| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 This is a standard Further Maths second-order differential equation question with straightforward steps: finding complex roots of the auxiliary equation, writing the complementary function with complex roots (requiring e^(ax)(Acos(bx) + Bsin(bx)) form), finding a particular integral (linear form), and applying initial conditions. While it's Further Maths content, it follows a completely routine procedure with no novel problem-solving required, making it slightly easier than average overall but typical for FP3. |
| Spec | 4.02i Quadratic equations: with complex roots4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((m+1)^2 = -1\) | M1 | Completing square or formula |
| \(m = -1 \pm i\) | A1 | Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| CF is \(e^{-x}(A\cos x + B\sin x)\) | M1 | If \(m\) is real give M0 |
| \(\{\)or \(e^{-x}A\cos(x+B)\}\) but not \(Ae^{(-1+i)x} + Be^{(-1-i)x}\}\) | A1\(\checkmark\) | On wrong \(a\)'s and \(b\)'s but roots must be complex |
| \(\{\)P.Int.\(\}\) try \(y = px + q\) | M1 | OE |
| \(2p + 2(px+q) = 4x\) | A1 | |
| \(p=2,\; q=-2\) | A1\(\checkmark\) | On one slip |
| GS: \(y = e^{-x}(A\cos x + B\sin x) + 2x - 2\) | B1\(\checkmark\) | 6 marks; Their CF + their PI with two arbitrary constants |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x=0,\; y=1 \Rightarrow A=3\) | B1\(\checkmark\) | Provided M1 gained in (b)(i) |
| \(y'(x) = -e^{-x}(A\cos x + B\sin x) + e^{-x}(-A\sin x + B\cos x) + 2\) | M1 | Product rule used |
| \(y'(0) = 2 \Rightarrow 2 = -A+B+2 \Rightarrow B=3\) | A1\(\checkmark\) | Slips |
| A1\(\checkmark\) | ||
| \(y = 3e^{-x}(\cos x + \sin x) + 2x - 2\) | Total: 4 marks |
# Question 1:
## Part 1(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(m+1)^2 = -1$ | M1 | Completing square or formula |
| $m = -1 \pm i$ | A1 | Total: 2 marks |
## Part 1(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| CF is $e^{-x}(A\cos x + B\sin x)$ | M1 | If $m$ is real give M0 |
| $\{$or $e^{-x}A\cos(x+B)\}$ but not $Ae^{(-1+i)x} + Be^{(-1-i)x}\}$ | A1$\checkmark$ | On wrong $a$'s and $b$'s but roots must be complex |
| $\{$P.Int.$\}$ try $y = px + q$ | M1 | OE |
| $2p + 2(px+q) = 4x$ | A1 | |
| $p=2,\; q=-2$ | A1$\checkmark$ | On one slip |
| GS: $y = e^{-x}(A\cos x + B\sin x) + 2x - 2$ | B1$\checkmark$ | 6 marks; Their CF + their PI with two arbitrary constants |
## Part 1(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x=0,\; y=1 \Rightarrow A=3$ | B1$\checkmark$ | Provided M1 gained in (b)(i) |
| $y'(x) = -e^{-x}(A\cos x + B\sin x) + e^{-x}(-A\sin x + B\cos x) + 2$ | M1 | Product rule used |
| $y'(0) = 2 \Rightarrow 2 = -A+B+2 \Rightarrow B=3$ | A1$\checkmark$ | Slips |
| | A1$\checkmark$ | |
| $y = 3e^{-x}(\cos x + \sin x) + 2x - 2$ | | Total: 4 marks |
---1
\begin{enumerate}[label=(\alph*)]
\item Find the roots of the equation $m ^ { 2 } + 2 m + 2 = 0$ in the form $a + i b$.\\
(2 marks)
\item \begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
\item Hence express $y$ in terms of $x$, given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$ when $x = 0$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q1 [12]}}