| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Show gradient expression then find coordinates |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring the chain rule (dy/dx = dy/dθ ÷ dx/dθ), double angle formula, and trigonometric identities. Part (i) involves routine differentiation and algebraic manipulation. Parts (ii) and (iii) are straightforward applications. Slightly above average due to the algebraic manipulation needed to reach the given form, but still a textbook-style exercise with no novel insight required. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain \(\frac{d\theta}{d\rho} = 2\sec^2\theta\) and \(\frac{d\theta}{d\varphi} = 6\cos 2\theta\) | B1 | |
| Use \(\cos 2\theta = 2\cos^2\theta - 1\) or equivalent | B1 | |
| Form expression for \(\frac{dy}{dx}\) in terms of \(\cos\theta\) | M1 | |
| Confirm \(6\cos^4\theta - 3\cos^2\theta\) with no errors seen | A1 | [4] |
| (ii) Equate first derivative to zero and obtain at least \(\cos\theta = \pm\frac{1}{\sqrt{2}}\) | B1 | |
| Obtain \(\theta = \frac{1}{4}\pi\) or equivalent | B1 | |
| Obtain (2, 3) | B1 | [3] |
| (iii) State or imply \(\theta = \frac{1}{3}\pi\) or equivalent | B1 | |
| Obtain \(-\frac{3}{8}\) or equivalent only | B1 | [2] |
(i) Obtain $\frac{d\theta}{d\rho} = 2\sec^2\theta$ and $\frac{d\theta}{d\varphi} = 6\cos 2\theta$ | B1 |
Use $\cos 2\theta = 2\cos^2\theta - 1$ or equivalent | B1 |
Form expression for $\frac{dy}{dx}$ in terms of $\cos\theta$ | M1 |
Confirm $6\cos^4\theta - 3\cos^2\theta$ with no errors seen | A1 | [4]
(ii) Equate first derivative to zero and obtain at least $\cos\theta = \pm\frac{1}{\sqrt{2}}$ | B1 |
Obtain $\theta = \frac{1}{4}\pi$ or equivalent | B1 |
Obtain (2, 3) | B1 | [3]
(iii) State or imply $\theta = \frac{1}{3}\pi$ or equivalent | B1 |
Obtain $-\frac{3}{8}$ or equivalent only | B1 | [2]5 A curve is defined by the parametric equations
$$x = 2 \tan \theta , \quad y = 3 \sin 2 \theta$$
for $0 \leqslant \theta < \frac { 1 } { 2 } \pi$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 4 } \theta - 3 \cos ^ { 2 } \theta$.\\
(ii) Find the coordinates of the stationary point.\\
(iii) Find the gradient of the curve at the point $\left( 2 \sqrt { } 3 , \frac { 3 } { 2 } \sqrt { } 3 \right)$.
\hfill \mbox{\textit{CAIE P2 2016 Q5 [9]}}