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4 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2 · Question 4(d)

The equation 3e^x = 1 - 2 ln(x) has a root in the interval (0, 1). Taking x_1 = 0.2 as the first approximation, use the Newton-Raphson method to find a second approximation, x_2, of the root in the interval (0, 1).

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  1. 4(a)Determine the coefficient of the term in x^7 in the expansion of (x^2 - 3/x)^8.[4 marks]
  2. 4(b)By expressing ^n C_r and ^n C_(r-1) in terms of factorials, show that ^n C_r + ^n C_(r-1) = ^(n+1) C_r.[6 marks]
  3. 4(c)(i)Use the intermediate value theorem to show that the equation 4 cos(x) - x^3 + 2 = 0 has a root in the interval (1, 1.5).[3 marks]
  4. 4(c)(ii)Use linear interpolation to approximate the value of the root of the equation 4 cos(x) - x^3 + 2 = 0 in the interval (1, 1.5), correct to two decimal places.[8 marks]

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