CAIE P3 2013 November — Question 1 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2013
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeExpress y in terms of x (ln/log equations)
DifficultyStandard +0.3 This is a straightforward application of logarithm laws (combining logs using subtraction and power rules) followed by solving a quadratic equation. It requires standard algebraic manipulation with no novel insight, making it slightly easier than average for A-level.
Spec1.06f Laws of logarithms: addition, subtraction, power rules

1 Given that \(2 \ln ( x + 4 ) - \ln x = \ln ( x + a )\), express \(x\) in terms of \(a\).

AnswerMarks Guidance
Apply at least one logarithm property correctly*M1
Obtain \(\frac{(x+4)^2}{x} = x + a\) or equivalent without logarithm involvedA1
Rearrange to express \(x\) in terms of \(a\)M1 d*M
Obtain \(\frac{16}{a-8}\) or equivalentA1 [4]
Apply at least one logarithm property correctly | *M1 |
Obtain $\frac{(x+4)^2}{x} = x + a$ or equivalent without logarithm involved | A1 |
Rearrange to express $x$ in terms of $a$ | M1 d*M |
Obtain $\frac{16}{a-8}$ or equivalent | A1 | [4]
1 Given that $2 \ln ( x + 4 ) - \ln x = \ln ( x + a )$, express $x$ in terms of $a$.

\hfill \mbox{\textit{CAIE P3 2013 Q1 [4]}}