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4 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2023 · Paper 2 · Question 4(b)(i)

Show that θₙ₊₁ = sin⁻¹(2 / (8 - θₙ²)) is a suitable iteration for the approximation of the roots of the equation cosec θ = 4 - (1/2)θ².

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  1. 4(a)(i)Use the intermediate value theorem to prove that x⁴ - 2x³ + x² + 2 = 0 has a root in the interval [-2, -1.5].[3 marks]
  2. 4(a)(ii)Hence, use two iterations of the Newton–Raphson method, with initial estimate x₁ = -1.5, to calculate a new estimate of the root of x⁴ - 2x³ + x² + 2 = 0 in…[6 marks]
  3. 4(b)(ii)Hence, using θ₁ = 0 as the first approximation, calculate a new estimate for the root of cosec θ = 4 - (1/2)θ², correct to three decimal places.[5 marks]
  4. 4(c)Determine the Maclaurin expansion of f(x) = (1 + x²) cos x up to and including the third non-zero term.[7 marks]

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