A-LevelMathematicsDifferential equations9709/33 · May/June 2022 · Q89 Marks

At time t days after the start of observations, the number of insects in a population is N. The variation in the number of insects is modelled by a differential equation of the form dN/dt = kN^(3/2) cos 0.02t, where k is a constant and N is a continuous variable. It is given that when t = 0, N = 100.

(a)Solve the differential equation, obtaining a relation between N, k and t.[5]
(b)Given also that N = 625 when t = 50, find the value of k.[2]
(c)Obtain an expression for N in terms of t, and find the greatest value of N predicted by this model.[2]

Mark scheme

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

[B1] Separate variables correctly
[B1] Obtain term 2/√N
[B1] Obtain term 50sin 0.02t
[M1] Use t = 0, N = 100 to evaluate a constant, or as limits, in a solution containing terms a/√N and bsin 0.02t, where ab≠ 0
[A1] Obtain correct solution in any form, e.g. 2/√N = 50k sin 0.02t − 0.2
[M1] Use the substitution N = 625 and t = 50 in expression of appropriate form to evaluate k
[A1] Obtain k = 0.00285[2148]
[M1] Rearrange and obtain N = 4/(0.2 − 0.142(607)sin 0.02t)²
[A1] Accept answers between 1209 and 1215

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