} \dfrac{7.81\ldots^2 + 11.18\ldots^2 - 10^2}{2 \times 7.81\ldots \times 11.18\ldots}\) or [60.49, 60.66] | M1 | oe eg \(\cos^{-1} [0.49,\ 0.4925]\) may be on diagram angle \(ABC\) |
| \((0.25 \times 7.81\ldots)^2 + 11.18\ldots^2\) \(- 2 \times (0.25 \times 7.81\ldots) \times 11.18\ldots\) \(\times \cos\) their [60.49, 60.66] | M1dep | |
| [107.3, 107.4] | A1 | |
| 10.36 | A1 | |
Additional guidance
| If 5.5 is from gradient \(BC\) | B0 |
| Alt 1 \(\ P\,(4.5,\ 3.75)\) \((9 - 4.5)^2 + (12 - 3.75)^2\) 9.397 | B0 M1 A1ft |