4 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2 · Question 4(b)(i)

Show that the binomial expansion of \left(1 + \frac{1}{8}x\right)^8 up to and including the term in x^4 is 1 + x + \frac{7}{16}x^2 + \frac{7}{64}x^3 + \frac{35}{2048}x^4.

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4(a)Eight boys and two girls are to be seated on a bench. How many seating arrangements are possible if the girls can neither sit together nor sit at the ends?[5 marks]
4(b)(ii)Use the expansion to approximate the value of (1.0125)^8.[4 marks]
4(c)(i)Use the intermediate value theorem to show that f(x) = \sqrt{x} - \cos x has a root in the interval [0, 1].[3 marks]
4(c)(ii)Use two iterations of the interval bisection method to approximate the root of f in the interval [0, 1].[4 marks]
4(d)(i)Show that x_{n+1} = \sqrt[3]{\frac{9 - 3x_n}{2}} is an appropriate iterative formula for finding the root of f(x) = -2x^3 - 3x + 9.[2 marks]
4(d)(ii)Apply the iterative formula with initial approximation x_1 = 1, to obtain a third approximation, x_3, of the root of the equation.[3 marks]

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