| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Moderate -0.8 Part (i) requires finding consecutive primes where a²+b² is divisible by 10 (easily found: 3,5 gives 9+25=34, or 5,7 gives 25+49=74). Part (ii) is straightforward exhaustion with only 2 values each for x and y (2,4), requiring simple arithmetic. Both parts are below-average difficulty, testing basic understanding of proof techniques rather than mathematical insight. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. \(7^2 + 11^2 = 170 \Rightarrow 170\) is a multiple of 10, so the statement is untrue * | M1A1* | M1: Attempts to find \(a^2 + b^2\) using two consecutive prime numbers \(a\) and \(b\). Note \(1^2 + 2^2 = ...\) is M0A0. A1*: Correctly evaluates sum of two consecutive primes squared; comments it is a multiple of 10; concludes not true. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Check if multiplied inequality by 4 to give \(4 < 4x^2 - xy < 60\). Table of values: \(x=2,y=2\): \(x^2-\frac{xy}{4}=3\); \(x=2,y=4\): \(x^2-\frac{xy}{4}=2\); \(x=4,y=2\): \(x^2-\frac{xy}{4}=14\); \(x=4,y=4\): \(x^2-\frac{xy}{4}=12\) | M1A1 | M1: Attempts at least two valid combinations of \(x\) and \(y\), evaluates \(x^2 - \frac{xy}{4}\). A1: All four valid combinations correctly evaluated. |
| Concludes \(1 < x^2 - \frac{xy}{4} < 15\) for all \(x\) and \(y\) positive even integers less than 6 * | A1* | Minimal conclusion that \(1 < x^2 - \frac{xy}{4} < 15\) (or condone \(4 < 4x^2 - xy < 60\)) for all valid \(x\), \(y\). Accept e.g. "hence statement is true", "proven", "QED". |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3\cos\theta(\tan\theta\sin\theta + 3) = 11 - 5\cos\theta\) leading to \(3\sin^2\theta + 9\cos\theta = 11 - 5\cos\theta\) | M1 | Substitutes or uses \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) to achieve equation in sine and cosine only. May be in another variable. |
| \(3(1 - \cos^2\theta) + 9\cos\theta = 11 - 5\cos\theta\) | dM1 | Attempts to use \(\pm\sin^2\theta \pm \cos^2\theta = \pm 1\) to achieve equation in cosine only. Dependent on previous M1. |
| \(3\cos^2\theta - 14\cos\theta + 8 = 0\) * | A1* | Achieves given answer with no errors seen including brackets. Withhold for poor notation such as \(3\cos\theta^2\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cos 2x = \frac{2}{3}\) | B1 | Ignore any reference to 4. |
| \(x = \frac{\cos^{-1}\left(\frac{2}{3}\right)}{2} = ...\) | M1 | Solves using correct order of operations to find at least one value of \(x\). |
| \(x = \text{awrt } 24.1, \text{ awrt } 155.9, \text{ awrt } 204.1, \text{ awrt } 335.9\) | A1A1 | A1: Two of awrt 24, awrt 156, awrt 204, awrt 336. A1: All four values awrt 24.1, 155.9, 204.1, 335.9 and no others in range. (In radians: awrt 0.42, 2.7, 3.6, 5.9) |
# Question 8(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. $7^2 + 11^2 = 170 \Rightarrow 170$ is a multiple of 10, so the statement is untrue * | M1A1* | M1: Attempts to find $a^2 + b^2$ using two consecutive prime numbers $a$ and $b$. Note $1^2 + 2^2 = ...$ is M0A0. A1*: Correctly evaluates sum of two consecutive primes squared; comments it is a multiple of 10; concludes not true. |
# Question 8(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Check if multiplied inequality by 4 to give $4 < 4x^2 - xy < 60$. Table of values: $x=2,y=2$: $x^2-\frac{xy}{4}=3$; $x=2,y=4$: $x^2-\frac{xy}{4}=2$; $x=4,y=2$: $x^2-\frac{xy}{4}=14$; $x=4,y=4$: $x^2-\frac{xy}{4}=12$ | M1A1 | M1: Attempts at least two valid combinations of $x$ and $y$, evaluates $x^2 - \frac{xy}{4}$. A1: All four valid combinations correctly evaluated. |
| Concludes $1 < x^2 - \frac{xy}{4} < 15$ for all $x$ and $y$ positive even integers less than 6 * | A1* | Minimal conclusion that $1 < x^2 - \frac{xy}{4} < 15$ (or condone $4 < 4x^2 - xy < 60$) for all valid $x$, $y$. Accept e.g. "hence statement is true", "proven", "QED". |
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# Question 9a:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\cos\theta(\tan\theta\sin\theta + 3) = 11 - 5\cos\theta$ leading to $3\sin^2\theta + 9\cos\theta = 11 - 5\cos\theta$ | M1 | Substitutes or uses $\tan\theta = \frac{\sin\theta}{\cos\theta}$ to achieve equation in sine and cosine only. May be in another variable. |
| $3(1 - \cos^2\theta) + 9\cos\theta = 11 - 5\cos\theta$ | dM1 | Attempts to use $\pm\sin^2\theta \pm \cos^2\theta = \pm 1$ to achieve equation in cosine only. Dependent on previous M1. |
| $3\cos^2\theta - 14\cos\theta + 8 = 0$ * | A1* | Achieves given answer with no errors seen including brackets. Withhold for poor notation such as $3\cos\theta^2$. |
# Question 9b:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos 2x = \frac{2}{3}$ | B1 | Ignore any reference to 4. |
| $x = \frac{\cos^{-1}\left(\frac{2}{3}\right)}{2} = ...$ | M1 | Solves using correct order of operations to find at least one value of $x$. |
| $x = \text{awrt } 24.1, \text{ awrt } 155.9, \text{ awrt } 204.1, \text{ awrt } 335.9$ | A1A1 | A1: Two of awrt 24, awrt 156, awrt 204, awrt 336. A1: All four values awrt 24.1, 155.9, 204.1, 335.9 and no others in range. (In radians: awrt 0.42, 2.7, 3.6, 5.9) |\begin{enumerate}
\item (i) A student writes the following statement:\\
"When $a$ and $b$ are consecutive prime numbers, $a ^ { 2 } + b ^ { 2 }$ is never a multiple of 10 "\\
Prove by counter example that this statement is not true.\\
(ii) Given that $x$ and $y$ are even integers greater than 0 and less than 6 , prove by exhaustion, that
\end{enumerate}
$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$
\hfill \mbox{\textit{Edexcel P2 2023 Q8 [5]}}