FP1 January 2013 Q5

5. \[\mathrm{f}(x) = (4x^2 + 9)(x^2 - 6x + 34)\]

(a) Find the four roots of \(\mathrm{f}(x) = 0\)
Give your answers in the form \(x = p + \mathrm{i}q\), where \(p\) and \(q\) are real. (5)

(b) Show these four roots on a single Argand diagram. (2)

Mark scheme (a)

Scheme	Marks
\(4x^2 + 9 = 0 \Rightarrow x = k\mathrm{i},\quad x = \pm\dfrac{3}{2}\mathrm{i}\) or equivalent	M1, A1
Solving 3-term quadratic by formula or completion of the square \(x = \dfrac{6 \pm \sqrt{36 - 136}}{2}\) or \((x - 3)^2 - 9 + 34 = 0\)	M1
\(= 3 + 5\mathrm{i}\) and \(3 - 5\mathrm{i}\)	A1 A1ft
	(5)

(a) Final A follow through conjugate of their first root.

Scheme	Marks

Two roots on imaginary axis	B1ft
Two roots – one the conjugate of the other	B1ft
Accept points or vectors
	(2)
	[7]