| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Standard +0.8 This is a standard second-order linear ODE with constant coefficients requiring both complementary function (repeated root case) and particular integral (using trial solution for sin t). While methodical, it involves multiple techniques (auxiliary equation, repeated roots, undetermined coefficients with differentiation) and careful algebra. Part (ii) requires understanding of transient vs steady-state behavior. Moderately above average for A-level but routine for Further Maths students. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral4.10g Damped oscillations: model and interpret |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | m2 +2m+1=0⇒ ( m+1 )2 =0 ⇒m=−1 | M1 |
| CF: ( A+Bt ) e−t | A1 | States CF. |
| PI: x= psint+qcost | M1 | Uses correct form of PI and differentiates |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒x(cid:5)= pcost−qsint ⇒ x=−psint−qcost | A1 | |
| −psint−qcost+2 ( pcost−qsint )+ psint+qcost=4sint | M1 | Compares coefficients and attempts to |
| Answer | Marks | Guidance |
|---|---|---|
| 2p=0⇒ p=0. −2q=4⇒q=−2. | A1 | |
| GS: x=( A+Bt ) e−t −2cost | A1FT | States general solution. FT on correct |
| Answer | Marks | Guidance |
|---|---|---|
| 4(ii) | x≈−2cost | B1FT |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | m2 +2m+1=0⇒ ( m+1 )2 =0 ⇒m=−1 | M1 | Forms and solves auxiliary equation.
CF: ( A+Bt ) e−t | A1 | States CF.
PI: x= psint+qcost | M1 | Uses correct form of PI and differentiates
twice.
¨
⇒x(cid:5)= pcost−qsint ⇒ x=−psint−qcost | A1
−psint−qcost+2 ( pcost−qsint )+ psint+qcost=4sint | M1 | Compares coefficients and attempts to
solve
2p=0⇒ p=0. −2q=4⇒q=−2. | A1
GS: x=( A+Bt ) e−t −2cost | A1FT | States general solution. FT on correct
form only
7
--- 4(ii) ---
4(ii) | x≈−2cost | B1FT
1
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 2\frac{\mathrm{d}x}{\mathrm{d}t} + x = 4\sin t.$$ [7]
\item State an approximate solution for large positive values of $t$. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2018 Q4 [8]}}