O-LevelAdditional MathematicsSeries4037/21 · May/June 2022 · Q57 Marks

Write down, in ascending powers of x, the first three terms in the expansion of (1+4x)ⁿ. Simplify each term.

(a)(i)Write down, in ascending powers of x, the first three terms in the expansion of (1+4x)ⁿ. Simplify each term.[2]
(a)(ii)In the expansion of (1+4x)ⁿ (1-4x) the coefficient of x² is 6032. Given that n > 0, find the value of n.[3]
(b)Find the term independent of x in the expansion of (x²/2 - 8/x⁴)¹⁰.[2]

Mark scheme

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

[B2] 1 + 4nx + 8n(n-1)x²
[B1] for any two correct terms or all 3 correct but listed not summed
[M1] FT from (i) identifying correct terms and combining their coefficient of x² - 4 × their coefficient of x, which evaluates to 8n(n-1) - 16n
[M1] Forms a 3-term quadratic = 0 and solves (except allow '= a constant' if they go on to complete to square)
[A1] n = 29 only
[M1] 10C2 X (x²/2)^8 X (-8/x⁴)² soi
[A1] 11.25 or 45/4

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About this question

Topic

Series, O-Level Additional Mathematics (syllabus 4037). Structured question, 7 marks.

Source

O-Level Additional Mathematics, May/June 2022 Paper 2 Variant 1, question 5.

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