A2 June 2023 Paper 2 Q13

AQACurrent spec11 marksUsing Roots of Polynomials

13 The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)

(a) Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]

(b) Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]

(c) Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 marks]

Scheme	Marks	AO
Obtains correct sum of roots.	B1	1.1b
Obtains correct product of roots.	B1	1.1b
	(2)

Mark scheme (b)

Scheme	Marks	AO
Expresses \(\alpha^2 + \beta^2\) in terms of \(\alpha + \beta\) and \(\alpha\beta\)	M1	1.1a
Obtains correct value of \(\alpha^2 + \beta^2\)	A1	1.1b
Expresses \(\alpha^4 + \beta^4\) in terms of sums and/or products of \(\alpha\), \(\beta\), \(\alpha^2\), \(\beta^2\)	M1	2.2a
Completes a reasoned argument to obtain the required result. CSO	R1	2.1
	(4)

Typical solution

\[\begin{aligned} \alpha^2 + \beta^2 &= (\alpha + \beta)^2 - 2\alpha\beta \\ &= 25 - 16 = 9 \end{aligned}\]\[\begin{aligned} \alpha^4 + \beta^4 &= \left(\alpha^2 + \beta^2\right)^2 - 2\alpha^2\beta^2 \\ &= 9^2 - 2 \times 8^2 \\ &= -47 \end{aligned}\]

Mark scheme (c)

Scheme	Marks	AO
Expresses the sum of roots of the new equation in terms of sums and/or products of \(\alpha\) and \(\beta\) or \(\alpha^2\) and \(\beta^2\)	M1	3.1a
Obtains correct sum of roots.	A1	1.1b
Expresses the product of roots of the new equation as \(= \alpha^3\beta^3 + \alpha^4 + \beta^4 + \alpha\beta\)	M1	1.1a
Obtains correct product of roots.	A1	1.1b
Deduces a correct quadratic equation with integer coefficients. Allow any variable.	A1	2.2a
	(5)
	(11 marks)

Typical solution

\[\begin{aligned} \text{Sum of roots} &= \alpha^3 + \beta^3 + \alpha + \beta \\ &= (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) + \alpha + \beta \\ &= 125 - 3 \times 8 \times 5 + 5 \\ &= 10 \end{aligned}\]\[\begin{aligned} \text{Product of roots} &= \left(\alpha^3 + \beta\right)\left(\beta^3 + \alpha\right) \\ &= \alpha^3\beta^3 + \alpha^4 + \beta^4 + \alpha\beta \\ &= 8^3 - 47 + 8 \\ &= 473 \end{aligned}\]\[z^2 - 10z + 473 = 0\]