| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Natural logarithm equation solving |
| Difficulty | Moderate -0.3 This is a straightforward C3 question testing basic logarithm manipulation and function composition. Part (i) is routine equation solving, part (ii) is standard curve sketching, and part (iii) involves simple function composition with algebraic simplification. All techniques are standard with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules |
5. The function $f$ is defined by
$$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$
(i) Solve the equation $\mathrm { f } ( x ) = 0$.\\
(ii) Sketch the curve $y = \mathrm { f } ( x )$.
The function g is defined by
$$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
(iii) Show that
$$\operatorname { fg } ( x ) = x + a - \ln b$$
where $a$ and $b$ are integers to be found.\\
\hfill \mbox{\textit{OCR C3 Q5 [7]}}