CAIE Further Paper 2 2021 November — Question 1 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2021
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeDirect multiplication of series
DifficultyChallenging +1.2 This requires computing derivatives of a product from first principles (product rule, chain rule) and evaluating at x=0, which is more involved than standard Maclaurin series recall. However, it only asks for terms up to x², limiting the algebraic complexity, and the technique is straightforward application of the definition rather than requiring novel insight.
Spec4.08a Maclaurin series: find series for function
1 Find the Maclaurin's series for \(e ^ { x } \tan x\) from first principles up to and including the term in \(x ^ { 2 }\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(e^x(\tan x + \sec^2 x)\)B1 Finds first derivative.
\(e^x(\tan x + 2\sec^2 x + 2\sec^2 x \tan x)\)M1 A1 Finds second derivative.
\(y(0)=0,\quad y'(0)=1,\quad y''(0)=2\)M1 Evaluates derivatives at \(x=0\).
\(y = x + x^2\)A1
Total5 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^x(\tan x + \sec^2 x)$ | B1 | Finds first derivative. |
| $e^x(\tan x + 2\sec^2 x + 2\sec^2 x \tan x)$ | M1 A1 | Finds second derivative. |
| $y(0)=0,\quad y'(0)=1,\quad y''(0)=2$ | M1 | Evaluates derivatives at $x=0$. |
| $y = x + x^2$ | A1 | |
| **Total** | **5** | |

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

\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q1 [5]}}
This paper (8 questions)
View full paper
Q1 5 Q2 6 Q3 8 Q4 10 Q5 11 Q6 10 Q7 11 Q8 14