Higher June 2019 Paper 1 Q22
22 \(ABC\) and \(ACD\) are triangles.
\(k\) is a constant.
Not drawn accurately
(a) Show that \(\quad \overrightarrow{CD} = 6\mathbf{a} + 4.5\mathbf{b}\) [1 mark]
(b) \(BCD\) is a straight line.
Work out the value of \(k\).
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(-3\mathbf{b} + 6\mathbf{a} + 7.5\mathbf{b}\ (= 6\mathbf{a} + 4.5\mathbf{b})\) or \(6\mathbf{a} + 7.5\mathbf{b} - 3\mathbf{b}\ (= 6\mathbf{a} + 4.5\mathbf{b})\) or \(6\mathbf{a} + 7.5\mathbf{b} - (6\mathbf{a} + 4.5\mathbf{b}) = 3\mathbf{b}\) | B1 | oe rearranged equation using all 5 terms |
Additional guidance
| \(3\mathbf{b} + 6\mathbf{a} + 4.5\mathbf{b} = 6\mathbf{a} + 7.5\mathbf{b}\) | B1 |
| \(6\mathbf{a} + 4.5\mathbf{b} + 3\mathbf{b} = 6\mathbf{a} + 7.5\mathbf{b}\) | B1 |
| \(7.5\mathbf{b} - 3\mathbf{b} = 4.5\mathbf{b}\), so \(6\mathbf{a} + 4.5\mathbf{b}\) | B0 |
| \(6\mathbf{a} + 7.5\mathbf{b} - 3\mathbf{b} = 4.5\mathbf{b}\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: equal ratios from \(k\mathbf{a} + 3\mathbf{b}\) and \(6\mathbf{a} + 4.5\mathbf{b}\) | ||
| \((BC =)\ k\mathbf{a} + 3\mathbf{b}\) or \(k : 6 = 3 : 4.5\) or \(k : 3 = 6 : 4.5\) | M1 | oe ratio |
| \(3 \times 6 \div 4.5\) or \(4\mathbf{a} + 3\mathbf{b}\) | M1dep | oe |
| 4 | A1 | |
| Alternative method 2: scale factor from \(k\mathbf{a} + 3\mathbf{b}\) and \(6\mathbf{a} + 4.5\mathbf{b}\) | ||
| \((BC =)\ k\mathbf{a} + 3\mathbf{b}\) or \(4.5 \div 3\) or \(\dfrac{3}{2}\) or \(3 \div 4.5\) or \(\dfrac{2}{3}\) or \(4.5 \div 6\) or \(\dfrac{3}{4}\) or \(6 \div 4.5\) or \(\dfrac{4}{3}\) | M1 | oe fractions or decimals |
| \(6 \div\) their \(\dfrac{3}{2}\) or \(6 \times\) their \(\dfrac{2}{3}\) or \(3 \div\) their \(\dfrac{3}{4}\) or \(3 \times\) their \(\dfrac{4}{3}\) or \(4\mathbf{a} + 3\mathbf{b}\) | M1dep | oe |
| 4 | A1 | |
| Alternative method 3: equal ratios from \((k + 6)\mathbf{a} + 7.5\mathbf{b}\) and \(6\mathbf{a} + 4.5\mathbf{b}\) | ||
| \((BD =)\ k\mathbf{a} + 6\mathbf{a} + 7.5\mathbf{b}\) or \((BD =)\ (k + 6)\mathbf{a} + 7.5\mathbf{b}\) or \((k + 6) : 6 = 7.5 : 4.5\) or \((k + 6) : 7.5 = 6 : 4.5\) | M1 | oe ratio |
| \(6 \times 7.5 \div 4.5 - 6\) or \(4\mathbf{a} + 3\mathbf{b}\) | M1dep | oe |
| 4 | A1 | |
| Alternative method 4: scale factor from \((k + 6)\mathbf{a} + 7.5\mathbf{b}\) and \(6\mathbf{a} + 4.5\mathbf{b}\) | ||
| \((BD =)\ k\mathbf{a} + 6\mathbf{a} + 7.5\mathbf{b}\) or \((BD =)\ (k + 6)\mathbf{a} + 7.5\mathbf{b}\) or \(7.5 \div 4.5\) or \(\dfrac{5}{3}\) or \(4.5 \div 7.5\) or \(\dfrac{3}{5}\) or \(4.5 \div 6\) or \(\dfrac{3}{4}\) or \(6 \div 4.5\) or \(\dfrac{4}{3}\) | M1 | oe fractions or decimals |
| \(6 \times\) their \(\dfrac{5}{3}\) \(-\ 6\) or \(6 \div\) their \(\dfrac{3}{5}\) \(-\ 6\) or \(7.5 \div\) their \(\dfrac{3}{4}\) \(-\ 6\) or \(7.5 \times\) their \(\dfrac{4}{3}\) \(-\ 6\) or \(4\mathbf{a} + 3\mathbf{b}\) | M1dep | oe |
| 4 | A1 | |
Additional guidance
| Check the diagram for working | |
| If working is not seen, only accept exact decimal values in place of fractions for method marks | |
| Answer 4 with no working or no incorrect working | M1M1A1 |
| Assumes that \(BC\) is \(3a + 2.25b\) (half the length of \(CD\)) or that \(BC\) is \(2a + 1.5b\) (one third of the length of \(CD\)) | M0M0A0 M0M0A0 |
| 4a on the answer line does not get the A mark, but may have scored the method marks |