| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with verification |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard A-level techniques: finding second derivative and solving for k (routine calculus), sign-change verification (basic numerical methods), Newton-Raphson iteration (standard algorithm application), and error bounds checking. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07p Points of inflection: using second derivative1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
A curve has equation $y = 2\ln(k - 3x) + x^2 - 3x$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Given that the curve has a point of inflection where $x = 1$, show that $k = 6$. [5]
It is also given that the curve intersects the $x$-axis at exactly one point.
\item Show by calculation that the $x$-coordinate of this point lies between 0.5 and 1.5. [2]
\item Use the Newton-Raphson method, with initial value $x_0 = 1$, to find the $x$-coordinate of the point where the curve intersects the $x$-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places. [3]
\item By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2020 Q4 [11]}}