| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (tan/sec/cot/cosec identities) |
| Difficulty | Standard +0.3 This is a standard parametric curves question requiring routine application of trigonometric identities (1 + cot²θ = cosec²θ), parametric differentiation using the chain rule, and finding a tangent equation. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation1.08e Area between curve and x-axis: using definite integrals |
12 A curve $C$ is given by the parametric equations $x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta$ for $0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi$.\\
(i) Show that the cartesian equation for $C$ is $\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y$.\\
(ii) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence show that $C$ has no stationary points.\\
(iii) $P$ is the point on $C$ where $\theta = \frac { 1 } { 4 } \pi$. The tangent to $C$ at $P$ intersects the $y$-axis at $Q$ and the $x$-axis at $R$. Find the exact area of triangle $O Q R$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q12}}