| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with trigonometric or exponential functions |
| Difficulty | Standard +0.3 This is a straightforward Newton-Raphson application with standard functions. Part (i) requires simple substitution to verify sign change, and part (ii) involves one iteration of the Newton-Raphson formula with f(x) = x² - arctan(x), requiring differentiation (f'(x) = 2x - 1/(1+x²)) and calculator work. While it involves inverse trig, the mechanics are routine for FP2 students with no conceptual challenges or multi-step reasoning required. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07i Differentiate x^n: for rational n and sums1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(0.8) = -0.03\), \(f(0.9) = +0.077\) (accept \(-0.02\) to \(-0.04\)) | B1 | |
| Explain change of sign, graph etc. | B1 | |
| SR | Use \(x = \sqrt{(\tan^{-1}x)}\) and compare \(x\) to \(\sqrt{J(\tan^{-1}x)}\) for \(x = 0.8, 0.9\) | |
| B1 | Explain "change in sign" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate two terms; get \(2x - 1/(1+x^2)\) | B1 | |
| Use correct form of Newton-Raphson with \(0.8\), using their \(f'(x)\) | M1 | \(0.8 - f(0.8)/f'(0.8)\) |
| Use N-R to give one more approximation to 3 d.p. minimum | M1\(\sqrt{}\) | |
| \(x = 0.835\) | A1 | 3 d.p. — accept answer which rounds |
## Question 2:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(0.8) = -0.03$, $f(0.9) = +0.077$ (accept $-0.02$ to $-0.04$) | B1 | |
| Explain change of sign, graph etc. | B1 | |
| | SR | Use $x = \sqrt{(\tan^{-1}x)}$ and compare $x$ to $\sqrt{J(\tan^{-1}x)}$ for $x = 0.8, 0.9$ |
| | B1 | Explain "change in sign" |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate two terms; get $2x - 1/(1+x^2)$ | B1 | |
| Use correct form of Newton-Raphson with $0.8$, using their $f'(x)$ | M1 | $0.8 - f(0.8)/f'(0.8)$ |
| Use N-R to give one more approximation to 3 d.p. minimum | M1$\sqrt{}$ | |
| $x = 0.835$ | A1 | 3 d.p. — accept answer which rounds |
---2 It is given that $\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x$.\\
(i) Show by calculation that the equation $\mathrm { f } ( x ) = 0$ has a root in the interval $0.8 < x < 0.9$.\\
(ii) Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation to this root. Give your answer correct to 3 decimal places.
\hfill \mbox{\textit{OCR FP2 2007 Q2 [6]}}