June 2025 Paper 2 Q13
13 Rearrange \(\quad x = \dfrac{3k^2}{5k^2 - 2} \quad\) to make \(k\) the subject. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: eliminates fraction | ||
| \(x(5k^2 - 2) = 3k^2\) | M1 | oe elimination of fraction eg \(5xk^2 - 2x = 3k^2\) |
| \(5xk^2 - 3k^2 = 2x\) | M1dep | oe expansion of bracket and collection of terms in \(k^2\) eg \(-2x = 3k^2 - 5xk^2\) |
| \(k^2(5x - 3) = 2x\) | M1dep | oe full factorisation eg \(-2x = k^2(3 - 5x)\) may be implied eg \(k^2 = \dfrac{2x}{5x - 3}\) or \(\sqrt{\dfrac{2x}{5x - 3}}\) |
| \(k = \sqrt{\dfrac{2x}{5x - 3}}\) | A1 | oe with \(k\) the subject eg \(k = \sqrt{\dfrac{-2x}{3 - 5x}}\) or \(k = \dfrac{\sqrt{2x}}{\sqrt{5x - 3}}\) or \(\left(\dfrac{2x}{5x - 3}\right)^{\frac{1}{2}} = k\) |
| Alternative method 2: takes reciprocal of both sides | ||
| \(\dfrac{1}{x} = \dfrac{5k^2 - 2}{3k^2}\) | M1 | oe reciprocal of both sides eg \(\dfrac{1}{x} = \dfrac{5k^2}{3k^2} - \dfrac{2}{3k^2}\) |
| \(\dfrac{2}{3k^2} = \dfrac{5x - 3}{3x}\) | M1dep | oe collection of term in \(\dfrac{1}{k^2}\) eg \(\dfrac{2}{3k^2} = \dfrac{5}{3} - \dfrac{1}{x}\) |
| \(\dfrac{3k^2}{2} = \dfrac{3x}{5x - 3}\) | M1dep | oe reciprocal of both sides may be implied eg \(k^2 = \dfrac{2x}{5x - 3}\) or \(\sqrt{\dfrac{2x}{5x - 3}}\) |
| \(k = \sqrt{\dfrac{2x}{5x - 3}}\) | A1 | oe with \(k\) the subject eg \(k = \sqrt{\dfrac{-2x}{3 - 5x}}\) or \(k = \dfrac{\sqrt{2x}}{\sqrt{5x - 3}}\) or \(\left(\dfrac{2x}{5x - 3}\right)^{\frac{1}{2}} = k\) |
Additional guidance
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Condone one transcription error for A1 after correct answer seen | |
| No miscopies for M marks | |
| Condone eg \(k = \sqrt{\dfrac{2x}{5x - 3}}\) in working with \(\sqrt{\dfrac{2x}{5x - 3}}\) on answer line | M3A1 |
| For A1 accept eg \(k = +\sqrt{\dfrac{2x}{5x - 3}}\) or \(k = \pm\dfrac{\sqrt{2x}}{\sqrt{5x - 3}}\) | |
| Correct answer followed by incorrect further work eg \(k = \sqrt{\dfrac{2x}{5x - 3}} = \dfrac{2x}{5x - 3}\) | M3A0 |
| For M1 the terms may all be on one side of an equation but must be recovered to award further marks eg Alt 1 \(\ x(5k^2 - 2) - 3k^2 = 0\) | M1 |
| Choose the alt that favours the student |