A-LevelMathematicsNumerical methods9709/31 · Oct/Nov 2022 · Q78 Marks

The equation of a curve is y = x/cos²x, for 0 ≤ x < π/2. At the point where x = a, the tangent to the curve has gradient equal to 12.

(a)Show that a = cos⁻¹(√(cos a + 2a sin a)/12).[3]
(b)Verify by calculation that a lies between 0.9 and 1.[2]
(c)Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] (a) Use correct product or quotient rule
[A1] (a) Obtain correct derivative in any form
[A1] (a) Equate derivative at x = a to 12 and obtain a = cos⁻¹(√(cos a + 2a sin a)/12) (AG)
[M1] (b) Evaluate a relevant expression or pair of expressions at a = 0.9 and a = 1
[A1] (b) Complete the argument correctly with correct calculated values
[M1] (c) Use the process aₙ₊₁ = cos⁻¹(√(cos aₙ + 2aₙ sin aₙ)/12) correctly at least once
[A1] (c) Obtain final answer 0.97
[A1] (c) Show sufficient iterations to 4 d.p. to justify 0.97 to 2 d.p., or show there is a sign change in the interval (0.965, 0.975)

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