4 The parametric equations of a curve are
$$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
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Question 4:
Answer Marks
Guidance
Answer Mark
Guidance
State \(\frac{dx}{d\theta} = \sin\theta\) or \(\frac{dy}{d\theta} = -\sin\theta + \frac{1}{2}\sin 2\theta\) B1
Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}\) M1
Obtain correct answer in any form A1
e.g. \(\dfrac{-\sin\theta + \frac{1}{2}\sin 2\theta}{\sin\theta}\)
Use double angle correctly to obtain \(\frac{dy}{dx}\) in terms of \(\theta\) M1
\(\sin 2\theta = 2\sin\theta\cos\theta\)
Obtain the given answer with no errors seen \(-2\sin^2\!\left(\frac{1}{2}\theta\right)\) A1
AG. Requires correct cancellation of ALL \(\sin\theta\) terms and \(\cos\theta = 1 - 2\sin^2\!\left(\frac{1}{2}\theta\right)\) seen. SC for incorrect signs, consistent throughout max. B0, M1, A0, M1, A1
Total: 5
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## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\frac{dx}{d\theta} = \sin\theta$ or $\frac{dy}{d\theta} = -\sin\theta + \frac{1}{2}\sin 2\theta$ | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}$ | M1 | |
| Obtain correct answer in any form | A1 | e.g. $\dfrac{-\sin\theta + \frac{1}{2}\sin 2\theta}{\sin\theta}$ |
| Use double angle correctly to obtain $\frac{dy}{dx}$ in terms of $\theta$ | M1 | $\sin 2\theta = 2\sin\theta\cos\theta$ |
| Obtain the given answer with no errors seen $-2\sin^2\!\left(\frac{1}{2}\theta\right)$ | A1 | AG. Requires correct cancellation of ALL $\sin\theta$ terms and $\cos\theta = 1 - 2\sin^2\!\left(\frac{1}{2}\theta\right)$ seen. SC for incorrect signs, consistent throughout max. B0, M1, A0, M1, A1 |
| **Total: 5** | | |
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4 The parametric equations of a curve are
$$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$.\\
\hfill \mbox{\textit{CAIE P3 2022 Q4 [5]}}