| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - simplify/express then solve |
| Difficulty | Moderate -0.3 Part (i) requires systematic application of log laws (power rule, combining logs, converting to exponential form) to solve an equation—standard P2 material with multiple steps but no novel insight. Part (ii) is a straightforward change of base/power manipulation. Both parts are routine exercises testing log law fluency, slightly easier than average due to being mechanical rather than requiring problem-solving creativity. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\log_2(2-x)\rightarrow\log_2(2-x)^2\) or \(4\rightarrow\log_2 16\) | B1 | One correct log law seen/used in relation to equation |
| Correct attempt to combine two original terms | M1 | e.g. \(4+\log_2(x+10)=\log_2 16+\log_2(x+10)=\log_2 16(x+10)\). Do not allow if they incorrectly divide by 2 first |
| \((2-x)^2=16(x+10)\) | A1 | Correct equation not involving logs from fully correct log work |
| \(x^2-20x-156=0 \Rightarrow (x+6)(x-26)=0\) | M1 | Correct attempt to solve 3TQ in \(x\) achieving a real root |
| \(x=-6\) only | A1 | \(x=-6\) only; \(x=26\) must be rejected if seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(12\) | B1 | 12 clearly identified as answer; condone \(a=12\) as long as 12 is clearly the answer |
# Question 3:
## Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\log_2(2-x)\rightarrow\log_2(2-x)^2$ or $4\rightarrow\log_2 16$ | B1 | One correct log law seen/used in relation to equation |
| Correct attempt to combine two original terms | M1 | e.g. $4+\log_2(x+10)=\log_2 16+\log_2(x+10)=\log_2 16(x+10)$. Do not allow if they incorrectly divide by 2 first |
| $(2-x)^2=16(x+10)$ | A1 | Correct equation not involving logs from fully correct log work |
| $x^2-20x-156=0 \Rightarrow (x+6)(x-26)=0$ | M1 | Correct attempt to solve 3TQ in $x$ achieving a real root |
| $x=-6$ only | A1 | $x=-6$ only; $x=26$ must be rejected if seen |
## Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $12$ | B1 | 12 clearly identified as answer; condone $a=12$ as long as 12 is clearly the answer |
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.\\
(i) Using the laws of logarithms, solve
\end{enumerate}
$$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
(ii) Find the value of
$$\log _ { \sqrt { a } } a ^ { 6 }$$
where $a$ is a positive constant greater than 1
\hfill \mbox{\textit{Edexcel P2 2024 Q3 [6]}}