2026 CCHY P2 Q11

1. Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{4x}{\sqrt{4x-2}}\right)=\dfrac{8(x-1)}{\sqrt{(4x-2)^3}}\).[3]

The diagram shows part of the curve \(y=\dfrac{8(x-1)}{\sqrt{(4x-2)^3}}\) and the line \(x=3\). The curve intersects the \(x\)-axis at the point \(P\). The line through \(P\) with gradient 1 intersects the curve again at the point \(Q\).

2. Show that the coordinates of \(Q\) are \(\left(\dfrac32,\dfrac12\right)\).[5]
3. Find the area of the shaded region bounded by the curve, the \(x\)-axis, the line \(PQ\), and the line \(x=3\).[3]