OCR MEI C2 — Question 11

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Functions
TypeLinear transformation to find constants
DifficultyModerate -0.3 This is a standard exponential modeling question requiring students to transform data using logarithms, plot a linear graph, and extract constants from gradient/intercept. While it involves multiple steps (calculating z values, taking logs, plotting, finding k and z₀, then using the model), each individual step is routine C2 material with clear scaffolding. The physical interpretation and algebraic manipulation are straightforward, making this slightly easier than average.
Spec1.06d Natural logarithm: ln(x) function and properties1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made. \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Concepts for Advanced Mathematics (C2)
    INSERT
    Wednesday
A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25.
(i) Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\). [7]
(ii) Show that the ratio of the nth terms of the two geometric progressions found in part (i) can be written as \(2^{n-2} : 3^{n-2}\). [3]
A geometric progression has first term $a$ and common ratio $r$. The second term is 6 and the sum to infinity is 25.

(i) Write down two equations in $a$ and $r$. Show that one possible value of $a$ is 10 and find the other possible value of $a$. Write down the corresponding values of $r$. [7]

(ii) Show that the ratio of the nth terms of the two geometric progressions found in part (i) can be written as $2^{n-2} : 3^{n-2}$. [3]

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11 Answer part (iii) of this question on the insert provided.\\
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time (minutes) & 10 & 20 & 30 & 40 & 50 \\
\hline
Temperature $\left( { } ^ { \circ } \mathrm { C } \right)$ & 68 & 53 & 42 & 36 & 31 \\
\hline
\end{tabular}
\end{center}

The room temperature is $22 ^ { \circ } \mathrm { C }$. The difference between the temperature of the drink and room temperature at time $t$ minutes is $z ^ { \circ } \mathrm { C }$. The relationship between $z$ and $t$ is modelled by

$$z = z _ { 0 } 10 ^ { - k t }$$

where $z _ { 0 }$ and $k$ are positive constants.\\
(i) Give a physical interpretation for the constant $z _ { 0 }$.\\
(ii) Show that $\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }$.\\
(iii) On the insert, complete the table and draw the graph of $\log _ { 10 } z$ against $t$.

Use your graph to estimate the values of $k$ and $z _ { 0 }$.\\
Hence estimate the temperature of the drink 70 minutes after it is made.

\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education

\section*{MEI STRUCTURED MATHEMATICS}

Concepts for Advanced Mathematics (C2)\\
INSERT\\
Wednesday\\

\hfill \mbox{\textit{OCR MEI C2  Q11}}
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