| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Moderate -0.8 This is a straightforward exponential decay question requiring only direct substitution (t=0), recognition of horizontal asymptote (as tââ), solving a simple exponential equation by taking logarithms, and sketching a standard shifted exponential curve. All steps are routine applications of C3 techniques with no problem-solving or novel insight required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Initial mass \(= 20 + 30e^0 = 50\) grams | M1A1 | |
| Long term mass \(= 20\) grams | B1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(30 = 20 + 30e^{-0.1t}\) | M1 | |
| \(e^{-0.1t} = \frac{1}{3}\) | M1 | anti-logging correctly |
| \(-0.1t = \ln(\frac{1}{3}) = -1.0986...\) | A1 | 11, 11.0, 10.99, 10.986 (not more than 3 d.p) |
| \(t = 11.0\) mins | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct shape through \((0, 50)\) | B1 | correct shape through \((0,50)\) â ignore negative values of \(t\) |
| \(\rightarrow 20\) as \(t \rightarrow \infty\) | B1 | |
| [2] |
# Question 6:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Initial mass $= 20 + 30e^0 = 50$ grams | M1A1 | |
| Long term mass $= 20$ grams | B1 | |
| | [3] | |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $30 = 20 + 30e^{-0.1t}$ | M1 | |
| $e^{-0.1t} = \frac{1}{3}$ | M1 | anti-logging correctly |
| $-0.1t = \ln(\frac{1}{3}) = -1.0986...$ | A1 | 11, 11.0, 10.99, 10.986 (not more than 3 d.p) |
| $t = 11.0$ mins | [3] | |
## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct shape through $(0, 50)$ | B1 | correct shape through $(0,50)$ â ignore negative values of $t$ |
| $\rightarrow 20$ as $t \rightarrow \infty$ | B1 | |
| | [2] | |
---6 In a chemical reaction, the mass $m$ grams of a chemical after $t$ minutes is modelled by the equation
$$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$
(i) Find the initial mass of the chemical.
What is the mass of chemical in the long term?\\
(ii) Find the time when the mass is 30 grams.\\
(iii) Sketch the graph of $m$ against $t$.
\hfill \mbox{\textit{OCR MEI C3 2008 Q6 [8]}}