A-LevelMathematicsDifferentiation9709/32 · Oct/Nov 2020 · Q57 Marks
The diagram shows the curve with parametric equations x = tan θ, y = cos² θ, for −π < θ < π. [Figure X.X]
- (a)Show that the gradient of the curve at the point with parameter θ is −2 sin θ cos³ θ.[3]
- (b)Find the exact value of the x-coordinate of P.[4]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.
- [B1] State dx/dθ = sec² θ or dy/dθ = -2 sin θ cos θ
- [M1] Use dy/dx = (dy/dθ) / (dx/dθ)
- [A1] Obtain dy/dx = -2 sin θ cos³ θ from correct working
- [M1] Use correct product rule to obtain d/dθ (±2 cos³ θ sin θ)
- [A1] Obtain correct derivative in any form (e.g. ±(-2cos⁴θ + 6sin²θcos²θ))
- [A1] Equate derivative to zero and obtain an equation in one trig ratio (e.g. 3 tan²θ=1, or 4sin²θ=1 or 4 cos²θ=3)
- [A1] Obtain answer x = 1/√3 or -1/√3
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