A2 June 2025 Paper 1 Q4

OCR ACurrent spec9 marksUsing Roots of Polynomials

4 The equation \(5x^3 - 4x^2 + 10 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

(a) Write down the values of \(\alpha + \beta + \gamma\), \(\alpha\beta + \beta\gamma + \gamma\alpha\) and \(\alpha\beta\gamma\). [2]

(b) By expanding \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), determine the value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\). [3]

(d) Hence find a cubic equation with integer coefficients that has roots \(\alpha^2\), \(\beta^2\), and \(\gamma^2\). [2]

Scheme	Marks	AO
\(\alpha + \beta + \gamma = \dfrac{4}{5} \qquad \alpha\beta + \beta\gamma + \gamma\alpha = 0 \qquad \alpha\beta\gamma = -2\)	B2	1.1 1.1
	[2]

Mark scheme (b)

Scheme	Marks	AO
\((\alpha\beta + \beta\gamma + \gamma\alpha)^2\) \(= \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma)\)	M1	2.1
\(0^2 = 2 \times (-2) \times \dfrac{4}{5} + \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\)	A1FT	1.1
\(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = \dfrac{16}{5}\)	A1	2.2a
	[3]

Notes

M1: For expanding \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\) - must be equivalent to nine terms but allow an error in at most two terms only. Or for using the correct identity \(\sum\alpha^2\beta^2 = \left(\sum\alpha\beta\right)^2 - 2\alpha\beta\gamma\sum\alpha\)

A1FT: Correct equation/expression for \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) using their results from part (a)

Mark scheme (c)

Scheme	Marks	AO
\((\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)\)	B1	1.1
\(\alpha^2 + \beta^2 + \gamma^2 = \dfrac{16}{25}\)	B1	2.2a
	[2]

Notes

B1: correct expansion of \((\alpha + \beta + \gamma)^2\)

B1: www but condone from \(\alpha + \beta + \gamma = -\frac{4}{5}\) stated in part (a) – this mark is independent of the first B1 mark

Mark scheme (d)

Scheme	Marks	AO
\(u^3 - \dfrac{16}{25}u^2 + \dfrac{16}{5}u - (-2)^2 \; (= 0)\)	M1	2.2a
\(25u^3 - 16u^2 + 80u - 100 = 0\)	A1	1.1
	[2]

Notes

M1: For a four term cubic equation/expression with two correct coefficients (not including the cubic term) FT their answers to part (b) and (c). Condone missing = 0. Can use any single unknown, including \(x\)

A1: Must include = 0. Can use any single unknown, including \(x\) but must be integer coefficients
SC B1 for \(x = (\pm)\sqrt{u}\) substitution with correct equation with integer coefficients