CAIE P3 2016 November — Question 1 3 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2016
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeExpress y in terms of x (requires exponentiating both sides)
DifficultyModerate -0.5 This is a straightforward application of logarithm laws (difference of logs becomes log of quotient) followed by basic exponential manipulation to make y the subject. It requires only standard algebraic rearrangement with no problem-solving insight, making it slightly easier than average.
Spec1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules
1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Use law of logarithm of a quotientM1
Remove logarithms and obtain correct equation, e.g. \(e^z = \frac{y+2}{y+1}\)A1
Obtain answer \(y = \frac{2-e^z}{e^z-1}\), or equivalentA1 [3] 
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use law of logarithm of a quotient | M1 | |
| Remove logarithms and obtain correct equation, e.g. $e^z = \frac{y+2}{y+1}$ | A1 | |
| Obtain answer $y = \frac{2-e^z}{e^z-1}$, or equivalent | A1 | [3] |

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1 It is given that $z = \ln ( y + 2 ) - \ln ( y + 1 )$. Express $y$ in terms of $z$.

\hfill \mbox{\textit{CAIE P3 2016 Q1 [3]}}
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