Question 9 - A-Level Maths

CAIE P3 2023 March — Question 9 7 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2023
Session	March
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Separable variables
Difficulty	Standard +0.3 This is a straightforward separable variables question requiring separation, integration using standard techniques (including the identity sin²(2x) = (1-cos(4x))/2), and applying initial conditions. While it involves multiple steps and careful algebraic manipulation, it follows a standard template with no novel problem-solving required, making it slightly easier than average.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

9 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$ It is given that $y = 0$ when $x = 0$.
Solve the differential equation and find the value of $y$ when $x = \frac { 1 } { 2 }$.

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Question 9:

Answer	Marks	Guidance
Answer	Mark	Guidance
Separate variables correctly and obtain $e^{-3y}$ and $\sin^2 2x$ on opposite sides	B1
Obtain term $-\frac{1}{3}e^{-3y}$	B1
Use correct double angle formula $\sin^2 2x = \frac{1}{2}[1-\cos 4x]$	M1
Obtain terms $\frac{1}{2}\left[x - \frac{1}{4}\sin 4x\right]$	A1	OE
Use $x=0$, $y=0$ to evaluate constant or as limits in solution containing terms of form $ax$ and $b\sin 4x$ and $ce^{\pm 3y}$	M1
Obtain correct answer in any form e.g. $-\frac{1}{3}e^{-3y} = \frac{1}{2}\left[x-\frac{1}{4}\sin 4x\right] - \frac{1}{3}$	A1
Substitute $x=\frac{1}{2}$ and obtain $y=0.175$ or $-\frac{1}{3}\ln\left(\frac{1}{4}+\frac{3}{8}\sin 2\right)$	A1	OE ISW
Total: 7

## Question 9:

| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables correctly and obtain $e^{-3y}$ and $\sin^2 2x$ on opposite sides | B1 | |
| Obtain term $-\frac{1}{3}e^{-3y}$ | B1 | |
| Use correct double angle formula $\sin^2 2x = \frac{1}{2}[1-\cos 4x]$ | M1 | |
| Obtain terms $\frac{1}{2}\left[x - \frac{1}{4}\sin 4x\right]$ | A1 | OE |
| Use $x=0$, $y=0$ to evaluate constant or as limits in solution containing terms of form $ax$ and $b\sin 4x$ and $ce^{\pm 3y}$ | M1 | |
| Obtain correct answer in any form e.g. $-\frac{1}{3}e^{-3y} = \frac{1}{2}\left[x-\frac{1}{4}\sin 4x\right] - \frac{1}{3}$ | A1 | |
| Substitute $x=\frac{1}{2}$ and obtain $y=0.175$ or $-\frac{1}{3}\ln\left(\frac{1}{4}+\frac{3}{8}\sin 2\right)$ | A1 | OE ISW |
| **Total: 7** | | |

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9 The variables $x$ and $y$ satisfy the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$

It is given that $y = 0$ when $x = 0$.\\
Solve the differential equation and find the value of $y$ when $x = \frac { 1 } { 2 }$.\\

\hfill \mbox{\textit{CAIE P3 2023 Q9 [7]}}

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