CAIE Further Paper 2 2020 November — Question 1 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2020
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeMaclaurin series of shifted function
DifficultyStandard +0.8 This requires applying the Maclaurin series formula to a shifted trigonometric function, involving multiple derivatives of tan(x + π/4) evaluated at x=0. Students must use the chain rule repeatedly and handle the algebra of tan derivatives, which is more demanding than standard Maclaurin series of simple functions but follows a systematic procedure.
Spec4.08a Maclaurin series: find series for function
1 Find the Maclaurin's series for \(\tan \left( x + \frac { 1 } { 4 } \pi \right)\) up to and including the term in \(x ^ { 2 }\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(f'(x) = \sec^2\left(x + \frac{1}{4}\pi\right)\)B1 Finds first derivative
\(f''(x) = 2\sec^2\left(x + \frac{1}{4}\pi\right)\tan\left(x + \frac{1}{4}\pi\right)\)B1 Finds second derivative
\(f(0) = 1, \quad f'(0) = 2, \quad f''(0) = 4\)M1 Evaluates derivatives at zero
\(\tan\left(x + \frac{1}{4}\pi\right) = 1 + 2x + 2x^2\)M1 A1
Total: 5 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = \sec^2\left(x + \frac{1}{4}\pi\right)$ | B1 | Finds first derivative |
| $f''(x) = 2\sec^2\left(x + \frac{1}{4}\pi\right)\tan\left(x + \frac{1}{4}\pi\right)$ | B1 | Finds second derivative |
| $f(0) = 1, \quad f'(0) = 2, \quad f''(0) = 4$ | M1 | Evaluates derivatives at zero |
| $\tan\left(x + \frac{1}{4}\pi\right) = 1 + 2x + 2x^2$ | M1 A1 | |
| **Total: 5** | | |
1 Find the Maclaurin's series for $\tan \left( x + \frac { 1 } { 4 } \pi \right)$ up to and including the term in $x ^ { 2 }$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q1 [5]}}
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Q1 5 Q2 6 Q3 4 Q4 8 Q5 8 Q6 11 Q7 7 Q8 10 Q9 16