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Integration and Its Applications · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SIndefinite integrals and their properties8 marksPaper 1~12 min
The rate of change of the volume of water in a tank, measured in litres per minute, is given by dVdt=12t−3t2\frac{dV}{dt} = 12t - 3t^2 where tt is the time in minutes, t≥0t \geq 0.
(a)
State the values of tt for which the volume of water in the tank is increasing. [1 mark]
(b)
Find V(t)V(t), given that the tank initially contains 5050 litres of water. [4 marks]
(c)
Determine the maximum volume of water in the tank and justify that is a maximum. [3 marks]
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Solutions

2MasterySAQ-SIndefinite integrals and their properties5 marksPaper 1~8 min
A particle moves along a straight line. Its velocity v m s−1v\,\text{m s}^{-1} at time tt seconds is given by v(t)=3t2−12t+9v(t) = 3t^2 - 12t + 9, for t≥0t \geq 0.
(a)
Show that the displacement s(t)s(t) metres of the particle from its starting point is s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t. [2 marks]
(b)
Calculate the total distance travelled by the particle in the first 44 seconds. [3 marks]
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Solutions

3ChallengeSAQ-LIndefinite integrals and their properties8 marksPaper 1~12 min
A biologist is modelling the growth of a bacterial colony. The rate of change of the mass mm (in grams) of the colony with respect to time tt (in hours) is given by dmdt=2t+3t2+3t,t>0.\frac{dm}{dt} = \frac{2t+3}{t^2+3t}, \quad t > 0. At time t=1t = 1 hour, the mass of the colony is 55 grams.
(a)
Determine an expression for m(t)m(t), the mass of the colony at time tt. Give your answer in the form m(t)=ln⁡ ⁣(g(t))+cm(t) = \ln\!\left(g(t)\right) + c, where g(t)g(t) is a function of tt and cc is a constant. [4 marks]
(b)
Calculate the mass of the colony after 44 hours, correct to 3 significant figures. [2 marks]
(c)
A second biologist claims this model predicts that the colony's mass will eventually exceed any fixed value, no matter how large. Evaluate this claim by analysing the behaviour of m(t)m(t) as t→∞t \to \infty, and comment on whether the model is realistic for large tt. [2 marks]
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Solutions

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