Edexcel P2 2022 January — Question 4 5 marks

Exam BoardEdexcel
ModuleP2 (Pure Mathematics 2)
Year2022
SessionJanuary
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeSolve log equation reducing to quadratic
DifficultyStandard +0.3 This is a straightforward application of logarithm laws requiring students to use the power rule (2log₃(1-x) = log₃(1-x)²), convert log form to exponential form, and solve a quadratic equation. While it involves multiple steps and a non-standard base, the techniques are routine for P2 level with no novel problem-solving required, making it slightly easier than average.
Spec1.06f Laws of logarithms: addition, subtraction, power rules
4. Using the laws of logarithms, solve $$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$
Question 4:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(2\log_3(1-x) = \log_3(1-x)^2\) or \(3 = \log_3 3^3\)B1 Correct power law used or implied
\(\log_3(32-12x) - \log_3(1-x)^2 = \log_3\frac{32-12x}{(1-x)^2}\)M1 Combines 2 log terms correctly
\(\frac{32-12x}{(1-x)^2} = 27\)A1 Obtains this equation in any form
\(27x^2 - 42x - 5 = 0 \Rightarrow x = \ldots\)M1 Solves 3TQ
\(x = -\frac{1}{9}\)A1 This value only; \(\frac{5}{3}\) must clearly be discarded if seen
Total: 5 marks
## Question 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\log_3(1-x) = \log_3(1-x)^2$ or $3 = \log_3 3^3$ | B1 | Correct power law used or implied |
| $\log_3(32-12x) - \log_3(1-x)^2 = \log_3\frac{32-12x}{(1-x)^2}$ | M1 | Combines 2 log terms correctly |
| $\frac{32-12x}{(1-x)^2} = 27$ | A1 | Obtains this equation in any form |
| $27x^2 - 42x - 5 = 0 \Rightarrow x = \ldots$ | M1 | Solves 3TQ |
| $x = -\frac{1}{9}$ | A1 | This value only; $\frac{5}{3}$ must clearly be discarded if seen |

**Total: 5 marks**

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4. Using the laws of logarithms, solve

$$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$

\hfill \mbox{\textit{Edexcel P2 2022 Q4 [5]}}
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Q1 7 Q2 8 Q3 7 Q4 5 Q5 8 Q6 8 Q7 8 Q8 9 Q9 10 Q10 5