CAIE P3 2019 November — Question 1 3 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2019
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeExpress y in terms of x (requires exponentiating both sides)
DifficultyStandard +0.3 This requires applying exponential function to both sides, rearranging to isolate e^(2y), then taking natural log and dividing by 2. It's a straightforward algebraic manipulation of logarithms and exponentials with clear steps, slightly above average due to the nested exponential but still routine for P3 level.
Spec1.06g Equations with exponentials: solve a^x = b
1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).
Question 1:
AnswerMarks Guidance
AnswerMark Guidance
State \(1 + e^{2y} = e^x\)B1
Make \(y\) the subjectM1 Rearrange to \(e^{2y} = \ldots\) and use logs
Obtain answer \(y = \frac{1}{2}\ln(e^x - 1)\)A1 OE
Total: 3 
**Question 1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| State $1 + e^{2y} = e^x$ | B1 | |
| Make $y$ the subject | M1 | Rearrange to $e^{2y} = \ldots$ and use logs |
| Obtain answer $y = \frac{1}{2}\ln(e^x - 1)$ | A1 | OE |
| **Total: 3** | | |

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1 Given that $\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x$, express $y$ in terms of $x$.\\

\hfill \mbox{\textit{CAIE P3 2019 Q1 [3]}}
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