<p>The curve \(C\) has cartesian equation \(y = \sqrt{4-2x^2}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the curve \(C\), labelling the exact coordinates of any axial intercepts.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the tangent to \(C\) at the point \(P(1, \sqrt{2})\), leaving your answer in exact form.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the substitution \(x = \sqrt{2} \sin \theta \), find the exact value of \(\int_{1}^{\sqrt{2}} \sqrt{4-2x^2} \, \mathrm{d}x\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the exact value of the area of the region bounded by \(C\), the tangent to \(C\) at \(P\), and the \(x\)-axis.</span><span class="inline-marks">[2]</span></div></li></ol>

The curve \(C\) has cartesian equation \(y = \sqrt{4-2x^2}\).

Sketch the curve \(C\), labelling the exact coordinates of any axial intercepts.

Find the equation of the tangent to \(C\) at the point \(P(1, \sqrt{2})\), leaving your answer in exact form.

Using the substitution \(x = \sqrt{2} \sin \theta \), find the exact value of \(\displaystyle \int_{1}^{\sqrt{2}} \sqrt{4-2x^2} \, \mathrm{d}x\).

Hence, find the exact value of the area of the region bounded by \(C\), the tangent to \(C\) at \(P\), and the \(x\)-axis.

2025 RVHS P1 Q11

2025 RVHS P1 Q11