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A-LevelMathematicsIntegration9709/22 · Feb/Mar 2025 · Q68 Marks

Find the quotient and remainder when 18x³ – 6x² – 30x+4 is divided by (3x-1).

  1. (a)Find the quotient and remainder when 18x³ – 6x² – 30x+4 is divided by (3x-1).[3]
  2. (b)Hence find ∫₂⁵ (18x³ – 6x² – 30x+4) / (3x-1) dx. Give your answer in the form a-lnb, where a and b are integers.[5]

Mark scheme

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

  1. [M1] Carry out division at least as far as 6x² + k₁
  2. [A1] Obtain quotient 6x² -10
  3. [A1] Obtain remainder -6
  4. [B1] Identify integrand as 6x²-10-6/(3x-1) FT Following their quotient and remainder.
  5. [*M1] Integrate to obtain at least 2x³ and term of the form k₂ ln(3x−1)
  6. [A1] Obtain 2x³ -10x – 2ln(3x – 1) FT Following their quotient and remainder.
  7. [DM1] Apply limits and appropriate logarithm properties
  8. [A1] Obtain 208–ln 49

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, Feb/Mar 2025 Paper 2 Variant 2, question 6.

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