plumiComplex numbers: Edexcel A‑Level Further Maths questions
Every Edexcel A-Level Further Maths exam question on CP.2 Complex numbers, 2018–2025, with its mark scheme and examiner’s report.
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All 39 questions on Complex numbers, as a list
- June 2025, Core Pure 1, Q3 (5 marks): The complex number z a b i where a and b are real constants. Given that z z* is purely imaginary, (a) show that a b 2 2 = (2) Given also that zz* = 50 (b) deter
- June 2025, Core Pure 1, Q6 (5 marks): f (z) = z3 + az2 + bz + c where a, b and c are real constants The roots of the equation f (z) = 0 are z1, z2 and z3 When plotted on an Argand diagram, the point
- June 2025, Core Pure 1, Q10 (12 marks): (a) Given that, for n ∈ z z n n n 1 2cos θ z z n n n 1 2isin θ show that 8 4 4 2 3 4 sin cos cos θ θ θ (5) O y R x Figure 1 Figure 2 Figure 1 shows the central
- June 2025, Core Pure 2, Q1 (6 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that z 2 2 3i and w 1 3
- 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, AS Core Pure, Q1 (6 marks): z = 3 – 3i (a) Write z in the form r(cos θ + i sin θ) where –π < θ π (2) (b) Show and label on a single Argand diagram (i) the point P representing z (ii) the
- June 2025, AS Core Pure, Q2 (7 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f(z) = 4z 3 – 12z 2 – 95z + 3
- 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 2024, Core Pure 1, Q1 (9 marks): f(z) = z z az bz 4 3 2 6 145 where a and b are real constants. Given that 2 + 5i is a root of the equation f(z) = 0 (a) determine the other roots of the equatio
- June 2024, Core Pure 1, Q4 (10 marks): The complex number z = e iθ, where θ is real. (a) Show that z n + 1 zn ≡2 cos nθ where n is a positive integer. (2) (b) Show that cos 5 θ = 1 16 (cos 5θ + 5 cos
- 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, Core Pure 2, Q7 (9 marks): (a) Determine the roots of the equation z 6 = 1 giving your answers in the form e iθ where 0 θ < 2π (2) (b) Show the roots of the equation in part (a) on a si
- 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, Core Pure 2, Q5 (9 marks): The points representing the complex numbers z1 = 35 – 25i and z2 = –29 + 39i are opposite vertices of a regular hexagon, H, in the complex plane. The centre of
- June 2023, Core Pure 2, Q8 (11 marks): Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane, (a) describe the possible lines the roots can
- June 2023, AS Core Pure, Q2 (7 marks): f (z) = z 3 + az 2 + bz + 175 where a and b are real constants Given that –3 + 4i is a root of the equation f(z) = 0 (a) determine the value of a and the value
- June 2023, AS Core Pure, Q4 (8 marks): (i) (a) Show that 2 3 5 1 i i i k( ) where k is a constant to be determined. (Solutions relying on calculator technology are not acceptable.) (3) Given that • n
- 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 2022, Core Pure 1, Q1 (6 marks): f(z) = z3 + az + 52 where a is a real constant Given that 2 – 3i is a root of the equation f(z) = 0 (a) write down the other complex root. (1) (b) Hence (i) sol
- June 2022, Core Pure 1, Q7 (7 marks): Given that z = a + bi is a complex number where a and b are real constants, (a) show that zz* is a real number. (2) Given that • zz* = 18 • z z* = 7 9 + 4 2 9 i
- June 2022, Core Pure 1, Q8 (12 marks): (a) Given zn + 1 zn = 2 cos nθ n ∈ show that 32 cos6 θ ≡ cos 6θ + 6 cos 4θ + 15 cos 2θ + 10 (5) y x O x = –4 x = 4 R Figure 1 Figure 2 Figure 1 shows a solid
- June 2022, Core Pure 2, Q1 (3 marks): A student was asked to answer the following: For the complex numbers z1 = 3 – 3i and z2 = 3 + i , find the value of arg z z 1 2 The student’s attempt
- 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, 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
- November 2021, Core Pure 2, Q1 (5 marks): Given that 1 2 3 cos isin 3 3 2 cos isin 12 12 π π z π π z = + = − (a) write down the ex
- 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, Core Pure 2, Q9 (8 marks): (a) Given that | z | < 1 , write down the sum of the infinite series 1 + z + z2 + z3 + … (1) (b) Given that z = 1 2 (cos θ + i sin θ) , (i) use the answer to pa
- November 2020, Core Pure 1, Q1 (10 marks): f ( z ) = 3z3 + pz2 + 57z + q where p and q are real constants. Given that 3 2 2 − i is a root of the equation f(z) = 0 (a) show all the roots of f(z) = 0 on a
- 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, Core Pure 2, Q4 (10 marks): (a) Use de Moivre’s theorem to prove that sin 7θ = 7 sin θ − 56 sin3 θ + 112 sin5 θ − 64 sin7 θ (5) (b) Hence find the distinct roots of the equation 1 + 7x − 5
- November 2020, AS Core Pure, Q2 (8 marks): Given that 1 1 2 1 2 2 3i 39 2 arg( ) 4 = + = = z z z π z z where z1 and z2 are complex numbers, (a) write z1 in the form r(cos θ + i sin θ) Give the exact valu
- November 2020, AS Core Pure, Q7 (6 marks): f(z) = z4 + az3 + bz2 + cz + d where a, b, c and d are real constants. The equation f(z) = 0 has complex roots z1, z2, z3 and z4 When plotted on an Argand diagr
- 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
- June 2019, Core Pure 1, Q1 (9 marks): I z z4 + az3 + bz2 + cz + d where a, b, c and d are real constants. *LYHQWKDWíLDQGíLDUHWZRURRWVRIWKHHTXDWLRQI z D VKRZDO
- June 2019, Core Pure 2, Q4 (8 marks): The infinite series C and S are defined by C = cos ș + 1 2 cos 5ș + 1 4 cos 9ș + 1 8 cos 13ș + … S = sin ș + 1 2 sin 5ș + 1 4 sin 9ș + 1 8 sin 13ș + … Given tha
- June 2019, Core Pure 2, Q6 (9 marks): In an Argand diagram, the points A, B and C are the vertices of an equilateral triangle with its centre at the origin. The point A represents the complex number
- 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 Core Pure, Q7 (7 marks): f (z) = z3 + z2 + pz + q where p and q are real constants. The equation f (z) = 0 has roots z1 , z2 and z3 When plotted on an Argand diagram, the points represe
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