| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 6 |
| Topic | Parametric differentiation |
| Type | Find gradient at given parameter |
| Difficulty | Challenging +1.2 This is a standard parametric differentiation question requiring the chain rule (dy/dx = (dy/dθ)/(dx/dθ)) and manipulation of trigonometric functions. Part (a) is routine A-level technique. Part (b) requires finding θ when y=8 (giving θ=π/6) and substituting, which adds modest problem-solving but remains a familiar exam pattern. The cosec³θ notation and exact value requirement elevate it slightly above average difficulty. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.07s Parametric and implicit differentiation |
The curve $C$ has parametric equations
$$x = \sin 2\theta \quad y = \cos\text{ec}^3 \theta \quad 0 < \theta < \frac{\pi}{2}$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac{dy}{dx}$ in terms of $\theta$
[3]
\item Hence find the exact value of the gradient of the tangent to $C$ at the point where $y = 8$
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q11 [6]}}