FP1 January 2009 Q9

EdexcelOld spec10 marksComplex Numbers

9. Given that \(z_1 = 3 + 2\mathrm{i}\) and \(z_2 = \dfrac{12 - 5\mathrm{i}}{z_1}\),

(a) find \(z_2\) in the form \(a + \mathrm{i}b\), where \(a\) and \(b\) are real. (2)

(b) Show on an Argand diagram the point \(P\) representing \(z_1\) and the point \(Q\) representing \(z_2\). (2)

The circle passing through the points \(O\), \(P\) and \(Q\) has centre \(C\). Find

(d) the complex number represented by \(C\), (2)

(e) the exact value of the radius of the circle. (2)

Mark scheme (a)

Scheme	Marks
\(z_2 = \dfrac{12 - 5\mathrm{i}}{3 + 2\mathrm{i}} \times \dfrac{3 - 2\mathrm{i}}{3 - 2\mathrm{i}} = \dfrac{36 - 24\mathrm{i} - 15\mathrm{i} - 10}{13}\)	M1
\(= 2 - 3\mathrm{i}\)	A1
	(2)

(a) \(\times\dfrac{3 - 2i}{3 - 2i}\) for M1

Scheme	Marks
\(P\): B1, \(Q\): B1ft	B1, B1ft
	(2)

Mark scheme (c)

Scheme	Marks
grad. \(OP \times\) grad. \(OQ = \dfrac{2}{3} \times -\dfrac{3}{2}\)	M1
\(= -1 \quad \Rightarrow \angle POQ = \dfrac{\pi}{2}\) (*)	A1
	(2)

Scheme	Marks
\(\angle POX = \tan^{-1}\tfrac{2}{3},\ \angle QOX = \tan^{-1}\tfrac{3}{2}\)
\(\tan(\angle POQ) = \dfrac{\frac{2}{3} + \frac{3}{2}}{1 - \frac{2}{3} \times \frac{3}{2}}\)	M1
\(\Rightarrow \angle POQ = \dfrac{\pi}{2}\) (*)	A1
	(2)

Scheme	Marks
\(r = \sqrt{\left(\dfrac{5}{2}\right)^2 + \left(-\dfrac{1}{2}\right)^2}\)	M1
\(= \dfrac{\sqrt{26}}{2}\) or exact equivalent	A1
	(2)
	[10]