◈ Oracle PrepSign Up Free
A-LevelFurther MathematicsIntegration9231/12 · May/June 2015 · Q911 Marks

The curve C has parametric equations x = 4t + 2t³, y = 4t - 2t², for 0 ≤ t ≤ 4. Find the arc length of C, giving your answer correct to 3 significant figures. Find the mean value of y with respect to x over the interval 0 ≤ x ≤ 32.

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

  1. [B1] ẋ = 4 + 3t², ẏ = 4 - 3t²
  2. [B1] ẋ² + ẏ² = 32 + 18t
  3. [M1A1] s = ∫₀⁴ (32+18t)¹/² dt = 1/27 (32+18t)³/² |₀⁴
  4. [M1A1] = 1/27 (104³/² - 32³/²) = 32.6 (CAO)
  5. [M1] ∫ y dx = ∫₀⁴ y (dx/dt) dt = ∫₀⁴ (4t-2t²)(4+3t²) dt = ∫₀⁴ (16t+4t²-6t³) dt
  6. [M1A1] = [8t²+4/3 t³ - 3/2 t⁴] |₀⁴ = 51.2
  7. [M1A1] MV = ∫ y dx / (b-a) = 51.2 / 32 = 1.6 (CAO)

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
Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 11 marks.
Source
A-Level Further Mathematics, May/June 2015 Paper 1 Variant 2, question 9.

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

IntegrationMay/June 201211m

The curve C has equation y = 2(x/3)^(3/2), where 0 ≤ x ≤ 3. Show that the arc length of C is 2(2√2 – 1). Find the coo…

IntegrationMay/June 20138m

Let I_n = ∫₀¹ 1/(1 + x²)^n dx. Prove that, for every positive integer n, 2nI_{n+1} = 2⁻ⁿ + (2n − 1)I_n. Given that I₁…

IntegrationMay/June 201314m

Answer only one of the following two alternatives. EITHER. The curve C has equation y = 2 sec x, for 0 ≤ x ≤ ¼π. Show…

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…

MatricesMay/June 20148m

The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = [ [2, -1, 1, 3], [2, 0, 0, 5], [6, -2…

Complex numbersMay/June 20149m

Use de Moivre's theorem to show that tan 5θ = (5t - 10t³ + t⁵) / (1 - 10t² + 5t⁴), where t = tan θ. Deduce that the r…

← Browse all past-paper questions

© 2026 Oracle Prep