| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Proof |
| Type | Identifying errors in proofs |
| Difficulty | Standard +0.3 This is a straightforward proof analysis question requiring students to find a counterexample (e.g., 3+4=9), identify that Charlie failed to consider m-n=1, and solve a simple difference of squares problem. All parts use standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.01c Disproof by counter example1.01d Proof by contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. \(1+3=4\) or \(4+5=9\) or \(9+7=16\) | B1 [1] | AO 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| If \(m-n=1\) (or \(-1\)) then \((m-n)(m+n)\) could be prime | E1 [1] | AO 2.3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow\) Other square number is \((n+1)^2\) | M1 | AO 3.1a |
| \(\Rightarrow 853=(n+1)^2-n^2=2n+1\) | M1 | AO 2.2a |
| \(\Rightarrow n=426\) | A1 | AO 1.1 |
| \(\Rightarrow S=181476\) | A1 [4] | AO 3.2a |
## Question 5:
### Part (i):
e.g. $1+3=4$ or $4+5=9$ or $9+7=16$ | **B1** [1] | AO 1.1 | or $25+11=36$ or any correct example
### Part (ii):
If $m-n=1$ (or $-1$) then $(m-n)(m+n)$ could be prime | **E1** [1] | AO 2.3 | or One of the factors of $p$ could be 1; (or if $m+n=1$)
### Part (iii):
Let $S=n^2$
$\Rightarrow$ Other square number is $(n+1)^2$ | **M1** | AO 3.1a | or Other square number is $(\sqrt{S}+1)^2$
$\Rightarrow 853=(n+1)^2-n^2=2n+1$ | **M1** | AO 2.2a | $\Rightarrow 853=(\sqrt{S}+1)^2-S=2\sqrt{S}+1$
$\Rightarrow n=426$ | **A1** | AO 1.1 | $\Rightarrow \sqrt{S}=426$
$\Rightarrow S=181476$ | **A1** [4] | AO 3.2a | $\Rightarrow S=181476$
Alternative: $m-n=1,\ m+n=853$ M1; $2m=854$ M1; $m=427,\ n=426$ A1; $n^2=181476$ A1
---5 Charlie claims to have proved the following statement.\\
"The sum of a square number and a prime number cannot be a square number."\\
(i) Give an example to show that Charlie's statement is not true.
Charlie's attempt at a proof is below.\\
Assume that the statement is not true.\\
⇒ There exist integers $n$ and $m$ and a prime $p$ such that $n ^ { 2 } + p = m ^ { 2 }$.\\
$\Rightarrow p = m ^ { 2 } - n ^ { 2 }$\\
$\Rightarrow p = ( m - n ) ( m + n )$\\
$\Rightarrow p$ is the product of two integers.\\
$\Rightarrow p$ is not prime, which is a contradiction.\\
⇒ Charlie's statement is true.\\
(ii) Explain the error that Charlie has made.\\
(iii) Given that 853 is a prime number, find the square number $S$ such that $S + 853$ is also a square number.
\hfill \mbox{\textit{OCR H240/02 2018 Q5 [6]}}