FP1 June 2016 Q7

EdexcelOld spec13 marksComplex Numbers

7. A complex number \(z\) is given by \[z = a + 2\mathrm{i}\] where \(a\) is a non-zero real number.

(a) Find \(z^2 + 2z\) in the form \(x + \mathrm{i}y\) where \(x\) and \(y\) are real expressions in terms of \(a\). (4)

Given that \(z^2 + 2z\) is real,

(b) find the value of \(a\). (1)

Using this value for \(a\),

(c) find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures. (3)

(d) Show the points \(P\), \(Q\) and \(R\), representing the complex numbers \(z\), \(z^2\) and \(z^2 + 2z\) respectively, on a single Argand diagram with origin \(O\). (3)

(e) Describe fully the geometrical relationship between the line segments \(OP\) and \(QR\). (2)

Mark scheme (a)

Scheme	Marks
\(z^2 = (a + 2\mathrm{i})(a + 2\mathrm{i}) = (a^2 - 4) + 4\mathrm{i}a\)	M1
So \(z^2 + 2z = (a^2 - 4 + 2a) + \mathrm{i}(4a + 4)\) or \(x = (a^2 + 2a - 4)\) and \(y = 4a + 4\)	M1 A1 A1
	(4)

Notes

M1: Squares \(z\) to produce at least 3 terms which can be implied by the correct answer.
M1: Adds \(2z\) to their \(z^2\)
A1: Correct \(x\) A1 Correct \(y\) accept \(4a\mathrm{i} + 4\mathrm{i}\)

Scheme	Marks
and so \(4a + 4 = 0 \to a = -1\)	B1
ALT (b) Substitute \(a = -1\) and show that \(y = 0\)	B1
	(1)

Scheme	Marks
\(\|z\| = \sqrt{5}\) or awrt 2.24	B1
\(\arctan, (-2) = 2.03\)	M1, A1 cao
	(3)

Argand diagram with Re axis from -6 to 6 and Im axis from -6 to 4: points P(-1, 2), Q(-3, -4) and R(-5, 0) plotted and labelled — Mark scheme (d)

Scheme	Marks
	M1 A1 B1ft
	(3)

Mark scheme (e)

Scheme	Marks
OP and QR are parallel, and QR is twice the length of OP Or Enlargement with Scale Factor 2 (centre O), followed by translation \(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\) Or Enlargement with Scale Factor 2, centre \((3,4)\) or centre \(3 + 4\mathrm{i}\) \(\overrightarrow{QR} = 2\overrightarrow{OP}\) with clear indication of vectors award B1B1, without vectors award B0B1	B1, B1
	(2)
	(13 marks)