A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2021 · Q58 Marks

Given that 2ln(x + 1) + lnx = ln(x + 9), show that x = √(9/(x + 2)).

(a)Given that 2ln(x + 1) + lnx = ln(x + 9), show that x = √(9/(x + 2)).[3]
(b)It is given that the equation x = √(9/(x + 2)) has a single root. Show by calculation that this root lies between 1.5 and 2.0.[2]
(c)Use an iterative formula, based on the equation in part (b), to find the root 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 (Feb/Mar 2021). The mark or mark type is shown in brackets.

[M1*] Use the power law correctly
[DM1] Use correct process to obtain equation with no logarithms
[A1] Confirm x = √(9/(x+2)) AG; condone absence of justification for choice of positive root.
[M1] Consider sign of x - √(9/(x+2)) or equivalent for 1.5 and 2
[A1] Obtain -0.1... and 0.5 or equivalents and justify conclusion
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 1.58
[A1] Show sufficient iterations to 5 sf to justify answer or show a sign change in interval [1.575, 1.585]

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