CAIE P3 2009 November — Question 1 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2009
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeSolve ln equation using subtraction law
DifficultyModerate -0.5 This question requires applying the subtraction law of logarithms (ln a - ln b = ln(a/b)) to form a quadratic equation, then solving it. While it involves multiple steps (simplifying logs, cross-multiplying, solving quadratic, checking domain), each step is routine and the approach is standard for this topic type. Slightly easier than average due to straightforward algebraic manipulation.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b

1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.

AnswerMarks Guidance
Use law of the logarithm of a product or quotient and remove logarithmsM1
Obtain quadratic equation \(x^2 - 5x + 5 = 0\), or equivalentA1
Solve 3-term quadratic obtaining 1 or 2 rootsA1
Obtain answers 1.38 and 3.62A1 [4]
Use law of the logarithm of a product or quotient and remove logarithms | M1 |
Obtain quadratic equation $x^2 - 5x + 5 = 0$, or equivalent | A1 |
Solve 3-term quadratic obtaining 1 or 2 roots | A1 |
Obtain answers 1.38 and 3.62 | A1 | [4]
1 Solve the equation

$$\ln ( 5 - x ) = \ln 5 - \ln x$$

giving your answers correct to 3 significant figures.

\hfill \mbox{\textit{CAIE P3 2009 Q1 [4]}}