| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Disprove statement by counterexample |
| Difficulty | Easy -1.2 This question tests basic mathematical reasoning rather than advanced techniques. Part (a) requires a simple counterexample (e.g., a=1, b=-2), part (b) tests understanding of the sine function's periodicity at GCSE/early A-level, and part (c) is a straightforward algebraic proof with consecutive terms. All parts are accessible with minimal steps and no sophisticated problem-solving. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01b Logical connectives: congruence, if-then, if and only if1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. \(1 > -2\), but \(1^2 < (-2)^2\) as \(1 < 4\) | B1 | Any correct counterexample, and contradiction identified. Initial inequality soi and then contradiction e.g. \(-3 > -4\) but \(9 < 16\) (or \(9 \not> 16\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. \(\sin 150° = 0.5\) as well | B1 | Any correct statement. Identifies that \(\sin x = 0.5\) could give values of \(x\) other than \(30°\). Either specific example or general statement e.g. 'many to one' function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sin x° = 0.5 \Leftarrow x° = 30°\) | B1 | Any correct relationship. If attempting to write general solution then must be fully correct e.g. \(x = 30° + 360n°\), \(x = 150° + 360n°\). Condone \(\leftarrow\) instead of \(\Leftarrow\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((4n) + (4n+4) + (4n+8) + (4n+12)\), where \(n\) is an integer | B1\* | Four consecutive multiples of 4 written correctly in terms of \(n\), or any other variable. Allow BOD if \(n\) not explicitly stated to be an integer. Sufficient to just list the 4 terms rather than as a sum. Not necessarily starting on \(4n\). Could also define \(k\) as a multiple of 4 and then have \(k, k+4\) etc |
| \(= 16n + 24 = 8(2n+3)\) | M1 dep\* | Correctly sum terms, and correctly take out common factor of 8 |
| \(2n+3\) is an integer, so \(8(2n+3)\) is a multiple of 8 | A1 | Conclude appropriately. Allow BOD if \(2n+3\) not explicitly stated to be an integer. If using \(k\)... expect \(8(0.5k+3)\) then justify \(0.5k\) as an integer, or \(4(k+6)\) then justify \(k+6\) is a multiple of 2 |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $1 > -2$, but $1^2 < (-2)^2$ as $1 < 4$ | **B1** | Any correct counterexample, and contradiction identified. Initial inequality soi and then contradiction e.g. $-3 > -4$ but $9 < 16$ (or $9 \not> 16$) |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $\sin 150° = 0.5$ as well | **B1** | Any correct statement. Identifies that $\sin x = 0.5$ could give values of $x$ other than $30°$. Either specific example or general statement e.g. 'many to one' function |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sin x° = 0.5 \Leftarrow x° = 30°$ | **B1** | Any correct relationship. If attempting to write general solution then must be fully correct e.g. $x = 30° + 360n°$, $x = 150° + 360n°$. Condone $\leftarrow$ instead of $\Leftarrow$ |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(4n) + (4n+4) + (4n+8) + (4n+12)$, where $n$ is an integer | **B1\*** | Four consecutive multiples of 4 written correctly in terms of $n$, or any other variable. Allow BOD if $n$ not explicitly stated to be an integer. Sufficient to just list the 4 terms rather than as a sum. Not necessarily starting on $4n$. Could also define $k$ as a multiple of 4 and then have $k, k+4$ etc |
| $= 16n + 24 = 8(2n+3)$ | **M1 dep\*** | Correctly sum terms, and correctly take out common factor of 8 |
| $2n+3$ is an integer, so $8(2n+3)$ is a multiple of 8 | **A1** | Conclude appropriately. Allow BOD if $2n+3$ not explicitly stated to be an integer. If using $k$... expect $8(0.5k+3)$ then justify $0.5k$ as an integer, or $4(k+6)$ then justify $k+6$ is a multiple of 2 |
---2
\begin{enumerate}[label=(\alph*)]
\item Given that $a$ and $b$ are real numbers, find a counterexample to disprove the statement that, if $a > b$, then $a ^ { 2 } > b ^ { 2 }$.
\item A student writes the statement that $\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }$.
\begin{enumerate}[label=(\roman*)]
\item Explain why this statement is incorrect.
\item Write a corrected version of this statement.
\end{enumerate}\item Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2022 Q2 [6]}}