OCR Further Pure Core 2 2020 November — Question 10 10 marks

Exam BoardOCR
ModuleFurther Pure Core 2 (Further Pure Core 2)
Year2020
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeMaclaurin series for inverse trigonometric functions
DifficultyStandard +0.8 This is a multi-part Further Maths question requiring differentiation of inverse trig functions, Maclaurin series construction, integration using series approximation, and integration by parts. While each technique is standard for FM students, the combination across multiple parts and the need to handle the second derivative of arcsin (requiring chain/quotient rule) elevates it above average FM difficulty but doesn't require exceptional insight.
Spec4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
10 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
    1. Determine \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
    3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
  2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.
Question 10:
AnswerMarks Guidance
10(a) (i)
f′(x)= from the formula book
( )1
1−x2 2
1
so f′′(x)=−1. . (−2x )
2 ( )3
1−x2 2
x
=
( )3
AnswerMarks
1−x2 2M1
A1
AnswerMarks Guidance
[2]1.1
1.1Formula from the Formula Booklet
and attempt differentiationTo within sign error
(a)(ii) f(0)=0, f′(0)=1and f′′(0)=0
3 1
2 2 3� 2 �2
′′′ (1−𝑥𝑥 ) −𝑥𝑥. 1−𝑥𝑥 .(−2𝑥𝑥)
f (𝑥𝑥)= 2 3
2
so f′′′(0)=1 and f(x(1)=−x𝑥𝑥+)1 x3+...
AnswerMarks
6B1
M1
A1
AnswerMarks
[3]1.1
3.1a
AnswerMarks
2.1or a = 0, a = 1 and a = 0
0 1 2
Differentiate and simplify far
enough to be able to justify value 1
AnswerMarks
Condone 3! In place of 6Ignore sign error in f′′(x)
Either full derivative or “zero
term” denoted as such
Not BC. If M0 then SC1 for
correct expansion
AnswerMarks Guidance
(a)(iii) 1 1
∫2f(x)dx≈∫2x+ 1 x3dx
0 0 6
=0.127604167...
AnswerMarks
=0.127604 to 6 dpM1
A1
AnswerMarks
[2]1.1
1.1Integral of their 2 term cubic with
limits
Could be BC
AnswerMarks
(b)∫1×sin−1xdx= xsin−1x−∫ x dx
1−x2
= xsin−1x+ ( 1−x2 )1 2 (+c)
1
2
𝜋𝜋 √3
AnswerMarks
� f(𝑥𝑥) = + − 1M1
A1
A1
AnswerMarks
[3]3.1a
1.1
AnswerMarks Guidance
1.1Attempt integration by parts ignore limits. Formula for parts
must be correct
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
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Question 10:
10 | (a) | (i) | 1
f′(x)= from the formula book
( )1
1−x2 2
1
so f′′(x)=−1. . (−2x )
2 ( )3
1−x2 2
x
=
( )3
1−x2 2 | M1
A1
[2] | 1.1
1.1 | Formula from the Formula Booklet
and attempt differentiation | To within sign error
(a) | (ii) | f(0)=0, f′(0)=1and f′′(0)=0
3 1
2 2 3� 2 �2
′′′ (1−𝑥𝑥 ) −𝑥𝑥. 1−𝑥𝑥 .(−2𝑥𝑥)
f (𝑥𝑥)= 2 3
2
so f′′′(0)=1 and f(x(1)=−x𝑥𝑥+)1 x3+...
6 | B1
M1
A1
[3] | 1.1
3.1a
2.1 | or a = 0, a = 1 and a = 0
0 1 2
Differentiate and simplify far
enough to be able to justify value 1
Condone 3! In place of 6 | Ignore sign error in f′′(x)
Either full derivative or “zero
term” denoted as such
Not BC. If M0 then SC1 for
correct expansion
(a) | (iii) | 1 1
∫2f(x)dx≈∫2x+ 1 x3dx
0 0 6
=0.127604167...
=0.127604 to 6 dp | M1
A1
[2] | 1.1
1.1 | Integral of their 2 term cubic with
limits
Could be BC
(b) | ∫1×sin−1xdx= xsin−1x−∫ x dx
1−x2
= xsin−1x+ ( 1−x2 )1 2 (+c)
1
2
𝜋𝜋 √3
� f(𝑥𝑥) = + − 1 | M1
A1
A1
[3] | 3.1a
1.1
1.1 | Attempt integration by parts | ignore limits. Formula for parts
must be correct
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored
10 Let $\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Determine $\mathrm { f } ^ { \prime \prime } ( x )$.
\item Determine the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$.
\item By considering the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$, find an approximation to $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer correct to 6 decimal places.
\end{enumerate}\item By writing $\mathrm { f } ( x )$ as $\sin ^ { - 1 } ( x ) \times 1$, determine the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in exact form.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 2 2020 Q10 [10]}}
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