| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | March |
| Marks | 11 |
| Topic | Partial Fractions |
| Type | Repeated linear factor with series expansion |
| Difficulty | Standard +0.8 This question combines partial fractions with a repeated linear factor (requiring three separate fractions A/(1-2x) + B/(2+x) + C/(2+x)²) with binomial series expansion of each term and validity conditions. While each component is A-level standard, the combination of algebraic manipulation, series expansion of multiple terms, and determining the intersection of validity ranges makes this moderately above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{A}{(1-2x)} + \frac{B}{(2+x)} + \frac{C}{(2+x)^2}\); \(A(2+x)^2 + B(1-2x)(2+x) + C(1-2x) = 5 + 4x - 3x^2\); \(A = 1\); \(B = 2\); \(C = -3\) | M1, M1, A1, A1, A1 | Correct partial fractions; Correct equation; Correct value for \(A\); Correct value for \(B\); Correct value for \(C\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 2x + 4x^2\); \(0.5(1 + 0.5x)^{-1}\) or \(0.25(1 + 0.5x)^{-2}\); \(1 - 0.5x + 0.25x^2\); \(1 - x + 0.75x^2\); \(\frac{5}{4} + \frac{9}{4}x + \frac{59}{16}x^2\) | B1, B1, M1, M1, A1FT | Correct expansion of \((1-2x)^{-1}\); At least one correct expression; Attempt expansion of \((1 + 0.5x)^{-1}\); Attempt expansion of \((1 + 0.5x)^{-2}\); Correct expansion, following their A, B and C; Any equiv, including decimals |
| Answer | Marks | Guidance |
|---|---|---|
| \( | x | < 0.5\) |
**(i)**
| $\frac{A}{(1-2x)} + \frac{B}{(2+x)} + \frac{C}{(2+x)^2}$; $A(2+x)^2 + B(1-2x)(2+x) + C(1-2x) = 5 + 4x - 3x^2$; $A = 1$; $B = 2$; $C = -3$ | M1, M1, A1, A1, A1 | Correct partial fractions; Correct equation; Correct value for $A$; Correct value for $B$; Correct value for $C$ |
**(ii)**
| $1 + 2x + 4x^2$; $0.5(1 + 0.5x)^{-1}$ or $0.25(1 + 0.5x)^{-2}$; $1 - 0.5x + 0.25x^2$; $1 - x + 0.75x^2$; $\frac{5}{4} + \frac{9}{4}x + \frac{59}{16}x^2$ | B1, B1, M1, M1, A1FT | Correct expansion of $(1-2x)^{-1}$; At least one correct expression; Attempt expansion of $(1 + 0.5x)^{-1}$; Attempt expansion of $(1 + 0.5x)^{-2}$; Correct expansion, following their A, B and C; Any equiv, including decimals |
**(iii)**
| $|x| < 0.5$ | B1 | Any equivalent notation |9 (i) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.\\
(ii) Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.\\
(iii) State the set of values for which the expansion in part (ii) is valid.
\hfill \mbox{\textit{OCR H240/01 2018 Q9 [11]}}