FP1 January 2010 Q6

6. Given that 2 and \(5 + 2\mathrm{i}\) are roots of the equation \[x^3 - 12x^2 + cx + d = 0, \qquad c,\ d \in \mathbb{R},\]

(a) write down the other complex root of the equation. (1)

(b) Find the value of \(c\) and the value of \(d\). (5)

Scheme	Marks
\(5 - 2\mathrm{i}\) is a root	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\big(x - (5 + 2\mathrm{i})\big)\big(x - (5 - 2\mathrm{i})\big) = x^2 - 10x + 29\)	M1 M1
\(x^3 - 12x^2 + cx + d = (x^2 - 10x + 29)(x - 2)\)	M1
\(c = 49, \qquad d = -58\)	A1, A1
	(5)

(b) 1^st M: Form brackets using \((x - \alpha)(x - \beta)\) and expand.

2^nd M: Achieve a 3-term quadratic with no i's.

Scheme	Marks
Substitute a complex root (usually 5+2i) and expand brackets	M1
\((5 + 2\mathrm{i})^3 - 12(5 + 2\mathrm{i})^2 + c(5 + 2\mathrm{i}) + d = 0\)
\((125 + 150\mathrm{i} - 60 - 8\mathrm{i}) - 12(25 + 20\mathrm{i} - 4) + (5c + 2c\mathrm{i}) + d = 0\) (2^nd M for achieving an expression with no powers of i)	M1
Equate real and imaginary parts	M1
\(c = 49, \qquad d = -58\)	A1, A1

Scheme	Marks
Conjugate pair in 1^st and 4^th quadrants (symmetrical about real axis)	B1
Fully correct, labelled	B1
	(2)
	[8]