SPS SPS FM Pure 2021 May — Question 9 12 marks

Exam BoardSPS
ModuleSPS FM Pure (SPS FM Pure)
Year2021
SessionMay
Marks12
TopicSystems of differential equations
TypeFind corresponding general solution for y
DifficultyStandard +0.8 This is a coupled differential equations problem requiring multiple techniques: verifying an exponential model from initial conditions, demonstrating conservation via sum of derivatives, and solving a first-order linear ODE with exponential forcing term. Part (iii) requires integrating factor method with careful algebra. More demanding than standard single DE questions but uses established A-level Further Maths techniques without requiring novel insight.
Spec4.10h Coupled systems: simultaneous first order DEs
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
  2. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). Comment on this result in the context of the industrial process. [4]
  3. Express \(y\) in terms of \(t\). [5]
During an industrial process substance $X$ is converted into substance $Z$. Some of the substance $X$ goes through an intermediate phase, and is converted to substance $Y$, before being converted to substance $Z$. The situation is modelled by

$$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$

where $x$, $y$ and $z$ are the amounts in kg of $X$, $Y$ and $Z$ at time $t$ hours after the process starts.

Initially there is 10 kg of substance $X$ and nothing of substance $Y$ and $Z$. The amount of substance $X$ decreases exponentially. The initial rate of decrease is 4 kg per hour.

\begin{enumerate}[label=(\roman*)]
\item Show that $x = Ae^{-0.4t}$, stating the value of $A$. [3]

\item Show that $\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0$. Comment on this result in the context of the industrial process. [4]

\item Express $y$ in terms of $t$. [5]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q9 [12]}}
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