Further complex numbers: Edexcel International A Level Further Maths questions

Every Edexcel International A Level Further Maths exam question on FP2.3 Further complex numbers, 2019–2025, with its mark scheme and examiner’s report.

All 29 questions on Further complex numbers, as a list

June 2025, Further Pure 2 (FP2), Q1 (6 marks): (a) Express –2 2 – (2 6 ) i in the form re iθ where –π < θ  π (b) Hence solve z 5 = –2 2 – (2 6 ) i Give your answers in the form pe iθ where p ∈  + and –π <
June 2025, Further Pure 2 (FP2), Q3 (6 marks): A complex number z is represented by the point P on an Argand diagram where |z| = 1 (a) Sketch the locus of P as z varies. The transformation T from the z–plane
June 2025, Further Pure 2 (FP2), Q9 (9 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Use de Moivre’s theorem t
June 2025 (A), Further Pure 2 (FP2), Q4 (8 marks): y x O R Figure 2 Figure 2 shows an Argand diagram for complex numbers of the form z = x + iy. The diagram is drawn accurately, although the scale is not shown o
June 2025 (A), Further Pure 2 (FP2), Q8 (12 marks): Given that z = cos θ + i sin θ (a) show that, for n ∈  1 2cos n n z nθ z (b) Hence show that cos 4 θ = 1 8 (cos 4 θ + a cos 2θ + b) where a and b are integers
January 2025, Further Pure 2 (FP2), Q7 (11 marks): (a) Use De Moivre’s theorem to (i) show that sin 5θ ≡ 5 cos 4 θ sin θ – 10 cos 2 θ sin 3 θ + sin 5 θ (ii) determine an expression for cos 5θ in terms of sin θ a
January 2025, Further Pure 2 (FP2), Q8 (13 marks): A transformation T from the z-plane, where z = x + iy to the w-plane, where w = u + iv , is given by 3 i 2 2 2 z w z z (a) Show that the real axis in the z-
June 2024, Further Pure 2 (FP2), Q1 (4 marks): The complex number z = x + iy satisfies the equation z z 3 4 1 i i (a) Determine an equation for the locus of z giving your answer in the form ax + by + c = 0 w
June 2024, Further Pure 2 (FP2), Q6 (7 marks): The transformation T from the z‑plane to the w‑plane is given by w z z z i 1 1 Given that T maps the imaginary axis in the z‑plane to the circle C in the w‑plan
June 2024, Further Pure 2 (FP2), Q9 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Use De Moivre’s theorem t
January 2024, Further Pure 2 (FP2), Q2 (8 marks): z 6 6 3i (a) (i) Determine the modulus of z (ii) Show that the argument of z is 3 Using de Moivre’s theorem, and making your method clear, (b) determine, in sim
January 2024, Further Pure 2 (FP2), Q7 (10 marks): A transformation T from the z‑plane, where z = x + iy, to the w‑plane, where w = u + iv is given by w z z z 3 2 2 i i The line in the z‑plane with equation y =
June 2023, Further Pure 2 (FP2), Q2 (10 marks): The complex number z1 is defined as 4 1 3 5π 5π cos i sin 12 12 π π cos i sin 3 3 z (a) Without using your calculator show that 1 2π 2π cos isin
June 2023, Further Pure 2 (FP2), Q5 (7 marks): The transformation T from the z-plane, where z = x + iy, to the w‑plane, where w = u + iv is given by w = z z 1 3 z ≠ 3 The straight line in the z‑plane with eq
January 2023, Further Pure 2 (FP2), Q6 (8 marks): A complex number z is represented by the point P in an Argand diagram. Given that z z − = − 2 3 i (a) sketch the locus of P. You do not need to find the coordin
January 2023, Further Pure 2 (FP2), Q7 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Use de Moivre’s theorem t
June 2022, Further Pure 2 (FP2), Q3 (8 marks): The transformation T from the z-plane to the w-plane is given by w z z z = + ≠− 4 4 i i The circle with equation z = 3 is mapped by T onto the circle C Determin
June 2022, Further Pure 2 (FP2), Q8 (14 marks): (a) Use de Moivre's theorem to show that sin 5θ ≡ 16 sin5 θ – 20 sin3 θ + 5 sin θ (b) Hence determine the five distinct solutions of the equation 16x 5 – 20x 3
January 2022, Further Pure 2 (FP2), Q1 (7 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Express the complex numbe
January 2022, Further Pure 2 (FP2), Q7 (8 marks): A transformation from the z‑plane to the w‑plane is given by w = ( ) ( ) 1 2 1 + + − − i i i z z z ≠ i The transformation maps points on the imaginary axis in t
October 2021, Further Pure 2 (FP2), Q1 (4 marks): Solve the equation z5 − 32i = 0 giving each answer in the form reiθ where 0 < θ < 2π
October 2021, Further Pure 2 (FP2), Q3 (6 marks): A transformation maps points from the z-plane, where z = x + iy , to the w-plane, where w = u + iv . The transformation is given by w = 2 4 + ( ) + − i i z z z
October 2021, Further Pure 2 (FP2), Q6 (9 marks): The complex number z on an Argand diagram is represented by the point P where |z + 1 − 13i| = 3|z − 7 − 5i| Given that the locus of P is a circle, (a) determine
June 2021, Further Pure 2 (FP2), Q2 (8 marks): The transformation T from the z-plane, where z = x +iy, to the w-plane, where w = u + iv, is given by w z z z = + − ≠ 2 i i The transformation T maps the circle
June 2021, Further Pure 2 (FP2), Q7 (9 marks): (a) Use de Moivre’s theorem to show that 3 2 4 4tan 4tan tan 4 1 6tan tan θ θ θ θ θ − ≡ − + (b) Use the identity given in part (a) to find the 2 positive roots
January 2021, Further Pure 2 (FP2), Q1 (3 marks): The transformation T from the z-plane, where z = x + iy , to the w-plane, where w = u + iv , is given by w = z p z + + i i 3 z ≠ 3i p   The point representing
January 2021, Further Pure 2 (FP2), Q8 (16 marks): Given that z = eiθ (a) show that z n + 1 zn = 2 cos nθ where n is a positive integer. (b) Show that cos6 θ = 1 32 (cos 6θ + 6 cos 4θ + 15 cos 2θ + 10) (c) Hence
October 2020, Further Pure 2 (FP2), Q4 (8 marks): (a) Express the complex number 18 3 − 18i in the form r (cos θ + i sin θ) −π < θ  π (b) Solve the equation z 4 = 18 3 − 18i giving your answers in the form rei
October 2020, Further Pure 2 (FP2), Q5 (7 marks): The transformation T from the z-plane to the w-plane is given by w = z z − + 3 2 i i z ≠ −2i The circle with equation | z | = 1 in the z-plane is mapped by T on

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