June 2016 Paper 2 Q8
8 \(D(-6, 4)\) and \(E(-2, 9)\) are joined by a straight line.
\(P\) is a point on \(DE\).
\[DP : PE = 3 : 5\]Work out the coordinates of \(P\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{5 \times -6 + 3 \times -2}{5 + 3}\) or \(\dfrac{5 \times 4 + 3 \times 9}{5 + 3}\) | M1 | oe |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
| Alternative method 2 | ||
| \(\dfrac{3}{5 + 3} \times (-2 - -6)\) or 1.5 or \(\dfrac{3}{5 + 3} \times (9 - 4)\) or 1.875 | M1 | oe eg \(\dfrac{3}{8} \times 4\) or \(\dfrac{3}{8} \times 5\) or \(\dfrac{4}{8} \times 3\) or \(\dfrac{5}{8} \times 3\) |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
| Alternative method 3 | ||
| \(\dfrac{5}{5 + 3} \times (-2 - -6)\) or 2.5 or \(\dfrac{5}{5 + 3} \times (9 - 4)\) or 3.125 | M1 | oe eg \(\dfrac{5}{8} \times 4\) or \(\dfrac{5}{8} \times 5\) or \(\dfrac{4}{8} \times 5\) |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
| Alternative method 4 | ||
| For \(x\): \(\dfrac{x - -6}{-2 - -6} = \dfrac{3}{5 + 3}\) or \(\dfrac{-2 - x}{-2 - -6} = \dfrac{5}{5 + 3}\) or \(\dfrac{x - -6}{-2 - x} = \dfrac{3}{5}\) For \(y\): \(\dfrac{y - 4}{9 - 4} = \dfrac{3}{5 + 3}\) or \(\dfrac{9 - y}{9 - 4} = \dfrac{5}{5 + 3}\) or \(\dfrac{y - 4}{9 - y} = \dfrac{3}{5}\) | M1 | oe eg both fractions inverted |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
| Alternative method 5 | ||
| \(\dfrac{3}{5 + 3} \times \sqrt{(9 - 4)^2 + (-2 - -6)^2}\) \(\times \sin\left(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\right)\) or 1.875 or \(\dfrac{3}{5 + 3} \times \sqrt{(9 - 4)^2 + (-2 - -6)^2}\) \(\times \cos\left(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\right)\) or 1.5 | M1 | \(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\) is the angle \(DE\) makes with the horizontal (= 51.3...) \(\sqrt{(9 - 4)^2 + (-2 - -6)^2}\) is \(DE\) (= \(\sqrt{41}\) or 6.4...) |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
| Alternative method 6 | ||
| \(\dfrac{5}{5 + 3} \times \sqrt{(9 - 4)^2 + (-2 - -6)^2}\) \(\times \sin\left(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\right)\) or 3.125 or \(\dfrac{5}{5 + 3} \times \sqrt{(9 - 4)^2 + (-2 - -6)^2}\) \(\times \cos\left(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\right)\) or 2.5 | M1 | \(\tan^{-1}\dfrac{9 - 4}{-2 - -6}\) is the angle \(DE\) makes with the horizontal (= 51.3...) \(\sqrt{(9 - 4)^2 + (-2 - -6)^2}\) is \(DE\) (= \(\sqrt{41}\) or 6.4...) |
| \(-4.5\) or 5.875 | A1 | oe |
| \((-4.5, 5.875)\) | A1 | oe eg \(\left(-\dfrac{9}{2}, \dfrac{47}{8}\right)\) SC2 \((5.875, -4.5)\) |
Additional guidance
\((-4.5, 5.9)\) or \((-4.5, 5.88)\) is M1 A1 A0 unless 5.875 seen in working
\((5.875, -4.5)\) is SC2
or M1 A1 A1 if \(x = -4.5\) and \(y = 5.875\) seen in working
\(-4.5\) in working that becomes 4.5 on answer line should not be regarded as choice so gains at least M1 A1
2 marks if one coordinate correct and 3 marks if both correct (possibly from accurate drawing or working with midpoints) with no incorrect working for that coordinate
Alts 5 and 6 also have equivalents where the angle \(DE\) makes with the vertical (= 38.6... or 38.7) is used. Mark using the principles of alts 5 and 6 or escalate