Challenging +1.2 This is a Further Maths parametric differentiation question requiring students to find dy/dx = (dy/dt)/(dx/dt), then prove it's always negative for the given domain. While it involves multiple trigonometric derivatives and requires algebraic manipulation to show the gradient is negative (likely factoring and using bounds on trig functions), it's a standard technique with a clear method. The 7 marks suggest extended working but no particularly novel insight—competent FM students should recognize the approach immediately.
In this question you must show detailed reasoning.
A curve has parametric equations
$$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$
Show that the gradient of the curve is always negative. [7]
In this question you must show detailed reasoning.
A curve has parametric equations
$$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$
Show that the gradient of the curve is always negative. [7]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q11 [7]}}