| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sign Change & Interval Methods |
| Type | Linear Interpolation Only |
| Difficulty | Moderate -0.3 This is a straightforward application of standard numerical methods from FP1. Part (a) requires simple sign change verification by substituting x=1 and x=2. Part (b) is a direct application of the linear interpolation formula with no complications. While it's a Further Maths topic, the execution is mechanical and requires no problem-solving insight, making it slightly easier than an average A-level question. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y - f(2) = \frac{f(2)-f(1)}{2-1}(x-2)\) or \(y - f(1) = \frac{f(2)-f(1)}{2-1}(x-1)\) or \(y = \frac{f(2)-f(1)}{2-1}x + c\) with attempt to find \(c\) | M1 | Correct straight line method. Must be a correct statement using their \(f(2)\) and \(f(1)\). Can be implied by working below. NB \(m = 4.011105235\) |
| \(y=0 \Rightarrow \alpha = \frac{f(2)}{f(1)-f(2)} + 2\) or \(\alpha = \frac{f(1)}{f(1)-f(2)}+1\) | A1\(\checkmark\) | Correct follow through expression to find \(\alpha\). Method can be implied. (Can be implied by awrt 1.61.) |
| \(= 1.611726037...\) | A1 | awrt 1.61 |
| [3] |
## Question 6(b) – Aliter Way 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y - f(2) = \frac{f(2)-f(1)}{2-1}(x-2)$ or $y - f(1) = \frac{f(2)-f(1)}{2-1}(x-1)$ or $y = \frac{f(2)-f(1)}{2-1}x + c$ with attempt to find $c$ | M1 | Correct straight line method. Must be a correct statement using their $f(2)$ and $f(1)$. Can be implied by working below. NB $m = 4.011105235$ |
| $y=0 \Rightarrow \alpha = \frac{f(2)}{f(1)-f(2)} + 2$ or $\alpha = \frac{f(1)}{f(1)-f(2)}+1$ | A1$\checkmark$ | Correct follow through expression to find $\alpha$. Method can be implied. (Can be implied by awrt 1.61.) |
| $= 1.611726037...$ | A1 | awrt 1.61 |
| | **[3]** | |
---6.
$$f ( x ) = \tan \left( \frac { x } { 2 } \right) + 3 x - 6 , \quad - \pi < x < \pi$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $[ 1,2 ]$.
\item Use linear interpolation once on the interval $[ 1,2 ]$ to find an approximation to $\alpha$. Give your answer to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2012 Q6 [5]}}