| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Multiple roots and starting value selection |
| Difficulty | Challenging +1.2 This is a Further Maths question testing understanding of Newton-Raphson method behavior and curve sketching from implicit equations. Part (i) requires conceptual understanding of convergence patterns (standard for FP2) rather than computation. Part (ii) involves sketching y² = f(x), which requires recognizing the relationship between f(x) and its square root, identifying where f(x) ≥ 0, and finding turning points—a moderately challenging visualization task. The question is above average difficulty due to the conceptual nature and being Further Maths content, but follows predictable FP2 patterns without requiring exceptional insight. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.09d Newton-Raphson method1.09e Iterative method failure: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) State \((x=) \, a\). None of roots | B1, B1 | No explanation needed |
| (b) Impossible to say. All roots can be derived | B1, B1 | Some discussion of values close to 1 or 2 or central leading to correct conclusion |
| (ii) Sketch showing: | B1 | Correct \(x\) for \(y=0\); allow 0.591, 1.59, 2.31 |
| B1 | Turning at \((1, 0.8)\) and/or \((1, -0.8)\) | |
| B1 | Meets \(x\)-axis at \(90°\) | |
| B1 | Symmetry in \(x\)-axis; allow |
(i) (a) State $(x=) \, a$. None of roots | B1, B1 | No explanation needed
(b) Impossible to say. All roots can be derived | B1, B1 | Some discussion of values close to 1 or 2 or central leading to correct conclusion
(ii) Sketch showing: | B1 | Correct $x$ for $y=0$; allow 0.591, 1.59, 2.31
| B1 | Turning at $(1, 0.8)$ and/or $(1, -0.8)$
| B1 | Meets $x$-axis at $90°$
| B1 | Symmetry in $x$-axis; allow\includegraphics{figure_5}
The diagram shows the curve with equation $y = f(x)$, where
$$f(x) = 2x^3 - 9x^2 + 12x - 4.36.$$
The curve has turning points at $x = 1$ and $x = 2$ and crosses the $x$-axis at $x = \alpha$, $x = \beta$ and $x = \gamma$, where $0 < \alpha < \beta < \gamma$.
\begin{enumerate}[label=(\roman*)]
\item The Newton-Raphson method is to be used to find the roots of the equation $f(x) = 0$, with $x_1 = k$.
\begin{enumerate}[label=(\alph*)]
\item To which root, if any, would successive approximations converge in each of the cases $k < 0$ and $k = 1$? [2]
\item What happens if $1 < k < 2$? [2]
\end{enumerate}
\item Sketch the curve with equation $y^2 = f(x)$. State the coordinates of the points where the curve crosses the $x$-axis and the coordinates of any turning points. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q5 [8]}}