| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find parameter value given gradient condition |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: differentiate both parametric equations, apply the chain rule formula dy/dx = (dy/dt)/(dx/dt), and simplify using basic trigonometric identities. Part (ii) involves solving cot t = 2, which is routine. The 5 marks for part (i) reflect algebraic manipulation rather than conceptual difficulty. Slightly above average due to the trigonometric simplification required, but still a standard textbook exercise. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| (i) State \(\frac{dx}{dt} = 1 - \sec^2 t\), or equivalent | B1 |
| Use chain rule | M1 |
| Obtain \(\frac{dy}{dt} = -\frac{\sin t}{\cos t}\), or equivalent | A1 |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 |
| Obtain the given answer correctly. | A1 |
| Total: 5 marks | |
| (ii) State or imply \(t = \tan^{-1}(1)\) | B1 |
| Obtain answer \(x = -0.0364\) | B1 |
| Total: 2 marks |
**(i)** State $\frac{dx}{dt} = 1 - \sec^2 t$, or equivalent | B1 |
Use chain rule | M1 |
Obtain $\frac{dy}{dt} = -\frac{\sin t}{\cos t}$, or equivalent | A1 |
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 |
Obtain the given answer correctly. | A1 |
| Total: 5 marks |
**(ii)** State or imply $t = \tan^{-1}(1)$ | B1 |
Obtain answer $x = -0.0364$ | B1 |
| Total: 2 marks |
---The parametric equations of a curve are
$$x = t - \tan t, \quad y = \ln(\cos t),$$
for $-\frac{1}{4}\pi < t < \frac{1}{4}\pi$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{dy}{dx} = \cot t$. [5]
\item Hence find the $x$-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q4 [7]}}