| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Pythagorean triples and number patterns |
| Difficulty | Standard +0.3 Part (i) is straightforward algebraic verification requiring students to substitute and expand $(2t)^2 + (t^2-1)^2$ to show it equals $(t^2+1)^2$. Part (ii) requires applying Pythagoras to find the third integer (29), then showing 20, 21, 29 cannot fit the given form by testing whether any integer t produces these values. This is slightly above average difficulty due to the proof element and the need to demonstrate a counterexample, but the algebra is routine and the logic is guided by the question structure. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks | Guidance |
|---|---|---|
| \(a^2 + b^2 = (2t)^2 + (t^2 - 1)^2\) | M1 | substituting for \(a, b\) and \(c\) in terms of \(t\), expanding brackets correctly, www |
| \(= 4t^2 + t^4 - 2t^2 + 1\) | M1 | |
| \(= t^4 + 2t^2 + 1\) | E1 | |
| \(= (t^2 + 1)^2 = c^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(c = \sqrt{(20^2 + 21^2)} = 29\) | B1 | Attempt to find \(t\), Any valid argument or E2 'none of 20, 21, 29 differ by two' |
| For example: \(2t = 20 \Rightarrow t = 10\) | M1 E1 | |
| \(\Rightarrow t^2 - 1 = 99\) which is not consistent with 21 | [6] |
**(i)**
$a^2 + b^2 = (2t)^2 + (t^2 - 1)^2$ | M1 | substituting for $a, b$ and $c$ in terms of $t$, expanding brackets correctly, www
$= 4t^2 + t^4 - 2t^2 + 1$ | M1 |
$= t^4 + 2t^2 + 1$ | E1 |
$= (t^2 + 1)^2 = c^2$ |
**(ii)**
$c = \sqrt{(20^2 + 21^2)} = 29$ | B1 | Attempt to find $t$, Any valid argument or E2 'none of 20, 21, 29 differ by two'
For example: $2t = 20 \Rightarrow t = 10$ | M1 E1 |
$\Rightarrow t^2 - 1 = 99$ which is not consistent with 21 | [6] |5 Positive integers $a , b$ and $c$ are said to form a Pythagorean triple if $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$.\\
(i) Given that $t$ is an integer greater than 1 , show that $2 t , t ^ { 2 } - 1$ and $t ^ { 2 } + 1$ form a Pythagorean triple.\\
(ii) 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 $2 t , t ^ { 2 } - 1$ and $t ^ { 2 } + 1$.
\hfill \mbox{\textit{OCR MEI C3 2006 Q5 [6]}}