| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Compare Newton-Raphson with linear interpolation |
| Difficulty | Moderate -0.3 This is a straightforward application of two standard numerical methods with all necessary information provided. Part (a) requires simple linear interpolation between two given points using the formula, and part (b) applies one iteration of Newton-Raphson with the gradient given. Both are routine FP1 procedures requiring only substitution into formulas, though the multi-part structure and need to understand the geometric interpretation adds slight complexity beyond pure recall. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Mark |
| (a)(i) | Line joining points A and B | B1 |
| (a)(ii) | \(x_P = 2 + w\), \(\frac{w}{10} = \frac{5-2}{22-(-10)}\) \(x_P = 2 + 10 \times \frac{3}{32}\) \(x_P = 2.9375 = 2.9\) (to 1dp) | M1 A1 A1 |
| (b)(i) | Tangent at A drawn | B1 |
| (b)(ii) | \(x_Q = 2 - \frac{-10}{8}\) ... \(= 3.25\) | M1 A1 |
| Part | Answer/Working | Mark | Guidance |
|------|---|---|---|
| (a)(i) | Line joining points A and B | B1 | 1 mark; Must not be linked to Q |
| (a)(ii) | $x_P = 2 + w$, $\frac{w}{10} = \frac{5-2}{22-(-10)}$ $x_P = 2 + 10 \times \frac{3}{32}$ $x_P = 2.9375 = 2.9$ (to 1dp) | M1 A1 A1 | 3 marks; OE eg correct equation for AB with y replaced by 0; $2 + 10 \times \frac{3}{32}$ OE; CAO Must be 2.9 |
| (b)(i) | Tangent at A drawn | B1 | 1 mark; At least as far as meeting the x-axis. Accept reasonable attempt. Must not be linked to P |
| (b)(ii) | $x_Q = 2 - \frac{-10}{8}$ ... $= 3.25$ | M1 A1 | 2 marks; PI by 3.25 or 26/8 OE; CAO Must be 3.25 |5 The diagram below (not to scale) shows a part of a curve $y = \mathrm { f } ( x )$ which passes through the points $A ( 2 , - 10 )$ and $B ( 5,22 )$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item On the diagram, draw a line which illustrates the method of linear interpolation for solving the equation $\mathrm { f } ( x ) = 0$. The point of intersection of this line with the $x$-axis should be labelled $P$.
\item Calculate the $x$-coordinate of $P$. Give your answer to one decimal place.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item On the same diagram, draw a line which illustrates the Newton-Raphson method for solving the equation $\mathrm { f } ( x ) = 0$, with initial value $x _ { 1 } = 2$. The point of intersection of this line with the $x$-axis should be labelled $Q$.
\item Given that the gradient of the curve at $A$ is 8 , calculate the $x$-coordinate of $Q$. Give your answer as an exact decimal.\\
\includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-3_876_1063_1779_523}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2012 Q5 [7]}}