21M.1.AHL.TZ2.7

pestleMathematicsAAHLPaper 121M· ahl-2-12-factor-and-remainder-theorems-sum-and-product-of-rootssource ↗

The cubic equation $x^{3} - k x^{2} + 3 k = 0$ where $k > 0$ has roots $α, β$ and $α + β$ .

Given that $α β = - \frac{k^{2}}{4}$ , find the value of $k$ .

Markscheme / solution

$α + β + α + β = k$ (A1)

$α + β = \frac{k}{2}$

$α β (α + β) = - 3 k$ (A1)

$(- \frac{k^{2}}{4}) (\frac{k}{2}) = - 3 k (- \frac{k^{3}}{8} = - 3 k)$ M1

attempting to solve $- \frac{k^{3}}{8} + 3 k = 0$ (or equivalent) for $k$ (M1)

$k = 2 \sqrt{6} (= \sqrt{24}) (k > 0)$ A1

Note: Award A0 for $k = \pm 2 \sqrt{6} (\pm \sqrt{24})$ .

[5 marks]

Examiners’ report

[N/A]