A curve is given by the parametric equations $x = 3t - 2\sin t$, $y = 5t + 4\cos t$, with $0 \leq t \leq 2\pi$. At the two points $P$ and $Q$ on the curve, the gradient equals $\frac{5}{2}$.
(a)[3]
Show that the $t$-values at $P$ and $Q$ satisfy the equation $10\cos t - 8\sin t = 5$.
(b)[3]
Express $10\cos t - 8\sin t$ in the form $R\cos(t + \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact value of $R$ and give $\alpha$ correct to 3 significant figures.
(c)[4]
Hence find the corresponding $t$-values at $P$ and $Q$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Find $\dfrac{dx}{dt}=3-2\cos t$ and $\dfrac{dy}{dt}=5-4\sin t$” …