| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Combining Algebraic Fractions |
| Difficulty | Moderate -0.8 This is a straightforward algebraic fractions question requiring routine techniques: finding common denominators, combining fractions, and simplifying by factoring and canceling. Part (i) is basic manipulation, and part (ii) builds directly on it with standard factorization. No problem-solving insight needed, just careful algebraic execution—easier than average A-level questions. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{2(1+x)+3(3-x)}{(3-x)(1+x)}\) | B1 | Or \(\frac{2(1+x)}{(3-x)(1+x)}+\frac{3(3-x)}{(3-x)(1+x)}\); allow recovery from omission of brackets; brackets may be expanded in numerator |
| \(\frac{11-x}{(3-x)(1+x)}\) oe isw | B1 | Numerator must be simplified; B2 if unsupported; denominator may be in expanded form at either stage e.g. \(3+2x-x^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{(x+11)(x-3)}{(11+x)(11-x)}\) or \(\frac{(x+11)(x-3)}{(121-x^2)}\) | M1* | Allow \((x-11)(x+3)\) for numerator and/or \((x-11)(x+11)\) in denominator |
| Their \(\frac{11-x}{(3-x)(1+x)} \times\) their \(\frac{(x+11)(x-3)}{(11+x)(11-x)}\) | M1*dep | With at least one pair of their terms correctly cancelled out; allow if RH fraction only partially factorised |
| \(\frac{-1}{(1+x)}\) oe cao | A1 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{2(1+x)+3(3-x)}{(3-x)(1+x)}$ | B1 | Or $\frac{2(1+x)}{(3-x)(1+x)}+\frac{3(3-x)}{(3-x)(1+x)}$; allow recovery from omission of brackets; brackets may be expanded in numerator |
| $\frac{11-x}{(3-x)(1+x)}$ oe isw | B1 | Numerator must be simplified; **B2** if unsupported; denominator may be in expanded form at either stage e.g. $3+2x-x^2$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{(x+11)(x-3)}{(11+x)(11-x)}$ or $\frac{(x+11)(x-3)}{(121-x^2)}$ | M1* | Allow $(x-11)(x+3)$ for numerator and/or $(x-11)(x+11)$ in denominator |
| Their $\frac{11-x}{(3-x)(1+x)} \times$ their $\frac{(x+11)(x-3)}{(11+x)(11-x)}$ | M1*dep | With at least one pair of their terms correctly cancelled out; allow if RH fraction only partially factorised |
| $\frac{-1}{(1+x)}$ oe cao | A1 | |
---1 (i) Express $\frac { 2 } { 3 - x } + \frac { 3 } { 1 + x }$ as a single fraction in its simplest form.\\
(ii) Hence express $\left( \frac { 2 } { 3 - x } + \frac { 3 } { 1 + x } \right) \times \frac { x ^ { 2 } + 8 x - 33 } { 121 - x ^ { 2 } }$ as a single fraction in its lowest terms.
\hfill \mbox{\textit{OCR C4 2015 Q1 [5]}}