| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Contradiction proof of irrationality |
| Difficulty | Standard +0.8 Part (i) is a standard exponential equation requiring substitution (u = e^x) and solving a quadratic—routine C3 material around difficulty 0.0. Part (ii) requires constructing a proof by contradiction for irrationality, which is significantly more demanding: students must assume log₂3 is rational, manipulate to reach 3^q = 2^p, and recognize the contradiction via unique prime factorization. This level of proof construction and logical reasoning is uncommon in standard C3 questions, elevating the overall difficulty notably above average. |
| Spec | 1.01d Proof by contradiction1.06g Equations with exponentials: solve a^x = b |
\begin{enumerate}
\item (i) Find, as natural logarithms, the solutions of the equation
\end{enumerate}
$$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$
(ii) Use proof by contradiction to prove that $\log _ { 2 } 3$ is irrational.\\
\hfill \mbox{\textit{OCR C3 Q5 [9]}}