A-LevelMathematicsNumerical methods9709/31 · May/June 2019 · Q79 Marks

The diagram shows the curves y = 4 cos ½x and y = 1/(4-x) for 0 ≤ x < 4. When x = a, the tangents to the curves are perpendicular.

(i)Show that a = 4 - √(2 sin (½a)).[4]
(ii)Verify by calculation that a lies between 2 and 3.[2]
(iii)Use an iterative formula based on the equation in part (i) to determine a correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B1] (i) State at least one correct derivative
[M1] (i) Equate product of derivatives to – 1 or equivalent
[A1] (i) Obtain a correct equation, e.g. 2sin(x/2)=(4-x)²
[AG] (i) Rearrange correctly to obtain a = 4-√(2sin(a/2))
[M1] (ii) Calculate values of a relevant expression or pair of expressions at a=2 and a=3
[A1] (ii) Complete the argument correctly with correct calculated values
[M1] (iii) Use the iterative formula a_(n+1) = 4-√(2sin(a_n/2)) correctly at least once
[A1] (iii) Obtain final answer 2.611
[A1] (iii) Show sufficient iterations to 5 d.p. to justify 2.611 to 3 d.p., or show there is a sign change in the interval (2.6105, 2.6115)

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