CAIE Further Paper 2 2024 June — Question 2 4 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2024
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeCombining or manipulating standard series
DifficultyModerate -0.3 This is a straightforward application of standard Maclaurin series with simple substitutions. Students need to recall e^x series, substitute (1+x²) and (1-x), then collect terms up to x². The algebra is routine and requires no problem-solving insight, making it slightly easier than average despite being Further Maths content.
Spec4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
2 Find the Maclaurin's series for \(\mathrm { e } ^ { 1 + x ^ { 2 } } + \mathrm { e } ^ { 1 - x }\) up to and including the term in \(x ^ { 2 }\).
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(e^{1+x^2} = e(e^{x^2}) = e(1+x^2+\ldots)\)B1 Changes \(e^{1+x^2}\) or \(e^{1-x}\) so that the formula given in the list of formulae (MF19) can be applied
\(e^{1-x} = e(1-x+\frac{1}{2}x^2+\ldots)\)B1
\(e(1+x^2+\ldots)+e(1-x+\frac{1}{2}x^2+\ldots)\)M1 Sums power series
\(e^{1+x^2}+e^{1-x} = 2e-ex+\frac{3}{2}ex^2+\ldots\)A1
Alternative method:
AnswerMarks Guidance
AnswerMarks Guidance
\(f'(x) = 2xe^{1+x^2}-e^{1-x}\)B1 Finds first derivative
\(f''(x) = 4x^2e^{1+x^2}+2e^{1+x^2}+e^{1-x}\)B1 Finds second derivative
\(f(0)=2e \quad f'(0)=-e \quad f''(0)=3e\)M1 Evaluates their derivatives at zero
\(e^{1+x^2}+e^{1-x} = 2e-ex+\frac{3}{2}ex^2+\ldots\)A1 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{1+x^2} = e(e^{x^2}) = e(1+x^2+\ldots)$ | B1 | Changes $e^{1+x^2}$ or $e^{1-x}$ so that the formula given in the list of formulae (MF19) can be applied |
| $e^{1-x} = e(1-x+\frac{1}{2}x^2+\ldots)$ | B1 | |
| $e(1+x^2+\ldots)+e(1-x+\frac{1}{2}x^2+\ldots)$ | M1 | Sums power series |
| $e^{1+x^2}+e^{1-x} = 2e-ex+\frac{3}{2}ex^2+\ldots$ | A1 | |

**Alternative method:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = 2xe^{1+x^2}-e^{1-x}$ | B1 | Finds first derivative |
| $f''(x) = 4x^2e^{1+x^2}+2e^{1+x^2}+e^{1-x}$ | B1 | Finds second derivative |
| $f(0)=2e \quad f'(0)=-e \quad f''(0)=3e$ | M1 | Evaluates their derivatives at zero |
| $e^{1+x^2}+e^{1-x} = 2e-ex+\frac{3}{2}ex^2+\ldots$ | A1 | |

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

\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q2 [4]}}
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