◈ Oracle PrepSign Up Free
A-LevelFurther MathematicsPolar coordinates9231/12 · May/June 2017 · Q1113 Marks

The curve C has polar equation r = a(1 + sin θ) for –π < θ < π, where a is a positive constant.

  1. (i)Sketch C.[2]
  2. (ii)Find the area of the region enclosed by C.[4]
  3. (iii)Show that the length of the arc of C from the pole to the point furthest from the pole is given by s = (√2)a ∫ (from -½π to ½π) √(1 + sin θ) dθ.[3]
  4. (iv)Show that the substitution u = 1 + sin θ reduces this integral for s to (√2)a ∫ (from 0 to 2) 1/√(2-u) du. Hence evaluate s.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

  1. [B1] Sketch of a cardioid
  2. [B1] Correct orientation and labelled
  3. [M1]
  4. [M1]
  5. [M1A1]
  6. [M1A1] Uses correct formula for arc length; AG
  7. [M1]
  8. [A1] Including limits
  9. [M1A1]

Write your answer and get it marked

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Sign up free →

About this question

Topic
Polar coordinates, A-Level Further Mathematics (syllabus 9231). Structured question, 13 marks.
Source
A-Level Further Mathematics, May/June 2017 Paper 1 Variant 2, question 11.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content

Related Further Mathematics questions

Polar coordinatesMay/June 20146m

The curve C has polar equation r = a(1 + sin θ), where a is a positive constant and 0 ≤ θ < 2π. Draw a sketch of C. F…

Polar coordinatesMay/June 20159m

The curves C₁ and C₂ have polar equations C₁: r = 1/√2, for 0 < θ < 2π, C₂: r = √(sin10), for 0 ≤ θ≤ π. Find the pola…

Polar coordinatesMay/June 20188m

The curve C has polar equation r = cos 2θ, for -¼π ≤ θ ≤ ¼π.

IntegrationOct/Nov 201312m

The curve C has parametric equations x = t², y=t-t³, for 0 ≤ t ≤ 1. Find the surface area generated when C is rotated…

IntegrationOct/Nov 201312m

The curve C has equation y = (px² + 4x + 1) / (x + 1), where p is a positive constant and p ≠ 3.

Complex numbersOct/Nov 201314m

Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form cos θ + i sin θ,…

← Browse all past-paper questions

© 2026 Oracle Prep