A-LevelMathematicsAlgebra9709/21 · Oct/Nov 2022 · Q69 Marks

The polynomial p(x) is defined by p(x) = 12x³ – 9x² + 8x – 4.

(a)Find the quotient when p(x) is divided by (4x – 3) and show that the remainder is 2.[3]
(b)Hence find ∫₂¹² (p(x)/(4x-3) - 3x²) dx, giving your answer in the form a + lnb.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] Carry out division at least as far as the 3x² + kx stage (where k should be zero.)
[A1] Obtain quotient 3x² + 2
[A1] Confirm remainder is 2 (AG – necessary detail needed. SC If M0A0 obtained then allow B1 if remainder theorem is used to obtain 2. Must see sufficient detail.)
[B1] Identify integrand as 2 + 2/(4x-3) (FT their quotient.)
[*M1] Integrate to obtain at least k ln(4x-3) term
[A1] Obtain correct 2x + ½ln(4x-3)
[DM1] Apply limits correctly
[M1] Use relevant logarithm properties (Must be in the form k ln(4x-3).)
[A1] Obtain 20+ln3

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2022 Paper 2 Variant 1, question 6.

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