A2 June 2024 Q5

EdexcelCurrent spec9 marksFurther Complex Numbers

(i) A circle \(C\) in the complex plane is defined by the locus of points satisfying\[|z - 3\mathrm{i}| = 2|z|\]

(a) Determine a Cartesian equation for \(C\), giving your answer in simplest form. (3)

(b) On an Argand diagram, shade the region defined by\[\left\{z \in \mathbb{C} : |z - 3\mathrm{i}| > 2|z|\right\}\] (2)

(ii) The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by\[w = z^3\]

(a) Describe the geometric effect of \(T\). (2)

The region \(R\) in the \(z\)-plane is given by\[\left\{z \in \mathbb{C} : 0 < \arg z < \frac{\pi}{4}\right\}\]

(b) On a different Argand diagram, sketch the image of \(R\) under \(T\). (2)

Mark scheme (i)(a)

Scheme	Marks	AO
\(\|x + (y - 3)\mathrm{i}\| = 2\|x + y\mathrm{i}\| \Rightarrow x^2 + (y - 3)^2 = 4\left(x^2 + y^2\right)\)	M1	3.1a
\(\Rightarrow x^2 + y^2 - 6y + 9 = 4x^2 + 4y^2 \Rightarrow x^2 + y^2 + 2y - 3 = 0\) in any order	M1 A1	1.1b 1.1b
	(3)

Notes

M1: Substitutes \(z = x + y\mathrm{i}\) into the equation and applies the modulus to obtain an equation with no i's. Must have dealt with the i\(^2\) correctly. Condone not squaring the 2.

M1: Expands and gathers terms. Must have an \(x^2\) and \(y^2\) after simplifying so that it is an equation of a circle.

A1: Cancels common factor 3 to obtain the equation shown. ISW if they make errors trying to complete the square.

Note: Any letters may be used not just \(x\) and \(y\)

Alt (a)

Scheme	Marks	AO
Points are twice as far from 3i as from 0 so i and −3i are diametrically opposite points.	M1	3.1a
So radius is \(\dfrac{\|\mathrm{i} - (-3\mathrm{i})\|}{2} = 2\) and centre is \(\dfrac{\mathrm{i} - 3\mathrm{i}}{2} = -\mathrm{i}\)	M1	1.1b
Hence equation is \(x^2 + (y + 1)^2 = 4\)	A1	1.1b
	(3)

M1: Identifies two diametrically opposite points on the circle by understanding the geometry of the situation.

M1: Finds the radius and centre of the circle from their points, dotted or solid line

A1: Correct equation need not expand, but should be in simplest form.

Argand diagram: circle centre −i through i and −3i, with the inside shaded — Mark scheme (i)(b)

Scheme	Marks	AO

Circle drawn with the inside shaded	M1	2.2a
Correct circle drawn for their equation. Implied by the position of the centre and radius. Inside shaded.	A1ft	3.1a
	(2)

Mark scheme (ii)(a)

Scheme	Marks	AO
A point \(z\) is mapped to a point with 3 times the argument… Rotate every point by 2 times the argument	B1	2.4
… and with modulus as the modulus of the cube of \(z\).	B1	2.5
	(2)

Notes

B1: Identifies that the argument triples in size.

B1: Identifies that the modulus scales according to the modulus of \(z\) cubed.

Argand diagram: shaded sector from the positive real axis anticlockwise to the half-line at angle 3π/4 — Mark scheme (ii)(b)

Scheme	Marks	AO

Sector from \(O\) along the real axis, indicated by some shading, in an anticlockwise direction	M1	1.1b
Correct sector, angle must be stated or implied by the diagram	A1	2.2a
	(2)
	(9 marks)