| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find parameter value given gradient condition |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring standard techniques: quotient rule for part (i), chain rule for dy/dx in part (ii), and solving a simple trigonometric equation in part (iii). All steps are routine applications of A-level calculus with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| Differentiate using chain or quotient rule | M1 |
| Obtain derivative in any correct form | A1 |
| Obtain given answer correctly | A1 |
| Answer | Marks |
|---|---|
| State \(\frac{dx}{d\theta} = \sec^2 \theta\), or equivalent | B1 |
| Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{dx}{d\theta}\) | M1 |
| Obtain given answer correctly | A1 |
| Answer | Marks |
|---|---|
| State that \(\theta = \frac{\pi}{6}\) | B1 |
| Obtain x-coordinate \(1 + \frac{1}{\sqrt{3}}\), or equivalent | B1 |
| Obtain y-coordinate \(\frac{2}{\sqrt{3}}\), or equivalent | B1 |
**(i)**
| Differentiate using chain or quotient rule | M1 |
| Obtain derivative in any correct form | A1 |
| Obtain given answer correctly | A1 |
**Total: 3 marks**
**(ii)**
| State $\frac{dx}{d\theta} = \sec^2 \theta$, or equivalent | B1 |
| Use $\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{dx}{d\theta}$ | M1 |
| Obtain given answer correctly | A1 |
**Total: 3 marks**
**(iii)**
| State that $\theta = \frac{\pi}{6}$ | B1 |
| Obtain x-coordinate $1 + \frac{1}{\sqrt{3}}$, or equivalent | B1 |
| Obtain y-coordinate $\frac{2}{\sqrt{3}}$, or equivalent | B1 |
**Total: 3 marks**
---5 (i) By differentiating $\frac { 1 } { \cos \theta }$, show that if $y = \sec \theta$ then $\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta$.\\
(ii) The parametric equations of a curve are
$$x = 1 + \tan \theta , \quad y = \sec \theta$$
for $- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin \theta$.\\
(iii) Find the coordinates of the point on the curve at which the gradient of the curve is $\frac { 1 } { 2 }$.
\hfill \mbox{\textit{CAIE P2 2005 Q5 [9]}}