| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent/normal meets curve again |
| Difficulty | Standard +0.8 This is a multi-step parametric question requiring differentiation using the chain rule, finding a normal equation, then solving a system by substitution leading to a quadratic in sin q. The algebraic manipulation in part (c) and the need to connect parametric form with the normal equation elevates this above routine tangent/normal questions, but it follows a clear structured path without requiring deep geometric insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
# Question 5
## (a) [3 marks]
M1: Find $\frac{dx}{dt} = -2\sin(2t)$ and $\frac{dy}{dt} = \cos(t)$
M1: At $t = \frac{\pi}{6}$: $\frac{dx}{dt} = -\sqrt{3}$, $\frac{dy}{dt} = \frac{\sqrt{3}}{2}$
A1: Gradient $= -\frac{1}{2}$
## (b) [3 marks]
M1: Gradient of normal is $2$
M1: Find coordinates of P: $(\frac{3}{4}, \frac{1}{2})$
A1: $y = 2x - 1$ (or $2x - y - 1 = 0$)
## (c) [5 marks]
M1: Substitute $y = 2x - 1$ into $y = \sin(q)$ and $x = \cos(2q)$
M1: Use $\sin(q) = 2\cos(2q) - 1$
M1: Substitute $\cos(2q) = 1 - 2\sin^2(q)$ to form quadratic
M1: Rearrange to $4\sin^2(q) + \sin(q) - 3 = 0$
M1: Solve to get $\sin(q) = -1$ (giving $x = -1$) or $\sin(q) = \frac{3}{4}$
A1: $x = -1$ or $x = -\frac{7}{16}$
---5 A curve is defined by the parametric equations $x = \cos 2 t , y = \sin t$.\\
The point $P$ on the curve is where $t = \frac { \pi } { 6 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the gradient at $P$.
\item Find the equation of the normal to the curve at $P$ in the form $y = m x + c$.
\item The normal at $P$ intersects the curve again at the point $Q ( \cos 2 q , \sin q )$.
Use the equation of the normal to form a quadratic equation in $\sin q$ and hence find the $x$-coordinate of $Q$.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2015 Q5 [11]}}