June 2024 Paper 2 Q2
2 The vector \(\begin{pmatrix} a \\ b \end{pmatrix}\) has magnitude 6 and direction \(60^\circ\) above the positive \(x\)-axis.
Determine the exact values of \(a\) and \(b\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(a^2 + b^2 = 36\) | M1* | 3.1a |
| \(\dfrac{b}{a} = \sqrt{3}\) | M1* | 1.2 |
| \(a^2 + 3a^2 = 36\) or \(\dfrac{b^2}{3} + b^2 = 36\) | M1 dep* | 1.1 |
| \(a = 3,\ b = 3\sqrt{3}\) | A1 | 1.1 |
| [4] |
Notes
M1*: Attempt at Pythagoras =36 oe, may see \(\sqrt{a^2 + b^2} = 6\)
M1*: oe or \(\dfrac{b}{a} = \tan 60\) or \(\dfrac{b}{a} = 1.73\), condone \(\dfrac{a}{b}\)
May see equivalent statements in sin or cos.
M1 dep*: Substitute both expressions (dependent on both previous M marks) – must reach an equation in \(a\) or \(b\) only.
A1: A0 for negative answers (if not disregarded)
Accept \(b = \sqrt{27}\)
If no (or insufficient) working then SC B1B1 (max 2/4) for each correct answer (must be exact).
Alternative method
| Scheme | Marks |
|---|---|
| \((a =)\,6\cos 60\) | M1 |
| \(a = 3\) | A1 |
| \((b =)\,6\sin 60\) | M1 |
| \(b = 3\sqrt{3}\) | A1 |
M1: Allow this mark for 6cos or 6sin of \(30, 60, 120^\circ\)
Must see this step oe.
M1: Allow this mark for 6cos or 6sin of \(30, 60, 120^\circ\) provided it is consistent with their other expression (i.e. not the same).
Must see this step oe.
If no (or insufficient) working then SC B1B1 (max 2/4) for each correct answer (must be exact).