A2 June 2025 Paper 1 Q5

5 The complex number \(z_1\) has argument \(\tan^{-1}\left(\dfrac{1}{5}\right)\)

The complex number \(z_2 = -2 - 4\mathrm{i}\)

Find \(\arg\left(\dfrac{z_1}{z_2}\right)\)

Give your answer as a number in the range \(-\pi \lt \alpha \lt \pi\), to two decimal places. [3 marks]

Mark scheme

Scheme	Marks	AO
Obtains a correct expression or value for \(\arg z_2\) Condone use of degrees. or Obtains \(\dfrac{z_1}{z_2} = k(-0.7 + 0.9\mathrm{i})\) PI AWRT 2.23 rad or AWRT 128 degrees	B1	1.1b
Uses \(\arg\left(\dfrac{z_1}{z_2}\right) = \arg z_1 - \arg z_2\) or Obtains argument of their \(\dfrac{z_1}{z_2}\) PI AWRT 2.23 rad or AWRT 128 degrees	M1	1.1a
Obtains AWRT 2.23	A1	1.1b
	(3 marks)

Typical solution

\[\arg z_1 = \tan^{-1}\left(\frac{1}{5}\right) = 0.19739\ldots\]\[\arg z_2 = \tan^{-1}\left(\frac{-4}{-2}\right) - \pi = -2.03444\ldots\]\[\arg\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2 = 2.23\]