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7 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2023 · Paper 2 · Question 3(a)

Determine the coefficient of x⁴ in the binomial expansion of (1 - 2x) / (1 + 3x)².

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  1. 3(b)Show that if the series 1 + (7 / (3x - 5)) + (7 / (3x - 5))² + (7 / (3x - 5))³ + ... converges, then its sum to infinity is (3x - 8) / (3x - 12).[8 marks]
  2. 3(c)Use mathematical induction to prove that dⁿ/dxⁿ (eˣ sin x) = 2ⁿ/² eˣ sin(x + nπ/4). You may use the fact that sin x + cos x = √2 sin(x + π/4) and that…[10 marks]

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