| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Iterative method for parameter value |
| Difficulty | Standard +0.8 This question requires finding dy/dx using the chain rule for parametric equations, algebraic manipulation to derive the iterative formula, then careful execution of the iteration process. While the calculus is standard A-level, the algebraic rearrangement to isolate the iterative form and the numerical method application make this moderately challenging, requiring multiple techniques and precision. |
| Spec | 1.07s Parametric and implicit differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain derivative of form \(k(2t + 1)^{-3}\) | M1 | |
| Obtain \(-4(2t + 1)^{-3}\) or equivalent as derivative of \(x\) | A1 | |
| Obtain \(\frac{1}{2}(t + 2)^{-\frac{1}{2}}\) or equivalent as derivative of \(y\) | B1 | |
| Equate attempt at \(\frac{dy}{dx}\) to \(-1\) | M1 | |
| Obtain \((2p + 1)^3 = 8(p + 2)^2\) or equivalent | A1 | |
| Confirm given answer \(p = (p + 2)^2 - \frac{3}{4}\) | A1 | [6] |
| (ii) Use iteration process correctly at least once | M1 | |
| Obtain final answer 0.678 | A1 | |
| Show sufficient iterations to 5 decimal places to justify answer or show a sign change in the interval \((0.6775, 0.6785)\) | A1 | [3] |
**(i)** Obtain derivative of form $k(2t + 1)^{-3}$ | M1 |
Obtain $-4(2t + 1)^{-3}$ or equivalent as derivative of $x$ | A1 |
Obtain $\frac{1}{2}(t + 2)^{-\frac{1}{2}}$ or equivalent as derivative of $y$ | B1 |
Equate attempt at $\frac{dy}{dx}$ to $-1$ | M1 |
Obtain $(2p + 1)^3 = 8(p + 2)^2$ or equivalent | A1 |
Confirm given answer $p = (p + 2)^2 - \frac{3}{4}$ | A1 | [6]
**(ii)** Use iteration process correctly at least once | M1 |
Obtain final answer 0.678 | A1 |
Show sufficient iterations to 5 decimal places to justify answer or show a sign change in the interval $(0.6775, 0.6785)$ | A1 | [3]
$[0.7 \to 0.68003 \to 0.67857 \to 0.67847 \to 0.67846]$6 A curve has parametric equations
$$x = \frac { 1 } { ( 2 t + 1 ) ^ { 2 } } , \quad y = \sqrt { } ( t + 2 )$$
The point $P$ on the curve has parameter $p$ and it is given that the gradient of the curve at $P$ is - 1 .\\
(i) Show that $p = ( p + 2 ) ^ { \frac { 1 } { 6 } } - \frac { 1 } { 2 }$.\\
(ii) Use an iterative process based on the equation in part (i) to find the value of $p$ correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places.
\hfill \mbox{\textit{CAIE P2 2012 Q6 [9]}}