| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Topic | Parametric differentiation |
| Type | Show gradient expression then find coordinates |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring standard chain rule application (dy/dx = dy/dθ ÷ dx/dθ), finding a tangent line equation, and eliminating the parameter using a trigonometric identity. All techniques are routine for A-level, though the multi-part structure and need to recall cos(2θ) = 1 - 2sin²(θ) places it slightly below average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
**(i)** Obtain $\text{d}y/\text{d}\theta = -2\sin 2\theta$ — B1
Obtain $\text{d}x/\text{d}\theta = 2\cos\theta$ — B1
Use $\text{d}y/\text{d}x = (\text{d}y/\text{d}\theta)/(\text{d}\theta/\text{d}x)$ and use identity for $\sin 2\theta$ — M1
Obtain $-2\sin\theta$ NIS — A1 **[4]**
**(ii)** Obtain $m = -2$, $x = 2$ and $y = -1$ — B1
Attempt equation of line — M1
Obtain $y = 3 - 2x$ — A1 **[3]**
**(iii)** Attempt $\cos 2\theta = \cos^2\theta - \sin^2\theta$ or equivalent — M1
Attempt to eliminate $\theta$ — M1
Obtain $y = 1 - x^2/2$ — A1 **[3]** **[10]**11 A curve has parametric equations given by
$$x = 2 \sin \theta , \quad y = \cos 2 \theta$$
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin \theta$.\\
(ii) Hence find the equation of the tangent to the curve at $\theta = \frac { 1 } { 2 } \pi$.\\
(iii) Find the cartesian equation of the curve.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q11 [10]}}