AS June 2025 Paper 1 Q2

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

\[\mathrm{f}(z) = 4z^3 - 12z^2 - 95z + 325\]

Given that \(\mathrm{f}(-5) = 0\)

(a) determine \(\mathrm{f}(z)\) in the form \((z + a)(bz^2 + cz + d)\) where \(a\), \(b\), \(c\) and \(d\) are integers. (3)

(b) Hence show that the complex roots of \(\mathrm{f}(z) = 0\) are \(\dfrac{8 \pm \mathrm{i}}{2}\) (2)

Mark scheme (a)

B1: Recalls the fact that if \(\mathrm{f}(-5) = 0\) then \((z + 5)\) is a factor

M1: Finds the quadratic factor with a \(4z^2\) term, any method. Condone using a factor of \((z - 5)\)

A1: Correct factorised expression for \(\mathrm{f}(z)\)

Mark scheme (b)

Scheme	Marks	AO
\(z = \dfrac{-(-32) \pm \sqrt{(-32)^2 - 4(4)(65)}}{2(4)} = \ldots\) Completing the square \(4(z \pm 4)^2 + q = 0 \Rightarrow z = \ldots\) where \(q \gt 0\)	M1	1.1b
\(z = \left\{\dfrac{32 \pm \sqrt{-16}}{8} = \right\}\dfrac{32 \pm 4\mathrm{i}}{8} = \dfrac{8 \pm \mathrm{i}}{2}\) *	A1*	2.1
	(2)

M1: Uses a correct method to show where the complex roots have come from.

Note just uses the calculator is M0

A1: Completes the method to show the complex roots. There needs to be an intermediate line of work with either 4i or \(4\sqrt{-1}\) term

Note: \(z = \dfrac{32 \pm \sqrt{-16}}{8} = \dfrac{8 \pm \mathrm{i}}{2}\) is M1 A0

Scheme	Marks	AO
Uses the roots to find the equation \(\left(z - \dfrac{8 + \mathrm{i}}{2}\right)\left(z - \dfrac{8 - \mathrm{i}}{2}\right) = z^2 - 8z + \dfrac{65}{4} = 0\) This could come from using the sum and product of roots	M1	1.1b
\(4z^2 - 32z + 65 = 0\) therefore roots are \(\dfrac{8 \pm \mathrm{i}}{2}\)	A1*	2.1
	(2)

(Corrected from the printed mark scheme: the alternative prints \(z^2 - 8z + \dfrac{64}{4} = 0\); the product of the roots is \(\dfrac{65}{4}\).)

M1: uses the roots and expands

A1: Achieves the correct equation with = 0 no errors and a conclusion

Mark scheme (c)

Scheme	Marks	AO
\(-2,\ \dfrac{10 \pm \mathrm{i}}{4}\) or \(-2,\ \dfrac{5}{2} \pm \dfrac{\mathrm{i}}{4}\) or \(-2,\ 2.5 \pm 0.25\mathrm{i}\)	B1 B1	2.2a 2.2a
	(2)
	(7 marks)

B1: Deduces one correct root

B1: Deduces all 3 correct roots