O-LevelAdditional MathematicsSeries4037/11 · May/June 2012 · Q98 Marks

Find the values of the positive constants p and q such that, in the binomial expansion of (p + qx)¹⁰, the coefficient of x⁵ is 252 and the coefficient of x³ is 6 times the coefficient of x².

Find the values of the positive constants p and q such that, in the binomial expansion of (p + qx)¹⁰, the coefficient of x⁵ is 252 and the coefficient of x³ is 6 times the coefficient of x².[8]

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

[B3,2,1,0] for correct binomial terms (10C5 p⁵ (qx)⁵ and 10C3 p⁷ (qx)³ and 10C2 p⁸ (qx)²) (-1 per element (of 4) incorrect in the terms/coefficients)
[B1] Coefficient of x⁵: 10C5 p⁵ q⁵ = 252 p⁵ q⁵. Since 10C5 = 252, then 252 p⁵ q⁵ = 252, leading to p⁵ q⁵ = 1, so pq = 1 (as p,q positive).
[M1] for setting up the equation for coefficients of x³ and x² (10C3 p⁷ q³ = 6 × 10C2 p⁸ q²)
[B1] for simplifying the equation to 4q = 9p (120p⁷q³ = 6 × 45p⁸q² => 120q = 270p => 4q = 9p)
[A1] for p = 2/3
[A1] for q = 3/2

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Topic: Series, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, May/June 2012 Paper 1 Variant 1, question 9.

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