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A-LevelMathematicsNumerical methods9709/31 · Oct/Nov 2017 · Q36 Marks

The equation x³ = 3x + 7 has one real root, denoted by α.

  1. (i)Show by calculation that α lies between 2 and 3.[2]
  2. (ii)Two iterative formulae, A and B, derived from this equation are as follows: xn+1 = (3xn + 7)³, (A) xn+1 = xn² - 7 / 3, (B) Each formula is used with initial value x₁ = 2.5. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

  1. [M1] (i) Calculate value of a relevant expression or expressions at x = 2 and x = 3
  2. [A1] (i) Complete the argument correctly with correct calculated values
  3. [M1] (ii) Use an iterative formula correctly at least once
  4. [A1] (ii) Show that (B) fails to converge
  5. [A1] (ii) Using (A), obtain final answer 2.43
  6. [A1] (ii) Show sufficient iterations to justify 2.43 to 2 d.p., or show there is a sign change in (2.425, 2.435)

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About this question

Topic
Numerical methods, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source
A-Level Mathematics, Oct/Nov 2017 Paper 3 Variant 1, question 3.

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