| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Find stationary point coordinate |
| Difficulty | Standard +0.3 This question requires quotient rule differentiation, setting the derivative to zero, and applying a given iterative formula. While it involves multiple steps, each component is standard A-level technique: the differentiation is routine quotient rule, the rearrangement is provided in the 'show that' format, and the iteration is straightforward calculator work with no conceptual difficulty. Slightly above average due to the quotient rule complexity and multi-part nature, but well within typical P2 scope. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use quotient rule (or product rule) to find first derivative | M1 | Quotient: Must have a difference in the numerator and \((x^2+1)^2\) in the denominator |
| Obtain \(\dfrac{\frac{4}{x}(x^2+1) - 8x\ln x}{(x^2+1)^2}\) or equivalent | A1 | Product: Must see an application of the chain rule |
| State \(\frac{4}{x}(x^2+1) - 8x\ln x = 0\) or equivalent | A1 | Condone missing brackets if correct use is implied by correct work later |
| Carry out correct process to produce equation without ln, without any incorrect working | M1 | |
| Confirm \(m = e^{0.5(1+m^{-2})}\) or \(x = e^{0.5(1+x^{-2})}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use iterative formula correctly at least once | M1 | Should not be attempting to use \(x_0 = 0\), but if used and 'recovered' then SC M1 A1 – usually see \(m_1 = 1.6487\) |
| Obtain final answer \(1.895\) | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval \((1.8945, 1.8955)\) | A1 |
## Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quotient rule (or product rule) to find first derivative | M1 | Quotient: Must have a difference in the numerator and $(x^2+1)^2$ in the denominator |
| Obtain $\dfrac{\frac{4}{x}(x^2+1) - 8x\ln x}{(x^2+1)^2}$ or equivalent | A1 | Product: Must see an application of the chain rule |
| State $\frac{4}{x}(x^2+1) - 8x\ln x = 0$ or equivalent | A1 | Condone missing brackets if correct use is implied by correct work later |
| Carry out correct process to produce equation without ln, without any incorrect working | M1 | |
| Confirm $m = e^{0.5(1+m^{-2})}$ or $x = e^{0.5(1+x^{-2})}$ | A1 | |
## Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use iterative formula correctly at least once | M1 | Should not be attempting to use $x_0 = 0$, but if used and 'recovered' then SC M1 A1 – usually see $m_1 = 1.6487$ |
| Obtain final answer $1.895$ | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval $(1.8945, 1.8955)$ | A1 | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648}
The diagram shows the curve $y = \frac { 4 \ln x } { x ^ { 2 } + 1 }$ and its stationary point $M$. The $x$-coordinate of $M$ is $m$.\\
(i) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence show that $m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }$.\\
(ii) Use an iterative formula based on the equation in part (i) to find the value of $m$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\hfill \mbox{\textit{CAIE P2 2016 Q5 [8]}}