| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Method of differences |
| Difficulty | Challenging +1.2 This is a standard reduction formula question requiring integration of tan^n(x), establishing a recurrence relation using the identity tan²x = sec²x - 1, then applying it iteratively. While it involves multiple steps and Further Maths content, the technique is well-practiced in FP2 and follows a predictable pattern with clear guidance ('considering I_n + I_{n-2}'). |
| Spec | 1.08i Integration by parts |
| Answer | Marks |
|---|---|
| (a) Obtain expression of form \(\frac{a \tan \alpha}{b + c \tan^2 \alpha}\) | M1 |
| State correct \(\frac{2 \tan \alpha}{1 - \tan^2 \alpha}\) | A1 |
| Attempt to produce polynomial equation in \(\tan \alpha\) | M1 |
| Obtain at least one correct value of \(\tan \alpha\) | A1 |
| Obtain \(41.8\) | A1 |
| Obtain \(138.2\) and no other values between 0 and 180 | A1 |
| (b)(i) State \(\frac{2}{6}\) | B1 |
| (ii) Attempt use of identity linking \(\cot^2 \beta\) and \(\cosec^2 \beta\) | M1 |
| Obtain \(\frac{13}{36}\) | A1 |
| | |
|---|---|
| (a) Obtain expression of form $\frac{a \tan \alpha}{b + c \tan^2 \alpha}$ | M1 |
| State correct $\frac{2 \tan \alpha}{1 - \tan^2 \alpha}$ | A1 |
| Attempt to produce polynomial equation in $\tan \alpha$ | M1 |
| Obtain at least one correct value of $\tan \alpha$ | A1 |
| Obtain $41.8$ | A1 |
| Obtain $138.2$ and no other values between 0 and 180 | A1 |
| (b)(i) State $\frac{2}{6}$ | B1 |
| (ii) Attempt use of identity linking $\cot^2 \beta$ and $\cosec^2 \beta$ | M1 |
| Obtain $\frac{13}{36}$ | A1 |5 It is given that, for $n \geqslant 0$,
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$
(i) By considering $I _ { n } + I _ { n - 2 }$, or otherwise, show that, for $n \geqslant 2$,
$$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
(ii) Find $I _ { 4 }$ in terms of $\pi$.
\hfill \mbox{\textit{OCR FP2 2008 Q5 [8]}}