CAIE Further Paper 2 2022 November — Question 1 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2022
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeFind series for logarithmic function
DifficultyStandard +0.3 This is a straightforward Maclaurin series question requiring computation of f(0), f'(0), and f''(0) for ln(1+e^x), then substituting into the standard formula. The derivatives involve chain rule and quotient rule but are routine calculations. While it's a Further Maths topic, it's a standard textbook exercise with no conceptual challenges, making it slightly easier than average overall.
Spec4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(y' = \dfrac{e^x}{1+e^x}\)B1 Finds first derivative.
\(y'' = \dfrac{e^x}{\left(1+e^x\right)^2}\)B1 Finds second derivative.
\(y(0) = \ln 2, \quad y'(0) = \frac{1}{2}, \quad y''(0) = \frac{1}{4}\)B1 Evaluates at \(x = 0\).
\(y = y(0) + y'(0)x + \frac{1}{2!}y''(0)x^2 + \ldots\)M1 Allow \(2!\) missing.
\(\ln 2 + \frac{1}{2}x + \frac{1}{8}x^2\)A1 Decimal used for \(\ln 2\) scores A0.
Total: 5 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $y' = \dfrac{e^x}{1+e^x}$ | **B1** | Finds first derivative. |
| $y'' = \dfrac{e^x}{\left(1+e^x\right)^2}$ | **B1** | Finds second derivative. |
| $y(0) = \ln 2, \quad y'(0) = \frac{1}{2}, \quad y''(0) = \frac{1}{4}$ | **B1** | Evaluates at $x = 0$. |
| $y = y(0) + y'(0)x + \frac{1}{2!}y''(0)x^2 + \ldots$ | **M1** | Allow $2!$ missing. |
| $\ln 2 + \frac{1}{2}x + \frac{1}{8}x^2$ | **A1** | Decimal used for $\ln 2$ scores A0. |
| **Total: 5** | | |

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1 Find the Maclaurin's series for $\ln \left( 1 + \mathrm { e } ^ { x } \right)$ up to and including the term in $x ^ { 2 }$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q1 [5]}}
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Q1 5 Q2 7 Q3 6 Q4 12 Q5 10 Q6 11 Q7 10 Q8 14