| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Parametric or substitution with partial fractions |
| Difficulty | Challenging +1.2 This is a Further Maths question combining partial fractions with the Weierstrass substitution (t = tan(x/2)), which is a standard FP2 technique. Part (i) is routine partial fractions decomposition. Part (ii) requires applying the given substitution formula, using the result from (i), and evaluating definite integrals—all standard procedures for this topic with no novel insight required. The multi-step nature and FP2 content place it above average difficulty, but it's a textbook application of well-known methods. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Write as \(\frac{A}{t} + \frac{B}{t^2} + \frac{Ct+D}{t^2+1}\) | M1 | Allow \(\frac{At+B}{t^2}\); justify \(\frac{B}{t^2} + \frac{D}{1+t^2}\) if only used |
| Equate \(At(t^2+1) + B(t^2+1) + (Ct+D)t^2\) to \(1-t^2\) | M1\(\sqrt{}\) | |
| Insert \(t\) values / equate coefficients | M1 | Lead to at least two constant values |
| \(A=C=0\), \(B=1\), \(D=-2\) | A1 | |
| SR | Other methods leading to correct PF can earn 4 marks; 2 M marks for reasonable method going wrong |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Derive or quote \(\cos x\) in terms of \(t\) | B1 | |
| Derive or quote \(dx = \frac{2\,dt}{1+t^2}\) | B1 | |
| Sub. into correct PF | M1 | Allow \(k\,\frac{(1-t^2)}{(t^2(t^2+t^2))}\) or equivalent |
| Integrate to \(-\frac{1}{t} - 2\tan^{-1}t\) | A1\(\sqrt{}\) | From their \(k\) |
| Use limits to clearly get AG | A1 |
## Question 7:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Write as $\frac{A}{t} + \frac{B}{t^2} + \frac{Ct+D}{t^2+1}$ | M1 | Allow $\frac{At+B}{t^2}$; justify $\frac{B}{t^2} + \frac{D}{1+t^2}$ if only used |
| Equate $At(t^2+1) + B(t^2+1) + (Ct+D)t^2$ to $1-t^2$ | M1$\sqrt{}$ | |
| Insert $t$ values / equate coefficients | M1 | Lead to at least two constant values |
| $A=C=0$, $B=1$, $D=-2$ | A1 | |
| | SR | Other methods leading to correct PF can earn 4 marks; 2 M marks for reasonable method going wrong |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Derive or quote $\cos x$ in terms of $t$ | B1 | |
| Derive or quote $dx = \frac{2\,dt}{1+t^2}$ | B1 | |
| Sub. into correct PF | M1 | Allow $k\,\frac{(1-t^2)}{(t^2(t^2+t^2))}$ or equivalent |
| Integrate to $-\frac{1}{t} - 2\tan^{-1}t$ | A1$\sqrt{}$ | From their $k$ |
| Use limits to clearly get AG | A1 | |
---7 (i) Express $\frac { 1 - t ^ { 2 } } { t ^ { 2 } \left( 1 + t ^ { 2 } \right) }$ in partial fractions.\\
(ii) Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that
$$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 - \cos x } \mathrm {~d} x = \sqrt { 3 } - 1 - \frac { 1 } { 6 } \pi$$
\hfill \mbox{\textit{OCR FP2 2007 Q7 [9]}}