A-LevelMathematicsSeries9709/12 · Feb/Mar 2025 · Q36 Marks

Find the complete expansion of (2x - 3/x)⁴.

(a)Find the complete expansion of (2x - 3/x)⁴.[4]
(b)Hence determine the coefficient of x² in the expansion of (x² + 5)(2x - 3/x)⁴.[2]

Mark scheme

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

[B1] (2x)⁴ and (-3/x)⁴ May be seen in a full expansion. This can be implied by 16x⁴ and + 81/x⁴ unless they are clearly using +-3/x throughout.
[B1] 4(2x)³(-3/x)+6(2x)²(-3/x)²+4(2x)(-3/x)³ Correct combination of numerical coefficients for the middle three terms. Can be implied by a correct full expansion.
[M1] [16x⁴]+k₁x² +k₂ [x⁰]+k₃x⁻² [+81x⁻⁴] OE Powers now simplified correctly with their k₁, k₂, k₃ ≠ 0.
[A1] 16x⁴-96x²+216-216x⁻² +81x⁻⁴ OE
[M1] Use of (their 216) + 5×(their –96) only, to arrive at the coefficient of x². Other terms may be seen.
[A1] -264 Accept -264x² as the final answer.

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Topic: Series, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Feb/Mar 2025 Paper 1 Variant 2, question 3.

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