2025 NYJC P2 Q4

The equation of a curve \(C\) is \(x^3 + xy + 2y^3 = k\), where \(k\) is a constant.

1. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(x\) and \(y\).[2]
2. It is given that \(C\) has a tangent which is parallel to the \(y\)-axis. Show that the \(y\)-coordinate of the points of contact of the tangent with \(C\) must satisfy \(216y^6 + 4y^3 + k = 0\). Hence show that \(k \leq \frac{1}{54}\).[6]
3. Find the possible values of \(k\) in the case where the line \(x = -6\) is a tangent to \(C\).[2]