CAIE FP1 2017 Specimen — Question 2 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionSpecimen
Marks6
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Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeStandard non-homogeneous with polynomial RHS
DifficultyStandard +0.8 This is a standard Further Maths second-order differential equation requiring both complementary function (repeated root case with auxiliary equation m²+4m+4=0) and particular integral (polynomial trial solution). While methodical, it involves multiple techniques and algebraic manipulation beyond standard A-level, placing it moderately above average difficulty.
Spec4.10e Second order non-homogeneous: complementary + particular integral
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = 7 - 2 t ^ { 2 }$$
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(m^2 + 4m + 4 = 0 \Rightarrow (m+2)^2 = 0 \Rightarrow m = -2\)1 M1
CF: \(Ae^{-2t} + Bte^{-2t}\) soi1 A1
PI: \(x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p\)1 M1
\(\Rightarrow 2p + 8pt + 4q + 4pt^2 + 4qt + 4r = 7 - 2t^2\)1 M1
\(\Rightarrow p = -\frac{1}{2},\ q = 1,\ r = 1\)1 A1
GS: \(x = Ae^{-2t} + Bte^{-2t} - \frac{1}{2}t^2 + t + 1\)1 A1
Total6 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2 + 4m + 4 = 0 \Rightarrow (m+2)^2 = 0 \Rightarrow m = -2$ | 1 | M1 |
| CF: $Ae^{-2t} + Bte^{-2t}$ soi | 1 | A1 |
| PI: $x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p$ | 1 | M1 |
| $\Rightarrow 2p + 8pt + 4q + 4pt^2 + 4qt + 4r = 7 - 2t^2$ | 1 | M1 |
| $\Rightarrow p = -\frac{1}{2},\ q = 1,\ r = 1$ | 1 | A1 |
| GS: $x = Ae^{-2t} + Bte^{-2t} - \frac{1}{2}t^2 + t + 1$ | 1 | A1 |
| **Total** | **6** | |

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2 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = 7 - 2 t ^ { 2 }$$

\hfill \mbox{\textit{CAIE FP1 2017 Q2 [6]}}
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Q1 4 Q2 6 Q3 6 Q4 7 Q5 8 Q6 10 Q7 10 Q8 11 Q9 12 Q10 12 Q11 EITHER Q11 OR