| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2022 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Find intersection points |
| Difficulty | Standard +0.3 Part (a) requires setting x=0 and solving a quadratic for t, then finding y-coordinates—straightforward parametric work. Part (b) requires finding dy/dx=0 when y=0, which involves standard differentiation of parametric equations and solving simultaneous conditions. Both parts are routine applications of parametric techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| 1 | 7 |
The parametric equations of the curve $C$ are
$$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$
a) Find the coordinates of the points where $C$ meets the $y$-axis.\\
b) Show that the $x$-axis is a tangent to the curve $C$.
\begin{center}
\begin{tabular}{ | l | l | }
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1 & 7 \\
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\end{tabular}
\end{center}
a) Prove that
$$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$
b) i) Hence show that $\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }$ can be expressed as $\cos \theta + \sin \theta$.\\
ii) Explain why $\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta$ when $\theta = \frac { 3 \pi } { 16 }$.
\hfill \mbox{\textit{WJEC Unit 3 2022 Q16}}