| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find tangent equation at parameter |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring standard techniques: find dy/dx using the chain rule, determine the parameter value from a given y-coordinate, then verify a tangent passes through a point. All steps are routine with no conceptual challenges beyond basic parametric calculus. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.06d Natural logarithm: ln(x) function and properties1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\frac{dx}{dt} = \frac{1}{2}t^{-\frac{1}{2}}\) or \(\frac{dy}{dt} = \frac{3}{t}\) | B1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 | |
| Use \(y = 6\) to find \(t\) | M1 | |
| Obtain \(t = e^2\) | A1 | |
| Obtain \(\frac{dy}{dx} = \frac{6}{e}\) | A1 | [5] |
| (ii) Obtain \(x\) and form equation of the tangent at their point | M1 | |
| Obtain correct equation for tangent \(y - 6 = -\frac{6}{e}(x - (1 + e))\) | A1 | |
| Show that tangent passes through \((1, 0)\) by substitution | A1 | [3] |
(i) State $\frac{dx}{dt} = \frac{1}{2}t^{-\frac{1}{2}}$ or $\frac{dy}{dt} = \frac{3}{t}$ | B1 |
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 |
Use $y = 6$ to find $t$ | M1 |
Obtain $t = e^2$ | A1 |
Obtain $\frac{dy}{dx} = \frac{6}{e}$ | A1 | [5]
(ii) Obtain $x$ and form equation of the tangent at their point | M1 |
Obtain correct equation for tangent $y - 6 = -\frac{6}{e}(x - (1 + e))$ | A1 |
Show that tangent passes through $(1, 0)$ by substitution | A1 | [3]5 The parametric equations of a curve are
$$x = 1 + \sqrt { } t , \quad y = 3 \ln t$$
(i) Find the exact value of the gradient of the curve at the point $P$ where $y = 6$.\\
(ii) Show that the tangent to the curve at $P$ passes through the point $( 1,0 )$.
\hfill \mbox{\textit{CAIE P2 2013 Q5 [8]}}