Standard +0.3 This is a straightforward recurrence relation question with standard techniques: part (a) requires explaining a given formula (routine), part (b) involves solving a first-order linear recurrence (standard A-level Further Maths method), part (c) finds a limit as n→∞ (simple substitution), and part (d) requires checking when E_n drops below 500mg within 2 periods (basic calculation). All parts follow textbook procedures with no novel insight required, making it slightly easier than average.
7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture.
To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.
Let \(n\) be the number of six-hour periods that have elapsed since the experiment began.
Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
In this question you must show detailed reasoning.
The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg .
Show that the new requirement ceases to be satisfied before 12 hours have elapsed.
\section*{END OF QUESTION PAPER}
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7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture.
To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.
\begin{enumerate}[label=(\alph*)]
\item Let $n$ be the number of six-hour periods that have elapsed since the experiment began.
Explain how the amount of enzyme, $\mathrm { E } _ { \mathrm { n } } \mathrm { mg }$, in the mixture is given by the recurrence system $E _ { 0 } = 1200$ and $E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500$ for $n \geqslant 0$.
\item Solve the recurrence system given in part (a) to obtain an exact expression for $\mathrm { E } _ { \mathrm { n } }$ in terms of $n$.
\item Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to $\mathbf { 3 }$ significant figures.
\item In this question you must show detailed reasoning.
The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg .
Show that the new requirement ceases to be satisfied before 12 hours have elapsed.
\section*{END OF QUESTION PAPER}
}{www.ocr.org.uk}) after the live examination series.\\
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.\\
For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.\\
OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2024 Q7 [12]}}