June 2022 Paper 2 Q15
15 Simplify fully \(\qquad \left(\dfrac{x}{2} + \dfrac{3x}{5}\right) \div \sqrt{\dfrac{x^6}{4}}\) [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Processes the brackets then divides | ||
| \(\dfrac{5x}{10} + \dfrac{6x}{10}\) | M1 | oe valid common denominator with both numerators correct eg \(\dfrac{10x}{20} + \dfrac{12x}{20}\) |
| \(\dfrac{11x}{10}\) | A1 | oe single term eg \(\dfrac{22x}{20}\) or \(1.1x\) may be implied eg by single term with roots evaluated that is equivalent to \(\dfrac{11}{5x^2}\) |
| \(\dfrac{x^{6 \div 2}}{2}\) or \(\dfrac{x^3}{2}\) | M1 | may be implied eg by multiplication by \(\dfrac{2}{x^3}\) |
| their \(\dfrac{11x}{10} \times \dfrac{2}{x^3}\) or \(\dfrac{22x}{10x^3}\) or \(\dfrac{22}{10x^2}\) or \(\dfrac{11x}{5x^3}\) or \(\dfrac{22}{10}x^{-2}\) | M1dep | oe multiplication eg \(\dfrac{11x}{10} \times 2x^{-3}\) their \(\dfrac{11x}{10}\) can be unprocessed dep on 2nd M1 |
| \(\dfrac{11}{5x^2}\) or \(\dfrac{11}{5}x^{-2}\) or \(2.2x^{-2}\) | A1 | allow \(2\dfrac{1}{5}x^{-2}\) or \(\dfrac{2.2}{x^2}\) |
| Alternative method 2 Divides then expands the brackets | ||
| \(\dfrac{x^{6 \div 2}}{2}\) or \(\dfrac{x^3}{2}\) | M1 | may be implied eg by multiplication by \(\dfrac{2}{x^3}\) |
| \(\left(\dfrac{x}{2} + \dfrac{3x}{5}\right) \times \dfrac{2}{x^3}\) | M1dep | oe multiplication eg \(\left(\dfrac{x}{2} + \dfrac{3x}{5}\right) \times 2x^{-3}\) |
| \(\dfrac{2x}{2x^3} + \dfrac{6x}{5x^3}\) or \(\dfrac{1}{x^2} + \dfrac{6}{5x^2}\) | M1dep | oe expansion of brackets |
| \(\dfrac{10x}{10x^3} + \dfrac{12x}{10x^3}\) or \(\dfrac{5}{5x^2} + \dfrac{6}{5x^2}\) or \(\dfrac{22x}{10x^3}\) or \(\dfrac{22}{10x^2}\) or \(\dfrac{11x}{5x^3}\) or \(\dfrac{22}{10}x^{-2}\) | M1dep | oe valid common denominator with both numerators correct eg \(\dfrac{10x^4}{10x^6} + \dfrac{12x^4}{10x^6}\) or \(\dfrac{22x^4}{10x^6}\) roots must be processed |
| \(\dfrac{11}{5x^2}\) or \(\dfrac{11}{5}x^{-2}\) or \(2.2x^{-2}\) | A1 | allow \(2\dfrac{1}{5}x^{-2}\) or \(\dfrac{2.2}{x^2}\) |
Additional guidance
| Any single fraction with roots evaluated that is equivalent to \(\dfrac{11}{5x^2}\) | 4 marks |
| Allow inclusion of \(\pm\) from the square root for up to 4 marks | |
| \(\dfrac{11}{5x^2}\) in working with answer \(\dfrac{11}{5}x^2\) | 4 marks |
| Alt 1 \(\dfrac{11x}{10}\) subsequently squared and not recovered | M1A1 M0M0A0 |