O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2023 · Q96 Marks

A particle travels in a straight line so that, t seconds after passing a fixed point, its velocity, v ms⁻¹, is given by v = e⁴ for 0 ≤ t ≤ 4, v = 16e⁴/t² for 4 < t ≤ k. The total distance travelled by the particle between t = 0 and t = k is 13.4 metres. Find the value of k.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[B3] ∫e⁴ dt = 4e⁴(+c) and ∫16e⁴/t² dt = -16e⁴/t (+c)
[B2] for either correct
[B1] for ∫e⁴ dt = ae⁴(+c) where a is a constant, a > 0 or ∫16e⁴/t² dt = -b/t (+c) where b is a constant b > 0
[M1] Correct plan: ∫₀⁴e⁴ dt + ∫⁴ᵏ(16e⁴/t²) dt = 13.4 soi
[A1] Correct equation: 4e⁴ - 4e⁰ + (-16e⁴/k + 16e⁴/4) = 13.4 OR [When t = 4 s = 4e⁴ - 4 When t = k s = -16e⁴/k + 8e⁴ and] 13.4 = -16e⁴/k + 8e⁴
[A1] k = 10 or awrt 10.0 dep on all previous marks awarded

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 6 marks.
Source: O-Level Additional Mathematics, May/June 2023 Paper 2 Variant 2, question 9.

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