| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Topic | Partial Fractions |
| Type | Partial fractions with quadratic in denominator |
| Difficulty | Standard +0.3 This is a standard partial fractions question with an irreducible quadratic factor, followed by routine integration. The decomposition is straightforward (equating coefficients or substitution), and the integration requires only basic techniques (ln and arctan). Slightly above average difficulty due to the quadratic factor, but still a textbook exercise with no novel problem-solving required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| - \(= 3\ln | x^2+1 | \) A1 FT Obtain correct integral, following *their* \(A\) |
| - \(+ \ln | x-2 | \) B1 FT Obtain correct integral of second fraction, following their \(C\) (allow brackets soi not modulus each time) |
| Answer | Marks | Guidance |
|---|---|---|
| - \((3\ln 2 + \ln 1) - (3\ln 1 + \ln 2)\) B1d\* Use or imply that \(\ln | -k | = \ln k\). B0 if using log laws with negative numbers |
**Question 8: Partial fractions of $\frac{7x^2-12x+1}{(x^2+1)(x-2)}$**
**(i)**
- $(Ax+B)(x-2) + C(x^2+1) = 7x^2 - 12x + 1$ **M1** Set up correct identity
- $A = 6$ **A1**
- $B = 0$ **A1**
- $C = 1$ **A1**
**(ii)**
- $\int\frac{7x^2-12x+1}{(x^2+1)(x-2)}\,\mathrm{d}x = \int\frac{6x}{x^2+1}\,\mathrm{d}x + \int\frac{1}{x-2}\,\mathrm{d}x$ **M1** Integrate first fraction to $k\ln(x^2+1)$
- $= 3\ln|x^2+1|$ **A1 FT** Obtain correct integral, following *their* $A$
- $+ \ln|x-2|$ **B1 FT** Obtain correct integral of second fraction, following their $C$ (allow brackets soi not modulus each time)
- **M1\*** Attempt correct use of limits in any changed function – could be just one of the two terms
- $(3\ln 2 + \ln 1) - (3\ln 1 + \ln 2)$ **B1d\*** Use or imply that $\ln|-k| = \ln k$. B0 if using log laws with negative numbers
- $= \ln 4$ **A1** Obtain $\ln 4$, or $2\ln 2$ (can follow B0)
**Total: 10 marks**8 (i) Express $\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }$ in the form $\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }$ where $A , B$ and $C$ are constants to be found.\\
(ii) Hence find the exact value of $\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2017 Q8 [10]}}