FP1 June 2017 Q4

(i) The complex number \(w\) is given by \[w = \frac{p - 4\mathrm{i}}{2 - 3\mathrm{i}}\] where \(p\) is a real constant.

(a) Express \(w\) in the form \(a + b\mathrm{i}\), where \(a\) and \(b\) are real constants.
Give your answer in its simplest form in terms of \(p\). (3)

Given that \(\arg w = \dfrac{\pi}{4}\)

(b) find the value of \(p\). (2)

(ii) The complex number \(z\) is given by \[z = (1 - \lambda\mathrm{i})(4 + 3\mathrm{i})\] where \(\lambda\) is a real constant.
Given that \[|z| = 45\] find the possible values of \(\lambda\)
Give your answers as exact values in their simplest form. (3)

Mark scheme (i)(a)

Scheme	Marks
Mark (i)(a) and (i)(b) together.
\(w = \dfrac{p - 4\mathrm{i}}{2 - 3\mathrm{i}} \qquad \arg w = \dfrac{\pi}{4}\)
Way 1 \(w = \dfrac{(p - 4\mathrm{i})}{(2 - 3\mathrm{i})} \times \dfrac{(2 + 3\mathrm{i})}{(2 + 3\mathrm{i})}\)	M1
\(= \left(\dfrac{2p + 12}{13}\right) + \left(\dfrac{3p - 8}{13}\right)\mathrm{i}\)	A1 A1
	(3)

Notes

M1: Multiplies by \(\dfrac{(2 + 3\mathrm{i})}{(2 + 3\mathrm{i})}\)

A1: At least one of either the real or imaginary part of \(w\) is correct. Must be expanded but could be unsimplified e.g. expressed as single fraction. Condone \(a + \mathrm{i}b\).

A1: Correct \(w\) in its simplest form.

(a) Way 2

Scheme	Marks
\((a + \mathrm{i}b)(2 - 3\mathrm{i}) = (p - 4\mathrm{i})\)
\(2a + 3b = p\) \(3a - 2b = 4\)	M1
\(= \left(\dfrac{2p + 12}{13}\right) + \left(\dfrac{3p - 8}{13}\right)\mathrm{i}\)	A1 A1
	(3)

M1: Multiplies out to obtain 2 equations in two unknowns.

A1: At least one of either the real or imaginary part of \(w\) is correct. Must be expanded but could be unsimplified e.g. expressed as single fraction. Condone \(a + \mathrm{i}b\).

A1: Correct \(w\) in its simplest form.

Mark scheme (i)(b)

Scheme	Marks
\(\left\{\arg w = \dfrac{\pi}{4} \Rightarrow\right\} \quad 2p + 12 = 3p - 8\) o.e. seen anywhere.	M1
\(\Rightarrow p = 20\)	A1
	(2)

Notes

M1: Sets the numerators of the real part of their \(w\) equal to the imaginary part of their \(w\) or if arctan used, require evidence of \(\tan\dfrac{\pi}{4} = 1\)

A1: \(p = 20\)

Mark scheme (ii)

Scheme	Marks
\(z = (1 - \lambda\mathrm{i})(4 + 3\mathrm{i})\) and \(\|z\| = 45\)
Way 1 \(\sqrt{1 + \lambda^2}\,\sqrt{4^2 + 3^2}\)	M1
\(\sqrt{1 + \lambda^2}\,\sqrt{4^2 + 3^2} = 45\)	A1
\(\left\{\lambda^2 = 9^2 - 1 \Rightarrow\right\}\ \lambda = \pm 4\sqrt{5}\)	A1
	(3)
	(8 marks)

Notes

M1: Attempts to apply \(|(1 - \lambda\mathrm{i})(4 + 3\mathrm{i})| = \sqrt{1 + \lambda^2}\,\sqrt{4^2 + 3^2}\)

A1: Correct equation.

A1: \(\lambda = \pm 4\sqrt{5}\)

M1: Also allow \((1 + \lambda^2)(4^2 + 3^2)\) for M1.

(ii) Way 2

Scheme	Marks
\(z = (4 + 3\lambda) + (3 - 4\lambda)\mathrm{i}\) \(\sqrt{(4 + 3\lambda)^2 + (3 - 4\lambda)^2}\)	M1
\((4 + 3\lambda)^2 + (3 - 4\lambda)^2 = 45^2\) or \(\sqrt{(4 + 3\lambda)^2 + (3 - 4\lambda)^2} = 45\)	A1
\(\left\{16 + 24\lambda + 9\lambda^2 + 9 - 24\lambda + 16\lambda^2 = 2025\right\}\)
\(\left\{25\lambda^2 = 2000 \Rightarrow\right\}\ \lambda = \pm 4\sqrt{5}\)	A1
	(3)

M1: Attempt to multiply out, group real and imaginary parts and apply the modulus.

A1: Correct equation. Condone if middle terms in expansions not explicitly stated.

A1: \(\lambda = \pm 4\sqrt{5}\)

M1: Also allow \((4 + 3\lambda)^2 + (3 - 4\lambda)^2\) for M1.