| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Simplify algebraic fractions by addition or subtraction |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation question requiring factorization and combining fractions over a common denominator. Part (i) is simple factorization and cancellation; part (ii) is routine fraction subtraction with no conceptual difficulty. Both are standard textbook exercises requiring only mechanical algebraic skills, making this easier than average for A-level. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 - 3x + 2 = (x-1)(x-2)\) or \((1-x)(2-x)\) | B1 | |
| Obtain \(-\frac{1}{x-2}\) or \(\frac{1}{2-x}\) or \(\frac{-1}{x-2}\) or \(\frac{1}{-(x-2)}\) | B1 | Not \(\frac{-1}{-(2-x)}\); Accept WW |
| Total: [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt single fraction or 2 fractions with same relevant denominator | M1 | e.g. \((x-1)(x-4)[(x-3)\) or \((x-3)^2]\) |
| Fully correct fraction(s) before any simplification | A1 | |
| Relevant numerator \(= 3x - 9\) or \(3x^2 - 18x + 27\) | B1 | Can award if no denominator |
| Final answer \(= \frac{3}{(x-1)(x-4)}\) or \(\frac{3}{x^2-5x+4}\) | A1 | |
| Total: [4] | ||
| S.R. If partial fractions used: \(-\frac{1}{x-1}+\frac{2}{x-3}\) | (M1)(A1) | |
| \(\frac{2}{x-3}-\frac{1}{x-4}\) | (A1) | |
| \(-\frac{1}{x-1}+\frac{1}{x-4}\) ISW | (A1) |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 3x + 2 = (x-1)(x-2)$ or $(1-x)(2-x)$ | B1 | |
| Obtain $-\frac{1}{x-2}$ or $\frac{1}{2-x}$ or $\frac{-1}{x-2}$ or $\frac{1}{-(x-2)}$ | B1 | Not $\frac{-1}{-(2-x)}$; Accept WW |
| **Total: [2]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt single fraction or 2 fractions with same relevant denominator | M1 | e.g. $(x-1)(x-4)[(x-3)$ or $(x-3)^2]$ |
| Fully correct fraction(s) before any simplification | A1 | |
| Relevant numerator $= 3x - 9$ or $3x^2 - 18x + 27$ | B1 | Can award if no denominator |
| Final answer $= \frac{3}{(x-1)(x-4)}$ or $\frac{3}{x^2-5x+4}$ | A1 | |
| **Total: [4]** | | |
| S.R. If partial fractions used: $-\frac{1}{x-1}+\frac{2}{x-3}$ | (M1)(A1) | |
| $\frac{2}{x-3}-\frac{1}{x-4}$ | (A1) | |
| $-\frac{1}{x-1}+\frac{1}{x-4}$ ISW | (A1) | |
---1 Simplify\\
(i) $\frac { 1 - x } { x ^ { 2 } - 3 x + 2 }$,\\
(ii) $\frac { ( x + 1 ) } { ( x - 1 ) ( x - 3 ) } - \frac { ( x - 5 ) } { ( x - 3 ) ( x - 4 ) }$.
\hfill \mbox{\textit{OCR C4 2012 Q1 [6]}}