| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Pythagorean triples and number patterns |
| Difficulty | Standard +0.3 This is a straightforward algebraic verification question. Part (i) requires expanding $(2t)^2 + (t^2-1)^2$ and showing it equals $(t^2+1)^2$ - routine algebra with no conceptual difficulty. Part (ii) involves solving $20^2 + 21^2 = c^2$ (basic arithmetic) then checking whether this triple fits the given form by testing if any integer $t$ works - a simple verification that requires minimal problem-solving. The question is slightly easier than average as it's mostly mechanical manipulation with clear instructions. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
Positive integers $a$, $b$ and $c$ are said to form a Pythagorean triple if $a^2 + b^2 = c^2$.
\begin{enumerate}[label=(\roman*)]
\item Given that $t$ is an integer greater than 1, show that $2t$, $t^2 - 1$ and $t^2 + 1$ form a Pythagorean triple. [3]
\item The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer.
Use this triple to show that not all Pythagorean triples can be expressed in the form $2t$, $t^2 - 1$ and $t^2 + 1$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q13 [6]}}