A-LevelFurther MathematicsPolar coordinates9231/12 · Oct/Nov 2025 · Q512 Marks

The curve C has polar equation r² = tan 2θ, where 0 < θ < π/4.

(a)Sketch C and state the greatest distance of a point on C from the pole.[2]
(b)Find the exact value of the area of the region bounded by C and the half-line θ = π/6.[4]
(c)Show that C has Cartesian equation x⁴ - 2xy - y⁴ = 0 given that 0 ≤ x ≤ cos(π/4) and 0 ≤ y ≤ sin(π/4).[4]
(d)Using your answer to (b), deduce the exact value of the area bounded by C, the x-axis and the line x = cos(π/4).[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[B1] Initial line drawn and correct shape.
[B1] Maximum distance of C from the pole is 1.
[M1] Uses 1/2 ∫ r² dθ with correct limits. (1/2 ∫(from 0 to π/6) tan 2θ dθ)
[M1 A1] Integrates a correct function to get -1/2 [ln cos 2θ] condoning the lack of limits and factor of 1/2 for M1A1.
[A1] OE, must be exact. ISW (1/2 ln 2)
[M1] Uses relevant identity for tan 2θ. (r² = 2 tan θ / (1 - tan² θ))
[M1 A1] For the M mark uses either r = √(x²+y²) or tan θ = y/x correctly. (x²+y² = 2(y/x) / (1 - (y/x)²))
[A1] At least one intermediate step before the answer given. Needs to be in this form as answer given.
[M1] Area of triangle minus their (b).
[A1] By double angle formula or otherwise. (1/2 - 1/2 ln 2 OR (√2-ln2)/2)

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