Higher June 2023 Paper 2 Q25
25 Show that \(\quad \dfrac{x - 5}{x - 2} + \dfrac{x + 5}{x + 2}\)
simplifies to \(\quad \dfrac{ax^2 - b}{x^2 - 4} \quad\) where \(a\) and \(b\) are integers. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{(x - 5)(x + 2)}{(x - 2)(x + 2)}\) and \(\dfrac{(x + 5)(x - 2)}{(x + 2)(x - 2)}\) | M1 | \((x - 2)(x + 2)\) or \(x^2 - 2x + 2x - 4\) must be seen (expansion may be seen in a grid) brackets in any order if the brackets are not shown for the numerators, expansions must be correct may be seen as a single fraction |
| \(x^2 - 5x + 2x - 10\) or \(x^2 - 3x - 10\) or \(x^2 + 5x - 2x - 10\) or \(x^2 + 3x - 10\) | M1 | correct expansion of \((x - 5)(x + 2)\) or \((x + 5)(x - 2)\) ignore denominators may be seen in a grid implied by \(2x^2 - 20\) if no errors seen in expansions |
| M2 seen with no errors and \(\dfrac{2x^2 - 20}{x^2 - 4}\) | A1 | allow M2 seen with no errors and \(a = 2 \quad b = 20\) |
Additional guidance
| Missing brackets must be recovered but condone missing closing bracket at the end of a numerator or denominator eg \(\dfrac{(x - 5)(x + 2)}{(x - 2)(x + 2)} + \dfrac{(x + 5)(x - 2)}{(x + 2)(x - 2}\) | 1st M1 |
| 2nd M1 is awarded for four correct terms even if subsequently simplified incorrectly | |
| For terms seen in a grid, signs must be correct (allow eg \(2x\) for \(+\,2x\)) | |
| For 1st M1 allow multiplication signs | |
| After M2A1 ignore incorrect values stated eg \(a = 2 \quad b = -20\) | |
| \(\dfrac{2x^2 - 20}{x^2 - 4}\) may come from wrong working or incomplete working eg \(\dfrac{(x - 5)(x + 2)}{(x - 2)(x + 2)} + \dfrac{(x + 5)(x - 2)}{(x + 2)(x - 2)}\) \(\dfrac{x^2 - 10 + x^2 - 10}{x^2 - 4} = \dfrac{2x^2 - 20}{x^2 - 4}\) | M1 M0A0 |