| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Iterative method for parameter value |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question followed by a standard iterative method application. Part (i) requires routine use of dy/dx = (dy/dt)/(dx/dt) and algebraic manipulation. Parts (ii) and (iii) are mechanical applications of iteration with no conceptual challenges—students simply substitute values and follow the algorithm. The question is slightly above average difficulty only because it combines multiple techniques, but each step is standard textbook material with no novel insight required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(\frac{dy}{dx} = \frac{y}{x}\) and equate \(\frac{dy}{dx}\) to 4 | M1 | |
| Obtain \(\frac{4p^3}{2p + 3} = 4\) or equivalent | A1 | |
| Confirm given result \(p = \sqrt[3]{2p + 3}\) correctly | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Evaluate \(p - \sqrt[3]{2p + 3}\) or \(p^3 - 2p - 3\) or equivalent at 1.8 and 2.0 | M1 | |
| Justify result with correct calculations and argument (\(-0.076\) and \(0.087\) or \(-0.77\) and \(1\) respectively) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative process correctly at least once with \(1.8 < p_n < 2.0\) | M1 | |
| Obtain final answer 1.89 | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify 1.89 or show sign change in interval (1.885, 1.895) | A1 | [3] |
**(i)**
Use $\frac{dy}{dx} = \frac{y}{x}$ and equate $\frac{dy}{dx}$ to 4 | M1 |
Obtain $\frac{4p^3}{2p + 3} = 4$ or equivalent | A1 |
Confirm given result $p = \sqrt[3]{2p + 3}$ correctly | A1 | [3]
**(ii)**
Evaluate $p - \sqrt[3]{2p + 3}$ or $p^3 - 2p - 3$ or equivalent at 1.8 and 2.0 | M1 |
Justify result with correct calculations and argument ($-0.076$ and $0.087$ or $-0.77$ and $1$ respectively) | A1 | [2]
**(iii)**
Use the iterative process correctly at least once with $1.8 < p_n < 2.0$ | M1 |
Obtain final answer 1.89 | A1 |
Show sufficient iterations to at least 4 d.p. to justify 1.89 or show sign change in interval (1.885, 1.895) | A1 | [3]4 A curve has parametric equations
$$x = t ^ { 2 } + 3 t + 1 , \quad y = t ^ { 4 } + 1$$
The point $P$ on the curve has parameter $p$. It is given that the gradient of the curve at $P$ is 4 .\\
(i) Show that $p = \sqrt [ 3 ] { } ( 2 p + 3 )$.\\
(ii) Verify by calculation that the value of $p$ lies between 1.8 and 2.0.\\
(iii) Use an iterative formula based on the equation in part (i) to find the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2015 Q4 [8]}}