| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Direct substitution into standard series |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question requiring standard techniques: part (i) uses the compound angle formula (routine A-level trigonometry), and part (ii) involves direct substitution into standard Maclaurin series for sin and cos from the formula booklet. While it's Further Maths content, it requires no problem-solving or novel insight—just methodical application of given formulas, making it easier than average overall. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Expand to \(\sin 2x \cos\frac{4\pi}{x} + \cos 2x \sin\frac{4\pi}{x}\) | B1 | |
| Clearly replace \(\cos\frac{4\pi}{x}\), \(\sin\frac{4\pi}{x}\) to A.G. | B1 | |
| (ii) Attempt to expand \(\cos 2x\) | M1 | Allow \(1 - 2x^2/2\) |
| Attempt to expand \(\sin 2x\) | M1 | Allow \(2x - 2x^3/3\) |
| Get \(\frac{1}{2}\sqrt{2}(1 + 2x - 2x^2 - 4x^3/3)\) | A1 | Four correct unsimplified terms in any order; allow bracket; AEEF SR Reasonable attempt at \(f'(0)\) for \(n = 0\) to 3 |
| Attempt to replace their values in Maclaurin | M1 | |
| Get correct answer only | A1 |
**(i)** Expand to $\sin 2x \cos\frac{4\pi}{x} + \cos 2x \sin\frac{4\pi}{x}$ | B1 |
Clearly replace $\cos\frac{4\pi}{x}$, $\sin\frac{4\pi}{x}$ to A.G. | B1 |
**(ii)** Attempt to expand $\cos 2x$ | M1 | Allow $1 - 2x^2/2$
Attempt to expand $\sin 2x$ | M1 | Allow $2x - 2x^3/3$
Get $\frac{1}{2}\sqrt{2}(1 + 2x - 2x^2 - 4x^3/3)$ | A1 | Four correct unsimplified terms in any order; allow bracket; AEEF SR Reasonable attempt at $f'(0)$ for $n = 0$ to 3 | M1 |
Attempt to replace their values in Maclaurin | M1 |
Get correct answer only | A1 |2 (i) Given that $\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)$, show that $\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )$.\\
(ii) Hence find the first four terms of the Maclaurin series for $\mathrm { f } ( x )$. [You may use appropriate results given in the List of Formulae.]
\hfill \mbox{\textit{OCR FP2 2007 Q2 [5]}}