| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Trigonometric method of differences |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on Maclaurin series requiring standard differentiation of arctan (using chain rule), evaluation at x=0, and formula application. While it involves Further Pure content, the execution is mechanical with no problem-solving insight needed—just careful application of known techniques across 8 marks. |
| Spec | 1.07l Derivative of ln(x): and related functions4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | (i) |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | B1 | 1.1 |
| Answer | Marks |
|---|---|
| (ii) | 1 1 |
| Answer | Marks |
|---|---|
| 1+( 1+x )2 2 | M1 |
| A1 | 2.1 |
| 1.1 | Diffn – Must be seen |
| Answer | Marks |
|---|---|
| (iii) | 1 1 |
| Answer | Marks |
|---|---|
| 4 2 | M1 |
| Answer | Marks |
|---|---|
| A1 | 2.1 |
| Answer | Marks |
|---|---|
| 2.1 | Diffn their f’(x) |
| Answer | Marks |
|---|---|
| (b) | x2 |
| Answer | Marks |
|---|---|
| 4 2 4 | M1 |
| A1 | 1.1 |
| 2.2a | Using the formula and substituting their value for f;(0) |
Question 2:
2 | (a) | (i) | π
f(0)=
4 | B1 | 1.1 | Not for 450
[1]
(ii) | 1 1
f '(x)= ⇒f '(0)=
1+( 1+x )2 2 | M1
A1 | 2.1
1.1 | Diffn – Must be seen
1
f '(x)= is M0
1+x2
[2]
(iii) | 1 1
f '(x)= =
1+( 1+x )2 2+2x+x2
1
⇒f ''(x)= ×(−1 )×( 2+2x )
( )2
2+2x+x2
−( 2+2x )
=
( )2
2+2x+x2
−2 1
⇒f ''(0)= =−
4 2 | M1
A1
A1 | 2.1
2.1
2.1 | Diffn their f’(x)
2 ( 1+x )
oe, e.g. f′′( x )=−
( )2
1+( 1+x )2
f’’(0) must be seen. The substitution must be seen
2
(implied by − )
4
AG
[3]
(b) | x2
f(x)=f(0)+f '(0)x+f ''(0)
2
π 1 1 x2
= + x− ×
4 2 2 2
π x x2
= + −
4 2 4 | M1
A1 | 1.1
2.2a | Using the formula and substituting their value for f;(0)
ft their values from (a)
[2]You are given that $\mathrm{f}(x) = \tan^{-1}(1 + x)$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $\mathrm{f}(0)$. [1]
\item Determine the value of $\mathrm{f}'(0)$. [2]
\item Show that $\mathrm{f}''(0) = -\frac{1}{2}$. [3]
\end{enumerate}
\item Hence find the Maclaurin series for $\mathrm{f}(x)$ up to and including the term in $x^2$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q2 [8]}}