| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Iterative method for parameter value |
| Difficulty | Standard +0.3 This is a standard parametric equations question with routine iterative method application. Part (a) is simple substitution, part (b) is straightforward evaluation, part (c) follows a standard iterative procedure with given initial value, and part (d) requires dy/dx = (dy/dt)/(dx/dt). All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate \(y\) to 3 and confirm \(p = \dfrac{1}{2\sin 2p}\) | B1 | AG |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider sign of \(p - \dfrac{1}{2\sin 2p}\) or equivalent for 0.5 and 0.6 | M1 | |
| Obtain \(-0.09...\) and \(0.06...\) or equivalents and justify conclusion | A1 | AG |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | Need to see \(0.55494...\) |
| Obtain final answer \(0.557\) only | A1 | Allow recovery. Allow if iterations are to 4sf. Allow if insufficient iterations seen. |
| Show sufficient iterations to 5 s.f. to justify answer or show sign change in interval \([0.5565,\ 0.5575]\) | A1 | If not starting at \(0.55\) then max marks M1A1A0 |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\dfrac{dx}{dt} = 4 + 2e^{2t}\) | B1 | |
| Use product rule to find \(\dfrac{dy}{dt}\) | M1 | Must be of the form \(p\sin 2t + qt\cos 2t\) |
| Obtain \(6\sin 2t + 12t\cos 2t\) | A1 | Allow unsimplified. |
| Divide to obtain \(\dfrac{dy}{dx}\) using *their* \(\dfrac{dy}{dt}\) and \(\dfrac{dx}{dt}\) correctly | DM1 | Must have either B1 or previous M1. |
| Obtain \(0.826\) | A1 | AWRT |
| Total: 5 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate $y$ to 3 and confirm $p = \dfrac{1}{2\sin 2p}$ | **B1** | AG |
| | **Total: 1** | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of $p - \dfrac{1}{2\sin 2p}$ or equivalent for 0.5 and 0.6 | **M1** | |
| Obtain $-0.09...$ and $0.06...$ or equivalents and justify conclusion | **A1** | AG |
| | **Total: 2** | |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | **M1** | Need to see $0.55494...$ |
| Obtain final answer $0.557$ only | **A1** | Allow recovery. Allow if iterations are to 4sf. Allow if insufficient iterations seen. |
| Show sufficient iterations to 5 s.f. to justify answer or show sign change in interval $[0.5565,\ 0.5575]$ | **A1** | If not starting at $0.55$ then max marks M1A1A0 |
| | **Total: 3** | |
## Question 7(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\dfrac{dx}{dt} = 4 + 2e^{2t}$ | **B1** | |
| Use product rule to find $\dfrac{dy}{dt}$ | **M1** | Must be of the form $p\sin 2t + qt\cos 2t$ |
| Obtain $6\sin 2t + 12t\cos 2t$ | **A1** | Allow unsimplified. |
| Divide to obtain $\dfrac{dy}{dx}$ using *their* $\dfrac{dy}{dt}$ and $\dfrac{dx}{dt}$ correctly | **DM1** | Must have either B1 or previous M1. |
| Obtain $0.826$ | **A1** | AWRT |
| | **Total: 5** | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-10_465_785_260_680}
The diagram shows the curve with parametric equations
$$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$
for $0 \leqslant t \leqslant 1$. The point $P$ on the curve has parameter $p$ and $y$-coordinate 3 .
\begin{enumerate}[label=(\alph*)]
\item Show that $p = \frac { 1 } { 2 \sin 2 p }$.
\item Show by calculation that the value of $p$ lies between 0.5 and 0.6 .
\item Use an iterative formula, based on the equation in part (a), to find the value of $p$ correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
\item Find the gradient of the curve at $P$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q7 [11]}}