June 2023 Paper 2 Q16
16 \((2x - 3)\) is a factor of \(\quad 6x^3 - 25x^2 + 28x - 6\)
Solve \(\quad 6x^3 - 25x^2 + 28x - 6 = 0\)
Give all solutions as exact values. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((2x - 3)(3x^2 - 8x \ldots)\) or \((2x - 3)(3x^2 \ldots + 2)\) | M1 | implied by algebraic division eg \(\begin{array}{r} 3x^2 - 8x \ldots \\ 2x - 3\,\overline{\big)\,6x^3 - 25x^2 + 28x - 6} \end{array}\) |
| \(3x^2 - 8x + 2 \;(= 0)\) or \(3x^2 - 8x = -2\) | M1dep | |
| \(\dfrac{- -8 \pm \sqrt{(-8)^2 - 4 \times 3 \times 2}}{2 \times 3}\) or \(\dfrac{4}{3} \pm \sqrt{\dfrac{10}{9}}\) or [2.38, 2.39] and [0.27, 0.28] | M1 | oe use of formula or completing the square or factorising their 3-term quadratic eg \(\dfrac{8 \pm \sqrt{40}}{6}\) or \(\dfrac{4}{3} \pm \dfrac{\sqrt{10}}{3}\) [2.38, 2.39] and [0.27, 0.28] implies M3 \(\dfrac{8 \pm \sqrt{40}}{6}\) oe implies M3 |
| \(\dfrac{3}{2}\) and \(\dfrac{4}{3} + \dfrac{\sqrt{10}}{3}\) and \(\dfrac{4}{3} - \dfrac{\sqrt{10}}{3}\) with no other solutions | A1 | oe values eg 1.5 and \(\dfrac{8 \pm \sqrt{40}}{6}\) or \(1\dfrac{1}{2}\) and \(\dfrac{4}{3} + \sqrt{\dfrac{10}{9}}\) and \(\dfrac{4}{3} - \sqrt{\dfrac{10}{9}}\) |
Additional guidance
| Up to M3 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |||||||||||||
| Up to the first two marks may be seen in a grid eg
| M1M1 | ||||||||||||
| 3rd M1 \(\;8^2\) is equivalent to \((-8)^2\) | |||||||||||||
| 3rd M1 Use of \(-8^2\) for \((-8)^2\) must be recovered | |||||||||||||
| 3rd M1 Completing the square must get as far as numerical answers which must be correct for their quadratic | |||||||||||||
| 3rd M1 3-term quadratic means \(ax^2 + bx + c\) with no zero coefficients | |||||||||||||
| 3rd M1 Allow correct substitution into quadratic formula for their 3-term quadratic even if it has no real solutions | |||||||||||||
| 3rd M1 can be implied by correct solutions of their 3-term quadratic truncated or rounded to 2 dp or better | |||||||||||||
| A quadratic from differentiation can be used to score the 3rd M1 | |||||||||||||
| Non-exact solutions cannot score full marks eg 1.5 and 2.38 and 0.28 | M3A0 | ||||||||||||
| Non-exact solutions on answer line is A0 |