A-LevelMathematicsDifferential equations9709/31 · Oct/Nov 2021 · Q79 Marks

Given that y = ln(lnx), show that `dy/dx = 1/(x lnx)`.

(a)Given that y = ln(lnx), show that `dy/dx = 1/(x lnx)`.[1]
(b)The variables x and t satisfy the differential equation x lnx + t (dx/dt) = 0. It is given that x = e when t = 2. Solve the differential equation obtaining an expression for x in terms of t, simplifying your answer.[7]
(c)Hence state what happens to the value of x as t tends to infinity.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] Show sufficient working to justify the given answer
[B1] Correct separation of variables
[B1] Obtain term `ln(lnx)`
[B1] Obtain term `-lnt`
[M1] Evaluate a constant or use x = e and t = 2 as limits in an expression involving `ln(lnx)`
[A1] Obtain correct solution in any form, e.g. `ln(lnx) = -lnt + ln 2`
[M1] Use log laws to enable removal of logarithms
[A1] Obtain answer `x = e^(2/t)`, or simplified equivalent
[B1] State that x tends to 1 coming from `x = e^(k/t)`

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2021 Paper 3 Variant 1, question 7.

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