| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find tangent equation at parameter |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring the chain rule (dy/dx = dy/dt ÷ dx/dt) and finding a tangent equation. The derivatives involve routine techniques (differentiating ln(tan t) and sin²t), though finding where x=0 requires solving ln(tan t)=0 to get t=π/4. Slightly above average due to the logarithmic/trigonometric combination, but still a textbook exercise with no novel insight required. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| State \(\frac{dx}{dt} = \sec^2 t / \tan t\), or equivalent | B1 |
| State \(\frac{dy}{dt} = 2 \sin t \cos t\), or equivalent | B1 |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 |
| Obtain correct answer in any form, e.g. \(2 \sin^2 t / \cos^2 t\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(y = e^{2t} / (1 + e^{2t})\), or equivalent | B1 | |
| Use correct quotient or product rule | M1 | |
| Obtain correct derivative in any form, e.g. \(2e^{2t} / (1 + e^{2t})^2\) | A1 | |
| Obtain correct derivative in terms of \(t\) in any form, e.g. \((2\tan^2 t) / (1 + \tan^2 t)^2\) | A1 | [4] |
| (ii) State or imply \(t = \frac{1}{4}\pi\) when \(x = 0\) | B1 | |
| Form the equation of the tangent at \(x = 0\) | M1 | |
| Obtain correct answer in any horizontal form, e.g. \(y = \frac{1}{2}x + \frac{1}{2}\) | A1 | [3] |
**(i) EITHER:**
State $\frac{dx}{dt} = \sec^2 t / \tan t$, or equivalent | B1 |
State $\frac{dy}{dt} = 2 \sin t \cos t$, or equivalent | B1 |
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 |
Obtain correct answer in any form, e.g. $2 \sin^2 t / \cos^2 t$ | A1 |
**OR:**
Obtain $y = e^{2t} / (1 + e^{2t})$, or equivalent | B1 |
Use correct quotient or product rule | M1 |
Obtain correct derivative in any form, e.g. $2e^{2t} / (1 + e^{2t})^2$ | A1 |
Obtain correct derivative in terms of $t$ in any form, e.g. $(2\tan^2 t) / (1 + \tan^2 t)^2$ | A1 | [4]
**(ii)** State or imply $t = \frac{1}{4}\pi$ when $x = 0$ | B1 |
Form the equation of the tangent at $x = 0$ | M1 |
Obtain correct answer in any horizontal form, e.g. $y = \frac{1}{2}x + \frac{1}{2}$ | A1 | [3]
[SR: If the OR method is used in part (i), give B1 for stating or implying $y = \frac{1}{2}$ or $\frac{dy}{dx} = \frac{1}{2}$ when $x = 0$.]5 The parametric equations of a curve are
$$x = \ln ( \tan t ) , \quad y = \sin ^ { 2 } t$$
where $0 < t < \frac { 1 } { 2 } \pi$.\\
(i) Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.\\
(ii) Find the equation of the tangent to the curve at the point where $x = 0$.
\hfill \mbox{\textit{CAIE P3 2011 Q5 [7]}}