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Differential Equations · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQSolving first-order differential equations6 marksPaper 3~9 min

A pharmaceutical company is developing a new drug. The rate at which the drug is absorbed into the bloodstream is modelled by the differential equation

\frac{dC}{dt} = k(20 - C),

where

C(t)

is the concentration of the drug (in

\text{mg\,L}^{-1}

) at time

t

hours after administration, and

k

is a positive constant. Initially,

C(0) = 0

(a)

Prove that the general solution to the differential equation is

C(t) = 20\!\left(1 - e^{-kt}\right). \quad \textbf{}

[5 marks]

(b)

After 3 hours, the concentration is measured as

12\,\text{mg\,L}^{-1}

. (i) Calculate the value of

k

, giving your answer correct to three significant figures. [2 marks]
(ii) A concentration of at least

19.9\,\text{mg\,L}^{-1}

is required for the drug to be therapeutically effective. Determine the time, to the nearest minute, at which this threshold is first reached. [2 marks]
(iii) The drug is considered unsafe if the concentration ever reaches

20\,\text{mg\,L}^{-1}

. Using your model, evaluate whether the drug can be administered safely for an indefinite period. Justify your answer with reference to the behaviour of

C(t)

t \to \infty

. [1 mark]

Solutions

Questions 2-8

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