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3 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2026 · Paper 2 · Question 4(c)(i)

This section covers roots of equations. An equation is given by cos x = xe^(-x).

Using the intermediate value theorem, show that cos x = xe^(-x) has a root between x = 1 and x = 1.5.

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  1. 4(a)Obtain the Maclaurin series expansion of f(x) = e^(2x) up to and including the term in x^4.[3 marks]
  2. 4(b)Obtain the binomial expansion of (8 + x)^(1/3) in ascending powers of x, up to and including the term in x^2. State the values of x for which the expansion is…[6 marks]
  3. 4(c)(ii)Hence, use the method of interval bisection to determine, correct to 1 decimal place, the approximate root of cos x = xe^(-x) which lies in the interval (1,…[6 marks]
  4. 4(d)(i)Use the Newton-Raphson formula to show that for n ≥ 1, with a given initial estimate x_n, x_(n+1) = (3x_n^4 + 13)/(4x_n^3 + 1).[4 marks]
  5. 4(d)(ii)Hence, or otherwise, use the Newton-Raphson method with the initial estimate x_1 = 2 to calculate, to 2 decimal places, a new estimate, x_2, of the root.[3 marks]

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