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6 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2 · Question 4(b)(ii)

Let f(x) = 2x^3 + 3x^2 - 4x - 1 = 0.

Using x_1 = 0.6 as a first approximation of a root T of f(x), execute FOUR iterations of the Newton–Raphson method to obtain a second approximation, x_2, of T.

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Other parts of this question

  1. 4(a)(i)Express \binom{n}{r} in terms of factorials.[1 mark]
  2. 4(a)(ii)Hence, show that \binom{n}{r} = \binom{n}{n - r}.[3 marks]
  3. 4(a)(iii)Find the coefficient of x^4 in \left(x^2 - \frac{3}{x}\right)^8.[5 marks]
  4. 4(a)(iv)Using the identity (1 + x)^{2n} = (1 + x)^n (1 + x)^n, show that \binom{2n}{n} = c_0^2 + c_1^2 + c_2^2 + \dots + c_{n-1}^2 + c_n^2, where…[8 marks]
  5. 4(b)(i)Use the intermediate value theorem to determine whether the equation f(x) has any roots in the interval [0.2, 2].[2 marks]

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