2024 DHS P2 Q5

Two chemicals \(X\) and \(Y\) react to form a chemical \(Z\) without any loss of mass. It is known that one part of \(X\) combines with two parts of \(Y\) to give three parts of \(Z\). For example, \(1.5\) g of \(X\) combines with \(3.0\) g of \(Y\) to give \(4.5\) g of \(Z\). Let \(z\) g be the mass of \(Z\) formed \(t\) minutes after the reaction started. The rate of change of \(z\) with respect to \(t\) is proportional to the product of the remaining masses of \(X\) and \(Y\) not reacted at any time \(t\). Initially, there are \(10\) g of \(X\), \(15\) g of \(Y\) and \(0\) g of \(Z\).

1. Show that
   \(\frac{\mathrm{d}z}{\mathrm{d}t}=k\left( 30-z \right)\left( 45-2z \right)\),
   where \(k\) is a positive constant.[2]
2. It is observed that the mass of \(Z\) is \(10\) g after \(5\) min. Solve the differential equation in part (a) and find \(z\) in terms of \(t\).[8]
3. State, with justification, the mass of \(Z\) that can be formed after a long time. Hence, or otherwise, find the corresponding remaining masses of \(X\) and \(Y\).[2]