CAIE P3 2022 November — Question 1 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2022
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeSolve ln equation using power law
DifficultyStandard +0.3 This question requires applying logarithm laws (power and subtraction rules) to simplify the equation, then solving a resulting quadratic. It's slightly above average difficulty due to the algebraic manipulation needed after applying log laws, but remains a standard textbook exercise with clear steps and no novel insight required.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b

1 Solve the equation \(\ln ( 2 x - 1 ) = 2 \ln ( x + 1 ) - \ln x\). Give your answer correct to 3 decimal places.

Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
Use law for the logarithm of a product, quotient or powerM1
Remove logarithms and state a correct equation, e.g. \(x(2x-1) = (x+1)^2\)A1
Solve a 3-term quadratic obtaining at least one rootM1
Obtain answer \(3.303\) onlyA1
Total4
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use law for the logarithm of a product, quotient or power | M1 | |
| Remove logarithms and state a correct equation, e.g. $x(2x-1) = (x+1)^2$ | A1 | |
| Solve a 3-term quadratic obtaining at least one root | M1 | |
| Obtain answer $3.303$ only | A1 | |
| **Total** | **4** | |
1 Solve the equation $\ln ( 2 x - 1 ) = 2 \ln ( x + 1 ) - \ln x$. Give your answer correct to 3 decimal places.\\

\hfill \mbox{\textit{CAIE P3 2022 Q1 [4]}}