2020 TJC P1 Q11

In a research laboratory, scientists carry out experiments to produce a new substance \(C\). \(C\) is produced when two substances \(A\) and \(B\) are reacted together in a chemical reaction. At time \(t\) hours, the amount of \(C\) is \(x\) grams, and the amounts of \(A\) and \(B\) present are \(\left( ax \right)\) grams and \(\left( bx \right)\) grams respectively, where \(a\) and \(b\) are real constants. At the start of the experiment, there are no traces of \(C\) found. At any instant, the rate at which the amount of \(C\) increases is proportional to the product of the amount of \(A\) and the amount of \(B\) present at that instant.

1. In an experiment, a scientist carries out the experiment with \(b=a\).
   1. Obtain a differential equation relating \(x\) and \(t\) solve for \(x\) in terms of \(t\).[4]
   2. Sketch the graph of \(x\) against \(t\).[1]
   3. Given that there is \(\frac{1}{2}a\) grams of \(C\) produced \(1\) hour after the experiment started, find the time needed for \(\frac{4}{5}a\) grams of \(C\) to be produced.[2]
2. Another scientist carries out the experiment with \(b < a\).
   1. Obtain a differential equation relating \(x\) and \(t\) and solve for \(x\) in terms of \(t\).[4]

   2. Using your answer in part (ii)(a)¸ find the amount of \(C\) produced after a long time.

      [2]