AS June 2022 Paper 1 Q8
8 The curve \(C\) has the polar equation
\[r = 4 - 2\cos\theta \qquad -\pi \lt \theta \leqslant \pi\](a) Verify that the point with polar coordinates \(\left(3, \dfrac{\pi}{3}\right)\) lies on \(C\) [1 mark]
(b) Find the exact polar coordinates of the point on \(C\) which is furthest from the pole, \(O\) [3 marks]
(c) Find the exact Cartesian coordinates of the point on \(C\) where \(\theta\) is \(\dfrac{\pi}{6}\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Verifies that \(r = 3\) and \(\theta = \dfrac{\pi}{3}\) satisfies the polar equation. Condone missing conclusion. Accept \(4 - 2 \times \dfrac{1}{2} = 3\) as sufficient verification. | B1 | 1.1b |
| (1) |
Typical solution
\[4 - 2\cos\left(\frac{\pi}{3}\right) = 4 - 2 \times \frac{1}{2} = 3\]\(\therefore\) \(\left(3, \dfrac{\pi}{3}\right)\) lies on \(C\)
| Scheme | Marks | AO |
|---|---|---|
| Selects a method to find the required polar coordinates, eg substitutes \(\cos\theta = -1\) to find \(r\) or solves \(\cos\theta = -1\) to find \(\theta\) PI by a correct value for \(r\) or \(\theta\) | M1 | 3.1a |
| Obtains a correct value for \(r\) or \(\theta\) | A1 | 1.1a |
| Obtains the correct polar coordinates. | A1 | 3.2a |
| (3) |
Typical solution
\[r = 4 - 2 \times (-1) = 6\]\[\cos\theta = -1 \ \Rightarrow \ \theta = \pi\]furthest from \(O\) is \((6, \pi)\)
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(\theta = \dfrac{\pi}{6}\) to find \(r\) PI by 2.27 or 1.96 or 1.13 or better | M1 | 1.1a |
| Obtains an expression for the \(x\) or the \(y\)-coordinate. PI by 1.96 or 1.13 or better | M1 | 2.2a |
| Obtains the correct exact Cartesian coordinates. ACF | A1 | 1.1b |
| (3) | ||
| (7 marks) |