2025 TGM P2 Q4

The following graph shows the two parts of the curve defined by the equation \(x^2 y = 5 - y^4\), and the normal to the curve at the point \(P \, (2,1)\).

1. Show that there are exactly two point on the curve where the gradient is zero.[3]
2. Find the equation of the normal to the curve at the point \(P\).[3]

3. The normal at \(P\) cuts the curve again at \(Q\). Find the \(x\)-coordinate of \(Q\).

   Leave your answer in \(5\) significant figures.

   [2]
4. The shaded region is rotated \(2\pi\) about the \(y\)-axis. Find the volume of the solid formed.[3]