CAIE Further Paper 2 2023 November — Question 3 6 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeMaclaurin series of shifted function
DifficultyChallenging +1.2 This is a straightforward Maclaurin series question requiring systematic application of the chain rule and standard derivatives. While it involves inverse hyperbolic functions and requires careful algebraic manipulation through multiple derivatives, it follows a standard procedure without requiring novel insight. The 6-mark allocation and three-term requirement make it moderately substantial but still routine for Further Maths students who have practiced Taylor series expansions.
Spec4.07f Inverse hyperbolic: logarithmic forms4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
Find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\frac{1}{2}e^t\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [6]
Question 3:
AnswerMarks
31
ex
dy
2
=
dx 1
1− e2x
AnswerMarks
4B1
 1 1  1  1 
1− e2x  ex − ex − e2x 
d2y  4 2  2  2 
=
dx2 2
 1 
1− e2x 
AnswerMarks
 4 B1
2 10
f '(0)= f ''(0)=
AnswerMarks Guidance
3 9M1 Evaluates derivatives at x=0.
1 1 3 
f(0)=tanh−1  = ln 2
AnswerMarks Guidance
2 2 2 M1 Uses logarithmic form of tanh−1.
1 2 5
ln3+ x+ x2
AnswerMarks Guidance
2 3 9M1 A1 1
Applies f(x)= f(0)+ f '(0)x+ f ''(0)x2
2!
6
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
3 | 1
ex
dy
2
=
dx 1
1− e2x
4 | B1
 1 1  1  1 
1− e2x  ex − ex − e2x 
d2y  4 2  2  2 
=
dx2 2
 1 
1− e2x 
 4  | B1
2 10
f '(0)= f ''(0)=
3 9 | M1 | Evaluates derivatives at x=0.
1 1 3 
f(0)=tanh−1  = ln 2
2 2 2  | M1 | Uses logarithmic form of tanh−1.
1 2 5
ln3+ x+ x2
2 3 9 | M1 A1 | 1
Applies f(x)= f(0)+ f '(0)x+ f ''(0)x2
2!
6
Question | Answer | Marks | Guidance
Find the first three terms in the Maclaurin's series for $\tanh^{-1}\left(\frac{1}{2}e^t\right)$ in the form $\frac{1}{2}\ln a + bx + cx^2$, giving the exact values of the constants $a$, $b$ and $c$. [6]

\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q3 [6]}}
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Q1 4 Q2 5 Q3 6 Q4 10 Q5 10 Q6 14 Q7 11 Q8 15