<p>A curve \(C\) has parametric equations</p>
<p style="text-align: center">\(x = 3t^2 + 2t, \quad y = t^2 + 2t^3 \quad \text{for}\hspace{0.5em} t \geqslant 0.\)</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}y}{\mathrm{d}x} = kt\), where&nbsp;\(k\) is a constant to be found.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">The tangent to \(C\) at a point \(P\) makes an angle of \(60^\circ\) with the \(x\)-axis. Find the exact coordinates of \(P\).<span class="inline-marks">[3]</span></div></li></ol>
<p>The point \(Q\) with coordinates \((16, 20)\) lies on \(C\).</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1787389552638-tujlqspb1" data-list-item-path="item-1787389552638-tujlqspb1"><div class="list-item-paragraph"><p>Using calculus, find the exact area of the region bounded by the curve \(C\), the \(x\)-axis and the line parallel to the \(y\)-axis through \(Q\).</p><span class="inline-marks">[5]</span></div></li></ol>

\(x = 3t^2 + 2t, \quad y = t^2 + 2t^3 \quad \text{for } t \geqslant 0.\)

Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = kt\), where \(k\) is a constant to be found.

The tangent to \(C\) at a point \(P\) makes an angle of \(60^\circ\) with the \(x\)-axis. Find the exact coordinates of \(P\).

The point \(Q\) with coordinates \((16, 20)\) lies on \(C\).

<p>Using calculus, find the exact area of the region bounded by the curve \(C\), the \(x\)-axis and the line parallel to the \(y\)-axis through \(Q\).</p>

N2024 P1 Q9

N2024 P1 Q9