| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Applied rate of change |
| Difficulty | Moderate -0.3 This is a straightforward applied exponential question requiring standard techniques: evaluating a limit as t→∞, solving an exponential equation using logarithms, and differentiating to find a rate of change. All three parts are routine C3 exercises with no conceptual challenges beyond direct application of learned methods. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(200°C\) | B1 | For value 200 |
| (ii) \(150 = 200 - 180e^{-0.1t} \Rightarrow e^{-0.1t} = \frac{50}{180}\) Hence \(-0.1t = \ln\frac{5}{18} \Rightarrow t = 12.8\) | M1, M1, A1 | For isolating the exponential term; For taking logs correctly; For correct value 12.8 (minutes) |
| (iii) \(\frac{d\theta}{dt} = 18e^{-0.1t}\) Hence rate is \(18e^{-0.1 \times 12.8} = 5.0°C\) per minute | M1, A1, M1, A1 | For differentiation attempt; For correct derivative; For using their value from (ii) in their \(\theta\); For value 5.0(0) |
**(i)** $200°C$ | B1 | For value 200
**(ii)** $150 = 200 - 180e^{-0.1t} \Rightarrow e^{-0.1t} = \frac{50}{180}$ Hence $-0.1t = \ln\frac{5}{18} \Rightarrow t = 12.8$ | M1, M1, A1 | For isolating the exponential term; For taking logs correctly; For correct value 12.8 (minutes)
**(iii)** $\frac{d\theta}{dt} = 18e^{-0.1t}$ Hence rate is $18e^{-0.1 \times 12.8} = 5.0°C$ per minute | M1, A1, M1, A1 | For differentiation attempt; For correct derivative; For using their value from (ii) in their $\theta$; For value 5.0(0)
**Total for Question 5: 8 marks**
---5 At time $t$ minutes after an oven is switched on, its temperature $\theta ^ { \circ } \mathrm { C }$ is given by
$$\theta = 200 - 180 \mathrm { e } ^ { - 0.1 t }$$
(i) State the value which the oven's temperature approaches after a long time.\\
(ii) Find the time taken for the oven's temperature to reach $150 ^ { \circ } \mathrm { C }$.\\
(iii) Find the rate at which the temperature is increasing at the instant when the temperature reaches $150 ^ { \circ } \mathrm { C }$.
\hfill \mbox{\textit{OCR C3 Q5 [8]}}