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3 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 2 · Question 3(a)

Determine the coefficient of the term in x³ in the binomial expansion of (3x + 2)⁵.

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  1. 3(b)(i)Show that the binomial expansion of (1 + x)^(1/2) + (1 - x)^(1/2) up to the term in x² is 2 - (3/16)x².[4 marks]
  2. 3(b)(ii)Hence, by letting x = 1/16, compute an approximation of √17 + √15, correct to 4 decimal places.[3 marks]
  3. 3(c)(i)Show that h(x) = 0 has a root on the interval [0, 1].[3 marks]
  4. 3(c)(ii)Use the iteration x_(n+1) = (x_n² + 1) / (2x_n + 1) with initial estimate x₁ = 0.7 to estimate the root of h(x) = 0, correct to 2 decimal places.[6 marks]
  5. 3(d)Use the Newton-Raphson method with initial estimate x₀ = 5.5 to approximate the root of g(x) = sin 3x in the interval [5, 6], correct to 2 decimal places.[6 marks]

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