The diagram shows the curve with parametric equations x = 3 cos 2θ, y = 4 sin θ, for π < θ < 3π/2. Points P and Q lie on the curve. The gradient of the curve at P is 2. The straight line 3x + y = 0 meets the curve at Q. [Figure 7.1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [B1] State dx/dθ = -6sin 2θ and dy/dθ = 4 cos θ
2. [M1] Use dy/dx = (dy/dθ) / (dx/dθ) and equate to 2
3. [M1] Use sin 2θ = 2sinθcosθ and attempt value of sinθ
4. [A1] Obtain sinθ = -1/6
5. [A1] Obtain θ = 3.31 only AWRT; and no second answer.
6. [M1] State or imply 9cos 2θ + 4sinθ = 0 and use identity to obtain quadratic in sinθ
7. [A1] Obtain 18sin²θ - 4sinθ - 9 = 0 OE
8. [DM1] Attempt solution to find negative value of sinθ
9. [A1] Obtain sinθ = -0.604... Or θ = 3.79... (2-√166)/18
10. [M1] Substitute value of sinθ (or their θ between π and 3π/2) in expression for first derivative
11. [A1] Obtain 0.551 AWRT