| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Prove formula by induction |
| Difficulty | Challenging +1.2 This is a structured Further Maths question requiring Leibnitz's theorem application and complex number form conversion. While it involves FP1 content (inherently harder than standard A-level), the question is highly guided with clear steps, standard techniques, and no novel insight required. The Leibnitz's theorem application is mechanical, and the R sin(x+α) conversion is a routine FP1 skill. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07q Product and quotient rules: differentiation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.08a Maclaurin series: find series for function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u'=3e^{3x},\ u''=9e^{3x},\ u'''=27e^{3x},\ u^{(4)}=81e^{3x}\) | M1, A1 | Correct form \(\alpha e^{3x}\); all four derivatives |
| \(v'=\cos x,\ v''=-\sin x,\ v'''=-\cos x,\ v^{(4)}=\sin x\) | M1, A1 | Correct derivatives of \(\sin x\); condone sign slips only |
| \(\dfrac{d^4y}{dx^4} = (e^{3x}\times\sin x)+(4\times 3e^{3x}\times{-}\cos x)+(6\times 9e^{3x}\times{-}\sin x)+(4\times 27e^{3x}\times\cos x)+(81e^{3x}\times\sin x)\) | M1 | Leibnitz theorem applied; binomial coefficients correct |
| \(\dfrac{d^4y}{dx^4} = 28e^{3x}\sin x + 96e^{3x}\cos x\) | A1* | Correct simplified result |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R = \sqrt{28^2+96^2} = 100\) | B1 | Correct value of \(R\) |
| \(\tan\alpha = \pm\dfrac{96}{28} \Rightarrow \alpha = \ldots\ (\alpha = 1.287\ldots)\) | M1 | Correct method for \(\alpha\) |
| \(\dfrac{d^4y}{dx^4} = 100e^{3x}\sin(x+1.29)\) | A1 | Correct final form |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u'=3e^{3x},\ u''=9e^{3x},\ u'''=27e^{3x},\ u^{(4)}=81e^{3x}$ | M1, A1 | Correct form $\alpha e^{3x}$; all four derivatives |
| $v'=\cos x,\ v''=-\sin x,\ v'''=-\cos x,\ v^{(4)}=\sin x$ | M1, A1 | Correct derivatives of $\sin x$; condone sign slips only |
| $\dfrac{d^4y}{dx^4} = (e^{3x}\times\sin x)+(4\times 3e^{3x}\times{-}\cos x)+(6\times 9e^{3x}\times{-}\sin x)+(4\times 27e^{3x}\times\cos x)+(81e^{3x}\times\sin x)$ | M1 | Leibnitz theorem applied; binomial coefficients correct |
| $\dfrac{d^4y}{dx^4} = 28e^{3x}\sin x + 96e^{3x}\cos x$ | A1* | Correct simplified result |
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## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{28^2+96^2} = 100$ | B1 | Correct value of $R$ |
| $\tan\alpha = \pm\dfrac{96}{28} \Rightarrow \alpha = \ldots\ (\alpha = 1.287\ldots)$ | M1 | Correct method for $\alpha$ |
| $\dfrac{d^4y}{dx^4} = 100e^{3x}\sin(x+1.29)$ | A1 | Correct final form |5.
$$y = \mathrm { e } ^ { 3 x } \sin x$$
\begin{enumerate}[label=(\alph*)]
\item Use Leibnitz's theorem to show that
$$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = 28 \mathrm { e } ^ { 3 x } \sin x + 96 \mathrm { e } ^ { 3 x } \cos x$$
\item Hence express $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$ in the form
$$\operatorname { Re } ^ { 3 \mathrm { x } } \sin ( \mathrm { x } + \alpha )$$
where $R$ and $\alpha$ are constants to be determined, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2024 Q5 [9]}}