| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Topic | Newton-Raphson method |
| Type | Show root in interval |
| Difficulty | Moderate -0.3 This is a straightforward application of standard A-level techniques: part (i) requires simple substitution to show a sign change (f(1) = -1, f(2) = 5), and part (ii) is a routine Newton-Raphson iteration starting from a given value. The function is simple (cubic polynomial), the derivative is easy, and the method is algorithmic with no conceptual challenges or novel insights required. Slightly easier than average due to its mechanical nature. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
**(i)** Attempt f(1) and f(2) — M1
Obtain $-1$ and $5$ and conclude correctly including reference to a root — A1 **[2]**
**(ii)** State derivative $= 3x^2 - 1$ — B1
Use correct Newton-Raphson formula — M1*
Obtain 1.5 and 1.3478 (or 1.348); 1.5, 1.3478, 1.3252, 1.3247, (1.3247) — A1
State 1.325 — A1 **[4]** **[6]**8 (i) Let $\mathrm { f } ( x ) = x ^ { 3 } - x - 1$. Use a sign change method to show that the equation $x ^ { 3 } - x - 1 = 0$ has a root between $x = 1$ and $x = 2$.\\
(ii) By taking $x = 1$ as a first approximation to this root, use the Newton-Raphson formula to find this root correct to 3 decimal places.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q8 [6]}}