| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Iterative method for special point |
| Difficulty | Challenging +1.2 This question combines implicit differentiation (standard A-level technique) with an iterative method to find a coordinate. Part (a) requires setting dy/dx=0 and solving e^(2x)=18, which is straightforward. Parts (b)-(d) involve algebraic rearrangement and applying a given iterative formula—mechanical processes requiring care but no novel insight. The multi-step nature and iteration component elevate it slightly above average difficulty. |
| Spec | 1.07s Parametric and implicit differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate \(y^3\) to obtain \(3y^2\dfrac{dy}{dx}\) | B1 | |
| Differentiate complete equation to produce at least one term involving \(\dfrac{dy}{dx}\) using implicit differentiation | M1 | |
| Obtain \(2e^{2x} - 18 + 3y^2\dfrac{dy}{dx} + \dfrac{dy}{dx} = 0\) | A1 | |
| Substitute \(\dfrac{dy}{dx}=0\) to obtain either \(p=\frac{1}{2}\ln 9\) or \(p=\ln 3\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute value of \(p\) in original equation and rearrange as far as \(y^3=\ldots\) or \(q^3=\ldots\) | M1 | Allow in terms of \(\ln 9\) |
| Obtain given result \(q=\sqrt[3]{2+18\ln 3 - q}\) or \(y=\sqrt[3]{2+18\ln 3 - y}\) with sufficient detail | A1 | AG |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider sign of \(q - \sqrt[3]{2+18\ln 3 - q}\) or equivalent for \(2.5\) and \(3.0\) | M1 | |
| Obtain \(-0.18...\) and \(0.34...\) with sufficient detail and justify conclusion | A1 | OE |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(q = 2.673\) | A1 | Answer required to exactly 4 s.f. |
| Show sufficient iterations to 6 sf to justify answer or show sign change in the interval \([2.6725, \ 2.6735]\) | A1 | |
| Total | 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate $y^3$ to obtain $3y^2\dfrac{dy}{dx}$ | B1 | |
| Differentiate complete equation to produce at least one term involving $\dfrac{dy}{dx}$ using implicit differentiation | M1 | |
| Obtain $2e^{2x} - 18 + 3y^2\dfrac{dy}{dx} + \dfrac{dy}{dx} = 0$ | A1 | |
| Substitute $\dfrac{dy}{dx}=0$ to obtain either $p=\frac{1}{2}\ln 9$ or $p=\ln 3$ | A1 | |
| **Total: 4** | | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute value of $p$ in original equation and rearrange as far as $y^3=\ldots$ or $q^3=\ldots$ | M1 | Allow in terms of $\ln 9$ |
| Obtain given result $q=\sqrt[3]{2+18\ln 3 - q}$ or $y=\sqrt[3]{2+18\ln 3 - y}$ with sufficient detail | A1 | AG |
| **Total: 2** | | |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of $q - \sqrt[3]{2+18\ln 3 - q}$ or equivalent for $2.5$ and $3.0$ | M1 | |
| Obtain $-0.18...$ and $0.34...$ with sufficient detail and justify conclusion | A1 | OE |
| **Total: 2** | | |
## Question 7(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | **M1** | |
| Obtain final answer $q = 2.673$ | **A1** | Answer required to exactly 4 s.f. |
| Show sufficient iterations to 6 sf to justify answer or show sign change in the interval $[2.6725, \ 2.6735]$ | **A1** | |
| **Total** | **3** | |7 The curve with equation $\mathrm { e } ^ { 2 x } - 18 x + y ^ { 3 } + y = 11$ has a stationary point at $( p , q )$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $p$.
\item Show that $q = \sqrt [ 3 ] { 2 + 18 \ln 3 - q }$.
\item Show by calculation that the value of $q$ lies between 2.5 and 3.0.
\item Use an iterative formula, based on the equation in (b), to find the value of $q$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.\\
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 2023 Q7 [11]}}