| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Direct multiplication of series |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring recall of standard series and multiplication of two known expansions. While it involves Further Maths content (Maclaurin series), the execution is mechanical: substitute into ln(1+x), then multiply term-by-term with e^x. No novel insight or complex manipulation required, just careful algebraic bookkeeping. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = \pm \sin x/(1+\cos x)\) | M1 | Reasonable attempt at chain at any stage |
| \(f''(x)\) using quotient/product rule | M1 | Reasonable attempt at quotient/product |
| \(f(0) = \ln 2, f'(0) = 0, f''(0) = -1/2\) | B1 | Any one correct from correct working |
| A1 | All three correct from correct working | |
| ALT: \(\ln(\cos 3x) = \ln(1 - 1/2(3x)^2) = -9/2 x^2\) | SC Use of standard cos and ln series can earn second M1 A1 | |
| \(\Rightarrow f(x) = -9/2 x^2\) | M1 | For use of Maclaurin soi |
| A1 | For correct series (condone \(a = -9/2 x^2\)) |
$f'(x) = \pm \sin x/(1+\cos x)$ | M1 | Reasonable attempt at chain at any stage
$f''(x)$ using quotient/product rule | M1 | Reasonable attempt at quotient/product
$f(0) = \ln 2, f'(0) = 0, f''(0) = -1/2$ | B1 | Any one correct from correct working
| A1 | All three correct from correct working
ALT: $\ln(\cos 3x) = \ln(1 - 1/2(3x)^2) = -9/2 x^2$ | | SC Use of standard cos and ln series can earn second M1 A1
$\Rightarrow f(x) = -9/2 x^2$ | M1 | For use of Maclaurin soi
| A1 | For correct series (condone $a = -9/2 x^2$)1 (i) Write down and simplify the first three non-zero terms of the Maclaurin series for $\ln ( 1 + 3 x )$.\\
(ii) Hence find the first three non-zero terms of the Maclaurin series for
$$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$
simplifying the coefficients.
\hfill \mbox{\textit{OCR FP2 Q1 [6]}}