| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2020 |
| Session | November |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Standard +0.3 Part (a) is trivial substitution (1^b = 1 < b^1 for b>1). Part (b) requires finding a counterexample, but the most obvious one (a=2, b=4 giving 2^4=4^2=16) is easily discovered through minimal trial. This is straightforward problem-solving with low conceptual demand, slightly easier than average. |
| Spec | 1.01c Disproof by counter example1.02a Indices: laws of indices for rational exponents |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a=1\) and \(b>1 \Rightarrow a^b = 1\) | B1 | Subbing values may score B1 but not E1 |
| and \(b^a = b\) hence \(a^b < b^a\) | E1 | Convincing completion; AG |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integer values of \(a\) and \(b\) with \(b > a > 1\) such that \(a^b\) not greater than \(b^a\) | B1 | Possible values: \(a=2, b=3\) or \(a=2, b=4\) |
| [1] |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a=1$ and $b>1 \Rightarrow a^b = 1$ | B1 | Subbing values may score B1 but not E1 |
| and $b^a = b$ hence $a^b < b^a$ | E1 | Convincing completion; AG |
| **[2]** | | |
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## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integer values of $a$ and $b$ with $b > a > 1$ such that $a^b$ not greater than $b^a$ | B1 | Possible values: $a=2, b=3$ or $a=2, b=4$ |
| **[1]** | | |
---9
\begin{enumerate}[label=(\alph*)]
\item Show that if $a = 1$ and $b > 1$ then $\mathrm { a } ^ { \mathrm { b } } < \mathrm { b } ^ { \mathrm { a } }$.
\item Find integer values of $a$ and $b$ with $b > a > 1$ and $\mathrm { a } ^ { \mathrm { b } }$ not greater than $\mathrm { b } ^ { \mathrm { a } }$ (a counter example to the conjecture given in lines 7-8).
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2020 Q9 [3]}}