| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 2 |
| Topic | Taylor series |
| Type | Taylor series about x=1: differential equation with given conditions at x=1 |
| Difficulty | Standard +0.8 This is a multi-part Taylor series question requiring systematic differentiation of a differential equation to find higher derivatives at x=1. While the method is standard for Further Maths students (substitute given values, differentiate the DE, repeat), it requires careful algebraic manipulation and multiple steps. The conceptual demand is moderate but the execution requires precision across four parts, making it somewhat above average difficulty for A-level Further Maths. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08a Maclaurin series: find series for function |
**(a)** Substituting $x = 1$, $f(1) = 2$ and $f'(1) = 3$ into $(*)$ $\Rightarrow f''(1) = 5$ **M1A1**
**Total: 2**
**(b)** Product Rule used twice; at least one bracket correct **M1**
$\{x^2f'''(x) + 2xf''(x)\} + \{(2x-1)f''(x) + 2f'(x)\} - 2f'(x) = 3e^{x-1}$ **A1**
Substituting $x = 1$, $f'(1) = 3$ and $f''(1) = 5$ into this $\Rightarrow f'''(1) = -12$
**ft** their $f''(1)$ **M1A1** (M1A1ft)
**Total: 4**
**(c)** $f(x) = f(1) + f'(1)(x-1) + \frac{1}{2}f''(1)(x-1)^2 + \frac{1}{6}f'''(1)(x-1)^3 + \ldots$
Use of the Taylor series **M1**
$= 2 + 3(x-1) + \frac{5}{2}(x-1)^2 - 2(x-1)^3 + \ldots$ 1st two terms CAO;
2nd two terms **ft (a)** & **(b)**'s answers **A1A1** (A1A1ft)
**Total: 3**
**(d)** Substituting $x = 1.1 \Rightarrow f(1.1) \approx 2.323$ to 3d.p. CAO **M1A1**
**Total: 2**7 The function f satisfies the differential equation
$$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$
and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.
\begin{enumerate}[label=(\alph*)]
\item Determine $f ^ { \prime \prime } ( 1 )$.
\item Differentiate ( $*$ ) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
\item Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
\item Deduce, to 3 decimal places, an approximation for f(1.1).
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2020 Q7 [2]}}