A-LevelMathematicsSeries9709/31 · May/June 2023 · Q34 Marks

Find the coefficient of x³ in the binomial expansion of (3 + x)√1 + 4x.

Mark scheme

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

[B1] State unsimplified term in x³, or its coefficient, in the expansion of (1+4x)^(1/2) e.g. (1/2)(-1/2)(-3/2) / (3*2*1) * (4x)³ (= 4) Must expand binomial coefficient.
[B1] State unsimplified term in x², or its coefficient, in the expansion of (1+4x)^(1/2) e.g. (1/2)(-1/2) / (2*1) * (4x)² (= -2) Must expand binomial coefficient.
[M1] Multiply by (3+x) and combine terms in x³, or their coefficients (3×4 - 1×2) Allow if they expanded with x rather than 4x.
[A1] Obtain answer 10 Accept 10x³

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Topic: Series, A-Level Mathematics (syllabus 9709). Structured question, 4 marks.
Source: A-Level Mathematics, May/June 2023 Paper 3 Variant 1, question 3.

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