Loci and regions in the Argand diagram: Edexcel A‑Level Further Maths questions

Edexcel A-Level Further Maths past-paper questions on 2.4 Loci and regions in the Argand diagram, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 26 questions on Loci and regions in the Argand diagram, as a list

June 2025, Core Pure 2, Q4 (10 marks): (i) z a b 1 i and z c d 2 i where a, b, c and d are real constants. Given that • b d > • z z 1 2 + is real • z1 13 = • z2 5 = • Re z z 2 1 2 show that a = 2 and
June 2025, Further Pure 2, Q3 (9 marks): The circle C has equation z z 5 4 2 2 i i where z = x + iy (a) Show that an equation for C is x 2 – 2x + y 2 –7 = 0 (3) The half‑line with equation 3 arg 4 z
June 2025, AS Core Pure, Q5 (8 marks): A complex number z is represented by the point P in the complex plane. Given that z satisfies | z – 1 | = 1 (a) sketch on an Argand diagram the locus of P as z
June 2025, AS Further Pure 2, Q5 (7 marks): In an Argand diagram, the curve C with equation 4 arg 2i 4 z π z represents an arc of a circle. Given that z = x + iy , where x and y are real numbers
June 2024, Core Pure 2, Q5 (9 marks): The locus C is given by z 4 4 The locus D is given by arg 3 π z (a) Sketch, on the same Argand diagram, the locus C and the locus D (4) The set of points A is d
June 2024, Further Pure 2, Q5 (9 marks): (i) A circle C in the complex plane is defined by the locus of points satisfying z z 3 2 i (a) Determine a Cartesian equation for C, giving your answer in simpl
June 2024, AS Core Pure, Q5 (10 marks): Given that on an Argand diagram the locus of points defined by | z + 5 – 12i | = 10 is a circle, (a) write down, (i) the coordinates of the centre of this circl
June 2023, Core Pure 1, Q3 (10 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. z1 = –4 + 4i (a) Express z1 in the for
June 2023, Further Pure 2, Q2 (6 marks): A complex number z is represented by the point P in the complex plane. Given that z satisfies z z 6 2 3i (a) show that the locus of P passes through the origin
June 2023, Further Pure 2, Q9 (5 marks): z2 z3 z1 Figure 1 Figure 1 shows a locus in the complex plane. The locus is an arc of a circle from the point represented by z1 3 2 i to the point represented b
June 2023, AS Core Pure, Q7 (9 marks): (i) Shade, on an Argand diagram, the set of points for which | z – 3 |  | z + 6i | (3) (ii) Determine the exact complex number w which satisfies both arg (w –
June 2023, AS Further Pure 2, Q3 (7 marks): A complex number z is represented by the point P on an Argand diagram. Given that 4 i arg 2 7i 2 z π z − −  =     − − (a) sketch the locus of P as z varie
June 2022, Core Pure 2, Q4 (6 marks): (i) Given that z1 = 6e i 3 π and z2 = 6 5 i 6 3e π show that z1 + z2 = 12e 2 i 3 π (3) (ii) Given that arg(z – 5) = 2 3 π determine the least value of ½z½ as z
June 2022, Further Pure 2, Q5 (6 marks): The locus of points z satisfies | z + a i | = 3 | z – a | where a is an integer. The locus is a circle with its centre in the third quadrant and radius 3 2 2 De
June 2022, Further Pure 2, Q8 (7 marks): The locus of points z = x + i y that satisfy 8 5i arg 2 5i 3 z π z − −  =   − −   is an arc of a circle C. (a) On an Argand diagram sketch the locus of z.
June 2022, AS Core Pure, Q2 (10 marks): (a) Express the complex number w = 4 3 − 4i in the form r (cos θ + i sin θ) where r > 0 and −π < θ  π (4) (b) Show, on a single Argand diagram, (i) the point r
June 2022, AS Further Pure 2, Q1 (4 marks): Sketch on an Argand diagram the region defined by ( ) ( ) { } : arg 2 : 1 Re 1 4 4 π π z z z z   ∈ − < + < ∩ ∈ −<     On your sketch • shade the part of t
November 2021, Core Pure 2, Q8 (11 marks): (i) The point P is one vertex of a regular pentagon in an Argand diagram. The centre of the pentagon is at the origin. Given that P represents the complex numbe
November 2021, Further Pure 2, Q5 (10 marks): The point P in the complex plane represents a complex number z such that ½z + 9½ = 4½z – 12i½ Given that, as z varies, the locus of P is a circle, (a) determine
November 2020, Core Pure 2, Q2 (9 marks): In an Argand diagram, the points A and B are represented by the complex numbers −3 + 2i and 5 − 4i respectively. The points A and B are the end points of a diam
November 2020, AS Core Pure, Q10 (7 marks): Given that there are two distinct complex numbers z that satisfy { } 3 : 3 5i 2 :arg( 2) 4 π z z r z z   − − = ∩ − =     determine the exact range of valu
November 2020, AS Further Pure 2, Q5 (6 marks): –4 –2 2 O 2 4 Re Im –2 Figure 1 Figure 1 shows an Argand diagram. The set of points, A, that lies within the shaded region, including its boundaries, is defined
June 2019, Further Pure 2, Q1 (5 marks): A complex number z = x + iy is represented by the point P in an Argand diagram. Given that | zí_ _z + 1 | (a) show that the locus of P has equation 15x
June 2019, AS Further Pure 2, Q3 (10 marks): A curve C in the complex plane is described by the equation z z −− = − 1 8 3 1 i (a) Show that C is a circle, and find its centre and radius. (4) (b) Using the
June 2018, AS Core Pure, Q3 (9 marks): (a) Shade on an Argand diagram the set of points { } ( ) 3 : 1 i 3 : arg 2 4 4 π π z z z z   − − ∩ −     C C ∈ ∈ (5) The complex number w satisfies ( ) 1
June 2018, AS Further Pure 2, Q5 (8 marks): A complex number z is represented by the point P on an Argand diagram. Given that 6i arg 3i 3 z π z −  =     − (a) sketch the locus of P as z varies, (3)

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