June 2018 Paper 2 Q12
12
(a) Write \(\quad \dfrac{7}{9x} + \dfrac{2}{3x^2} \quad\) as a single fraction in its simplest form. [3 marks]
(b) Show that \(\qquad \dfrac{x^4}{x + 4} \times \dfrac{x + 2}{x} \div \dfrac{x^2}{3x + 12}\)
simplifies to the form \(\ ax^2 + bx \ \) where \(a\) and \(b\) are integers. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Valid common denominator with at least one numerator correct | M1 | eg \(\dfrac{7x}{9x^2}\) and \(\dfrac{\mathrm{a}}{9x^2}\) or \(\dfrac{7x + \mathrm{a}}{9x^2}\) or \(\dfrac{\mathrm{b}}{9x \times 3x^2}\) and \(\dfrac{2 \times 9x}{9x \times 3x^2}\) numerators and denominators may be seen as products a can be numerical or algebraic b can be numerical or algebraic |
| Valid common denominator with both numerators correct | M1dep | \(\dfrac{7x}{9x^2}\) and \(\dfrac{6}{9x^2}\) or \(\dfrac{7 \times 3x^2}{9x \times 3x^2}\) and \(\dfrac{2 \times 9x}{9x \times 3x^2}\) numerators and denominators may be seen as products |
| \(\dfrac{7x + 6}{9x^2}\) or \(\dfrac{7x + 6}{(3x)^2}\) with no further work | A1 |
Additional guidance
| \(\dfrac{21x^2 + 18x}{27x^3}\) or \(\dfrac{21x + 18}{27x^2}\) or \(\dfrac{7x^2 + 6x}{9x^3}\) | M2A0 |
| \(\dfrac{7x^{-1} + 6x^{-2}}{9}\) | M2A0 |
| \(7x + 6 / 9x^2\) | M2A0 |
| Answer | Mark | Comments |
|---|---|---|
| Changes division to multiplication and inverts to \(\dfrac{3x + 12}{x^2}\) | M1 | may be implied |
| \((3x + 12 =)\ 3(x + 4)\) | M1 | may be implied |
| Correct expression written as a single fraction or a product must have factor \((x + 4)\) in a numerator and denominator \(x + 4\) or correct expression written as a single fraction or a product must have denominator \(x^3\) or \(x^2\) or \(x\) or 1 | A1 | may be implied by final A1 eg \(\ \dfrac{3x(x + 2)(x + 4)}{x + 4}\) or \(\dfrac{(3x^2 + 6x)(x + 4)}{x + 4}\) or \(\dfrac{x}{x + 4} \times \dfrac{x + 2}{1} \times 3(x + 4)\) or \(\dfrac{x}{x + 4} \times 3(x + 2)(x + 4)\) or \(\dfrac{3x^4(x + 2)}{x^3}\) or \(x^4 \times \dfrac{x + 2}{x} \times \dfrac{3}{x^2}\) or \(\dfrac{(x + 2)}{x^3} \times 3x^4\) or \(\dfrac{3x^3(x + 2)}{x^2}\) or \(\dfrac{3x^2(x + 2)}{x}\) or \(\dfrac{3x(x + 2)}{1}\) or \(x \times (x + 2) \times 3\) or \(3x(x + 2)\) |
| \(3x^2 + 6x\) | A1 | SC2 \(\ \dfrac{x(x + 2)(3x + 12)}{x + 4}\) |
Additional guidance
| The list of examples in the first A1 is not exhaustive | |
| \(3x^2 + 6x\) with no incorrect working | 4 marks |