3 The parametric equations of a curve are
$$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$
for \(0 < \theta < \frac { 1 } { 2 } \pi\).
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Question 3:
Answer Marks
Guidance
Answer Marks
Guidance
State or imply \(\frac{dx}{d\theta} = 2\sin 2\theta\) or \(\frac{dy}{d\theta} = 2 + 2\cos 2\theta\) B1
Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}\) M1
Obtain correct answer \(\frac{dy}{dx} = \frac{2 + 2\cos 2\theta}{2\sin 2\theta}\) A1
OE
Use correct double angle formulae M1
Obtain the given answer correctly \(\frac{dy}{dx} = \cot\theta\) A1
AG. Must have simplified numerator in terms of \(\cos\theta\)
Alternative method:
Answer Marks
Guidance
Answer Marks
Guidance
Start by using both correct double angle formulae e.g. \(x = 3 - (2\cos^2\theta - 1)\), \(y = 2\theta + 2\sin\theta\cos\theta\) M1
\(\frac{dx}{d\theta}\) or \(\frac{dy}{d\theta}\) B1
\(\frac{dy}{dx} = \frac{2 + 2(\cos^2\theta - \sin^2\theta)}{4\cos\theta\sin\theta}\) M1 A1
Simplify to given answer correctly \(\frac{dy}{dx} = \cot\theta\) A1
AG
Alternative method (substitution \(t = 2\theta\)):
Answer Marks
Guidance
Answer Marks
Guidance
Set \(t = 2\theta\). State \(\frac{dx}{dt} = \sin t\) or \(\frac{dy}{dt} = 1 + \cos t\) B1
Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) M1
Obtain correct answer \(\frac{dy}{dx} = \frac{1 + \cos t}{\sin t}\) A1
OE
Use correct double angle formulae M1
Obtain the given answer correctly \(\frac{dy}{dx} = \cot\theta\) A1
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## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\frac{dx}{d\theta} = 2\sin 2\theta$ or $\frac{dy}{d\theta} = 2 + 2\cos 2\theta$ | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}$ | M1 | |
| Obtain correct answer $\frac{dy}{dx} = \frac{2 + 2\cos 2\theta}{2\sin 2\theta}$ | A1 | OE |
| Use correct double angle formulae | M1 | |
| Obtain the given answer correctly $\frac{dy}{dx} = \cot\theta$ | A1 | AG. Must have simplified numerator in terms of $\cos\theta$ |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Start by using both correct double angle formulae e.g. $x = 3 - (2\cos^2\theta - 1)$, $y = 2\theta + 2\sin\theta\cos\theta$ | M1 | |
| $\frac{dx}{d\theta}$ or $\frac{dy}{d\theta}$ | B1 | |
| $\frac{dy}{dx} = \frac{2 + 2(\cos^2\theta - \sin^2\theta)}{4\cos\theta\sin\theta}$ | M1 A1 | |
| Simplify to given answer correctly $\frac{dy}{dx} = \cot\theta$ | A1 | AG |
**Alternative method (substitution $t = 2\theta$):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Set $t = 2\theta$. State $\frac{dx}{dt} = \sin t$ or $\frac{dy}{dt} = 1 + \cos t$ | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 | |
| Obtain correct answer $\frac{dy}{dx} = \frac{1 + \cos t}{\sin t}$ | A1 | OE |
| Use correct double angle formulae | M1 | |
| Obtain the given answer correctly $\frac{dy}{dx} = \cot\theta$ | A1 | |
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3 The parametric equations of a curve are
$$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$
for $0 < \theta < \frac { 1 } { 2 } \pi$.\\
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$.\\
\hfill \mbox{\textit{CAIE P3 2020 Q3 [5]}}