WMA12/01 Oct 2019, Question 3: The Binomial Expansion
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks
The question, in short
Expand (1 + x/4)^12 up to the x³ term with simplified coefficients, then find the constant term in ((x² + 8)/x⁵)(1 + x/4)^12.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 3 marks
Give the first four terms of the binomial expansion of (1 + x/4)^12 in ascending powers of x, with each coefficient simplified.
- 1(1 + x/4)¹² = 1 + 12(x/4) + (12×11/2!)(x/4)² + (12×11×10/3!)(x/4)³ + …
- 2= 1 + 3x + 66(x²/16) + 220(x³/64) + …
- 3= 1 + 3x + (33/8)x² + (55/16)x³ + …
Answer1 + 3x + (33/8)x² + (55/16)x³
Part (b) · 3 marks
Using the expansion, find the term with no x in ((x² + 8)/x⁵)(1 + x/4)^12.
- 1(x² + 8)/x⁵ = x⁻³ + 8x⁻⁵
- 2x⁻³ needs the x³ term: x⁻³ × (55/16)x³ = 55/16
- 38x⁻⁵ needs the x⁵ term: ¹²C₅(1/4)⁵x⁵ = 792/1024 x⁵ = (99/128)x⁵
- 48x⁻⁵ × (99/128)x⁵ = 99/16
- 5Term independent of x = 55/16 + 99/16 = 77/8
Answer77/8
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