| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring partial fractions with an irreducible quadratic factor, followed by integration involving arctan. While the partial fractions decomposition is systematic, students must handle the quadratic term correctly and recognize the arctan integral form. This is moderately challenging but standard for FP2, placing it above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Express as \(\frac{A}{(x-1)} + \frac{Bx+C}{(x^2+9)}\) | M1 | Allow \(C=0\) here |
| Equate \((x^2+9x) = A(x^2+9) + (Bx+C)(x-1)\) | M1√ | May imply above line; on their P.F. |
| Sub. for \(x\) or equate coeff. | M1 | Must lead to at least 3 coeff.; allow cover-up method for \(A\) |
| Get \(A=1, B=0, C=9\) | A1 | cao from correct method |
| (ii) Get \(A\ln(x-1)\) | B1√ | On their \(A\) |
| Get \(C/3 \tan^{-1}(x/3)\) | B1√ | On their \(C\); condone no constant; ignore any \(B \neq 0\) |
**(i)** Express as $\frac{A}{(x-1)} + \frac{Bx+C}{(x^2+9)}$ | M1 | Allow $C=0$ here
Equate $(x^2+9x) = A(x^2+9) + (Bx+C)(x-1)$ | M1√ | May imply above line; on their P.F.
Sub. for $x$ or equate coeff. | M1 | Must lead to at least 3 coeff.; allow cover-up method for $A$
Get $A=1, B=0, C=9$ | A1 | cao from correct method
**(ii)** Get $A\ln(x-1)$ | B1√ | On their $A$
Get $C/3 \tan^{-1}(x/3)$ | B1√ | On their $C$; condone no constant; ignore any $B \neq 0$3 It is given that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( x - 1 ) \left( x ^ { 2 } + 9 \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence find $\int f ( x ) \mathrm { d } x$.
\hfill \mbox{\textit{OCR FP2 2007 Q3 [6]}}