A-LevelMathematicsNumerical methods9709/21 · Oct/Nov 2019 · Q59 Marks

It is given that ∫₀^a (3x² + 4 cos 2x – sin x) dx = 2, where a is a constant.

(i)Show that a = √(3 – 2 sin 2a – cos a).[4]
(ii)Using the equation in part (i), show by calculation that 0.5 < a < 0.75.[2]
(iii)Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

Mark scheme

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

[*M1] Integrate to obtain form x³ + k₁ sin 2x + k₂ cos x
[A1] Obtain correct x³ + 2sin2x + cos x
[DM1] Apply limits correctly and equate to 2
[A1] Confirm given result AG; necessary detail needed
[M1] Consider sign of a – √(3 – 2sin 2a – cos a) or equivalent for 0.5 and 0.75
[A1] Obtain -0.26 and 0.10 or equivalents and justify conclusion AG; necessary detail needed
[M1] Use iterative process correctly at least once Need to see a correct x₃, may be implied by x₁ = 0.5 so x₃ = 0.65256 or x₁ = 0.75 so x₃ = 0.64897 OE Must be working with radians
[A1] Obtain final answer 0.651
[A1] Show sufficient iterations to 5sf to justify answer or show a sign change in the interval [0.6505, 0.6515]

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