<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express \(\frac{1}{Q(500 - Q)}\) in partial fractions.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A group of scientists is monitoring the population of a particular species of birds on an island. At time \(t\) years after the start of the monitoring, the number of birds on the island is \(Q\). The scientists observe that the birth rate is proportional to the bird's population and the birds are dying at a rate proportional to the square of the bird's population. There were \(1000\) birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are \(500\) birds.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that the differential equation relating \(Q\) and \(t\) is given by\[\frac{\mathrm{d}Q}{\mathrm{d}t} = kQ(500 - Q)\]where \(k\) is a positive constant.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the differential equation, expressing \(Q\) in terms of \(k\) and \(t\).</span><span class="inline-marks">[5]</span></div></li></ol></li></ol>

Express \(\frac{1}{Q(500 - Q)}\) in partial fractions.

A group of scientists is monitoring the population of a particular species of birds on an island. At time \(t\) years after the start of the monitoring, the number of birds on the island is \(Q\). The scientists observe that the birth rate is proportional to the bird's population and the birds are dying at a rate proportional to the square of the bird's population. There were \(1000\) birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are \(500\) birds.

Show that the differential equation relating \(Q\) and \(t\) is given by\n\[\frac{\mathrm{d}Q}{\mathrm{d}t} = kQ(500 - Q)\]\nwhere \(k\) is a positive constant.

Hence, solve the differential equation, expressing \(Q\) in terms of \(k\) and \(t\).

2025 HCI P1 Q10

2025 HCI P1 Q10

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