Question 7 - A-Level Maths

CAIE P3 2023 June — Question 7 8 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2023
Session	June
Marks	8
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 standard techniques: separate variables, integrate both sides (using substitution for sin³3y and recognizing tan2x/cos2x = sin2x/cos²2x), apply initial condition, then solve for x. The trigonometric functions add minor complexity but the method is routine and well-practiced at A-level.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

7 The variables $x$ and $y$ satisfy the differential equation $$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$ where $0 \leqslant x < \frac { 1 } { 4 } \pi$. It is given that $y = 0$ when $x = \frac { 1 } { 6 } \pi$.
Solve the differential equation to obtain the value of $x$ when $y = \frac { 1 } { 6 } \pi$. Give your answer correct to 3 decimal places.

Show mark scheme Show mark scheme source

Question 7:

Answer	Marks	Guidance
Answer	Marks	Guidance
Correct separation of variables	B1	$\int \sin^2 3y\, dy = \int 4\sec 2x \tan 2x\, dx$ or equivalent. Condone missing integral signs or $dx$ and $dy$.
Integrate to obtain $k\sec 2x$	M1
Obtain $2\sec 2x$	A1
Use double angle formula and integrate to obtain $py + q\sin 6y$	M1	Or two cycles of integration by parts.
Obtain $\frac{1}{2}y - \frac{1}{12}\sin 6y$	A1
Use $y=0$, $x=\frac{\pi}{6}$ in a solution containing terms $\lambda\sec 2x$ and $\mu\sin 6y$ to find the constant of integration	M1
Obtain $\frac{1}{2}y - \frac{1}{12}\sin 6y = 2\sec 2x - 4$	A1	Or equivalent seen or implied by $\frac{\pi}{2}\!\left(-\frac{1}{12}\sin\pi\right) = 2\sec 2x - 4$
Obtain $x = 0.541$	A1	From correct working (not by using the calculator to integrate).
Total	8

## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct separation of variables | B1 | $\int \sin^2 3y\, dy = \int 4\sec 2x \tan 2x\, dx$ or equivalent. Condone missing integral signs or $dx$ and $dy$. |
| Integrate to obtain $k\sec 2x$ | M1 | |
| Obtain $2\sec 2x$ | A1 | |
| Use double angle formula and integrate to obtain $py + q\sin 6y$ | M1 | Or two cycles of integration by parts. |
| Obtain $\frac{1}{2}y - \frac{1}{12}\sin 6y$ | A1 | |
| Use $y=0$, $x=\frac{\pi}{6}$ in a solution containing terms $\lambda\sec 2x$ and $\mu\sin 6y$ to find the constant of integration | M1 | |
| Obtain $\frac{1}{2}y - \frac{1}{12}\sin 6y = 2\sec 2x - 4$ | A1 | Or equivalent seen or implied by $\frac{\pi}{2}\!\left(-\frac{1}{12}\sin\pi\right) = 2\sec 2x - 4$ |
| Obtain $x = 0.541$ | A1 | From correct working (not by using the calculator to integrate). |
| **Total** | **8** | |

---

Show LaTeX source

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

$$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$

where $0 \leqslant x < \frac { 1 } { 4 } \pi$. It is given that $y = 0$ when $x = \frac { 1 } { 6 } \pi$.\\
Solve the differential equation to obtain the value of $x$ when $y = \frac { 1 } { 6 } \pi$. Give your answer correct to 3 decimal places.\\

\hfill \mbox{\textit{CAIE P3 2023 Q7 [8]}}

This paper (10 questions)

View full paper

Q1 3 Q2 4 Q3 4 Q4 6 Q5 8 Q6 10 Q7 8 Q8 10 Q9 10 Q10 12