Moderate -0.5 This is a straightforward parametric differentiation question requiring standard application of the chain rule (dy/dx = (dy/dθ)/(dx/dθ)) with routine trigonometric differentiation and simplification. The algebra is clean and the result follows directly from basic trig identities, making it slightly easier than average for A-level.
4 The parametric equations of a curve are
$$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
State or imply \(\frac{dx}{d\theta} = a(2 - 2\cos 2\theta)\) or \(\frac{dy}{d\theta} = 2a\sin 2\theta\)
B1
Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{dx}{d\theta}\)
M1
Obtain \(\frac{dy}{dx} = \frac{\sin 2\theta}{(1-\cos 2\theta)}\), or equivalent
A1
Make use of correct \(\sin 2\theta\) and \(\cos 2\theta\) formulae
M1
Obtain the given result following sufficient working
A1
[5]
[SR: An attempt which assumes \(\theta\) is the parameter and \(\theta\) a constant can only earn the two M marks. One that assumes \(\theta\) is the parameter and \(a\) is a function of \(\theta\) can earn B1M1A0M1A0.]
[SR: For an attempt that gives \(a\) a value, e.g. 1, or ignores \(a\), give B0 but allow the remaining marks.]
State or imply $\frac{dx}{d\theta} = a(2 - 2\cos 2\theta)$ or $\frac{dy}{d\theta} = 2a\sin 2\theta$ | B1 |
Use $\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{dx}{d\theta}$ | M1 |
Obtain $\frac{dy}{dx} = \frac{\sin 2\theta}{(1-\cos 2\theta)}$, or equivalent | A1 |
Make use of correct $\sin 2\theta$ and $\cos 2\theta$ formulae | M1 |
Obtain the given result following sufficient working | A1 | [5] |
[SR: An attempt which assumes $\theta$ is the parameter and $\theta$ a constant can only earn the two M marks. One that assumes $\theta$ is the parameter and $a$ is a function of $\theta$ can earn B1M1A0M1A0.] |
[SR: For an attempt that gives $a$ a value, e.g. 1, or ignores $a$, give B0 but allow the remaining marks.] |
4 The parametric equations of a curve are
$$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$.
\hfill \mbox{\textit{CAIE P3 2008 Q4 [5]}}