| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential growth/decay model setup |
| Difficulty | Moderate -0.3 This is a straightforward exponential model question requiring standard techniques: finding a constant from given values, solving a logarithmic equation, and differentiating an exponential function. All parts are routine C3 exercises with no problem-solving insight required, making it slightly easier than average but not trivial due to the multi-step nature. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form1.06i Exponential growth/decay: in modelling context1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| \(t\) | 0 | 10 | 20 |
| \(X\) | 275 | 440 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: State proportion \(\frac{440}{275}\) | B1 | |
| Attempt calculation involving proportion | M1 | Involving multiplication and \(X\) value |
| Obtain 704 | A1 | 3 |
| Or: Use formula of form \(275e^{kt}\) or \(275a^t\) | M1 | Or equiv |
| Obtain \(k = 0.047\) or \(a = \sqrt[10]{1.6}\) | A1 | Or equiv |
| Obtain 704 | A1 | (3) Allow \(\pm 0.5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt correct process involving logarithm | M1 | Or equiv including systematic trial and improvement attempt |
| Obtain \(\ln\frac{20}{80} = -0.02t\) | A1 | Or equiv |
| Obtain 69 | A1 | 3 Or greater accuracy; scheme for T&I: M1A2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate to obtain \(ke^{-0.02t}\) | M1 | Any constant \(k\) different from 80 |
| Obtain \(-1.6e^{-0.02t}\) (or \(1.6e^{-0.02t}\)) | A1 | Or unsimplified equiv |
| Obtain 0.88 | A1 | 3 Or greater accuracy; allow \(-0.88\) |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Either: State proportion $\frac{440}{275}$ | B1 | |
| Attempt calculation involving proportion | M1 | Involving multiplication and $X$ value |
| Obtain 704 | A1 | **3** |
| Or: Use formula of form $275e^{kt}$ or $275a^t$ | M1 | Or equiv |
| Obtain $k = 0.047$ or $a = \sqrt[10]{1.6}$ | A1 | Or equiv |
| Obtain 704 | A1 | **(3)** Allow $\pm 0.5$ |
## Question 6(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt correct process involving logarithm | M1 | Or equiv including systematic trial and improvement attempt |
| Obtain $\ln\frac{20}{80} = -0.02t$ | A1 | Or equiv |
| Obtain 69 | A1 | **3** Or greater accuracy; scheme for T&I: M1A2 |
## Question 6(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate to obtain $ke^{-0.02t}$ | M1 | Any constant $k$ different from 80 |
| Obtain $-1.6e^{-0.02t}$ (or $1.6e^{-0.02t}$) | A1 | Or unsimplified equiv |
| Obtain 0.88 | A1 | **3** Or greater accuracy; allow $-0.88$ |
---6
\begin{enumerate}[label=(\alph*)]
\item \begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$t$ & 0 & 10 & 20 \\
\hline
$X$ & 275 & 440 & \\
\hline
\end{tabular}
\end{center}
The quantity $X$ is increasing exponentially with respect to time $t$. The table above shows values of $X$ for different values of $t$. Find the value of $X$ when $t = 20$.
\item The quantity $Y$ is decreasing exponentially with respect to time $t$ where
$$Y = 80 \mathrm { e } ^ { - 0.02 t }$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $t$ for which $Y = 20$, giving your answer correct to 2 significant figures.
\item Find by differentiation the rate at which $Y$ is decreasing when $t = 30$, giving your answer correct to 2 significant figures.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C3 2006 Q6 [9]}}