| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential growth/decay model setup |
| Difficulty | Moderate -0.8 This is a straightforward exponential decay question requiring only routine application of the formula m = m₀e^(kt). Part (i) involves finding k from given data points and substituting to find missing values. Part (ii) requires solving a logarithmic equation. All steps are standard C3 techniques with no problem-solving insight needed, making it easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context |
| \(t\) | 0 | 5 | 10 | 25 |
| \(m\) | 200 | 160 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain 128 for value corresponding to 10 | B1 | Allow any value rounding to 128 |
| Obtain 65.5 for value corresponding to 25 | B1 | Allow any value rounding to 65 or 66; whether obtained using powers of 0.8 or by use of formula |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to find formula for \(m\) of form \(200k^t\) or \(200 \times r^t\) | M1 | Whether attempted in part (i) or (ii) |
| Obtain \(200e^{(0.2 \ln 0.8)t}\) or \(200e^{-0.0446t}\) or \(200 \times 0.8^{0.2}\) or \(200 \times 0.956^t\) | A1 | Or equiv |
| Show correct process for solving equation of form \(200e^{kt} = 50\) or \(200r^{t} = 50\) | M1 | |
| Obtain 31 | A1 | Or greater accuracy rounding to 31; ignore any units given; second M1 is implied by correct answer |
| [4] |
## Part i
Obtain 128 for value corresponding to 10 | B1 | Allow any value rounding to 128
Obtain 65.5 for value corresponding to 25 | B1 | Allow any value rounding to 65 or 66; whether obtained using powers of 0.8 or by use of formula
| [2] |
## Part ii
Attempt to find formula for $m$ of form $200k^t$ or $200 \times r^t$ | M1 | Whether attempted in part (i) or (ii) | If formula attempted in part (i), marks earned must be recorded in part (ii)
Obtain $200e^{(0.2 \ln 0.8)t}$ or $200e^{-0.0446t}$ or $200 \times 0.8^{0.2}$ or $200 \times 0.956^t$ | A1 | Or equiv
Show correct process for solving equation of form $200e^{kt} = 50$ or $200r^{t} = 50$ | M1 |
Obtain 31 | A1 | Or greater accuracy rounding to 31; ignore any units given; second M1 is implied by correct answer | Special case: no formula anywhere and answer 31 (or greater accuracy) given, award B2 (i.e. 2/4 for part (ii))
| [4] |3 The mass of a substance is decreasing exponentially. Its mass is $m$ grams at time $t$ years. The following table shows certain values of $t$ and $m$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$t$ & 0 & 5 & 10 & 25 \\
\hline
$m$ & 200 & 160 & & \\
\hline
\end{tabular}
\end{center}
(i) Find the values missing from the table.\\
(ii) Determine the value of $t$, correct to the nearest integer, for which the mass is 50 grams.
\hfill \mbox{\textit{OCR C3 2016 Q3 [6]}}