plumiComplex numbers: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on B Complex numbers, 2019–2025, with its mark scheme and examiner’s report.
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All 57 questions on Complex numbers, as a list
- June 2025, Paper 1, Q1 (1 mark): The equation + + = 2 4 0 x x c has non-real roots. Find the range of possible values of c Circle your answer. [1 mark] < 4 c ≤4 c ≥4 c > 4 c
- June 2025, Paper 1, Q2 (1 mark): The point A on the Argand diagram represents the complex number z The point B is the reflection in the imaginary axis of the point A A B Re Im O Which complex n
- June 2025, Paper 1, Q5 (3 marks): The complex number 1z has argument tan 1 1 5 − The complex number 2z = –2 – 4i Find 1 2 arg z z Give your answer as a number in the range
- June 2025, Paper 1, Q13 (7 marks): The function f is defined by f ( ) = + + + 3 2 4 92 z z rz z s where r and s are real numbers. One of the roots of the equation f ( ) = 0 z is i −+ 4 3 13 (a) F
- June 2025, Paper 2, Q5 (2 marks): The complex number ( )( ) i i = + + 3 4 5 z c , where c is an integer. It is given that ( ) = Re 7 z Find the value of c [2 marks]
- June 2025, Paper 2, Q9 (9 marks): (a) It is given that, for the complex number z, = + 1 1 z z Find ( ) Re z [3 marks] 9 (b) Show that the only solutions of the equation = + 3 1 1 w w
- June 2025, Paper 2, Q13 (10 marks): The matrix M is defined by − = 1 3 3 1 M 13 (a) The matrix M represents an anticlockwise rotation about the origin through an angle θ, where 0 ≤
- June 2025, AS Paper 1, Q1 (1 mark): Calculate the product (3 + i)(2 – i) Circle your answer. [1 mark] 5 – i 5 + i 7 – i 7 + i
- June 2025, AS Paper 1, Q2 (1 mark): The complex number 3 + 5i is a root of the quadratic equation z2 + az + b = 0 where a and b are real constants. Find the other root of the equation. Circle your
- June 2025, AS Paper 1, Q6 (4 marks): The complex numbers w and z are defined as follows: w = 2(cos 0.4 + i sin 0.4) and z = 6(cos 1.2 + i sin 1.2) 6 (a) Express z2 in the form r(cos θ + i sin θ) [2
- June 2025, AS Paper 1, Q16 (8 marks): A circle C is drawn on an Argand diagram. The roots of the equation w2 + 4w + 9 = 0 lie on C 16 (a) Show that the roots of the equation w2 + 4w + 9 = 0 are –2 +
- June 2024, Paper 1, Q2 (1 mark): The complex number z = e iπ 3 Which one of the following is a real number? Circle your answer. [1 mark] _ z4 z5 z6 z7
- June 2024, Paper 1, Q7 (5 marks): The complex numbers z and w satisfy the simultaneous equations _ z + w* = 5 _ 3z* – w = 6 + 4i Find z and w [5 marks]
- June 2024, Paper 1, Q10 (6 marks): The complex numbers z and w are defined by z = cos π 4 + i sin π 4 _ and w = cos π 6 + i sin π 6 By evaluating the product zw, show that tan 5π 12 = 2 + √ 3 [6
- June 2024, Paper 1, Q13 (9 marks): (a) Use de Moivre’s theorem to show that cos3 θ = 4cos3 θ – 3cos θ [3 marks] 13 (b) Use de Moivre’s theorem to express sin3 θ in terms of sin θ [2 marks] 13 (c)
- June 2024, Paper 2, Q4 (1 mark): The function f is a quartic function with real coefficients. The complex number 5i is a root of the equation f(x) = 0 Which one of the following must be a facto
- June 2024, Paper 2, Q11 (3 marks): Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation ha
- June 2024, Paper 2, Q17 (9 marks): The Argand diagram below shows a circle C Re Im – 1 – 1 5 6 7 8 9 10 4 3 2 1 5 6 7 8 4 3 2 1 O 17 (a) Write down the equation of the locus of C in the form │z –
- June 2024, AS Paper 1, Q8 (7 marks): (a) The complex number z is given by z = x + iy where x, y ∈ ℝ 8 (a) (i) Write down the complex conjugate z* in terms of x and y [1 mark] ����������������������
- June 2024, AS Paper 1, Q11 (3 marks): The matrices A and B are given by A = 3i –2 a –i and B = 4 5 –2i –1 where a is a real number. Calculate the product AB in terms of a Give your answer in its sim
- June 2024, AS Paper 1, Q17 (9 marks): The circle C represents the locus of points satisfying the equation |z – ai| = b where a and b are real constants. The circle C intersects the imaginary axis at
- June 2023, Paper 1, Q2 (1 mark): The diagram below shows a locus on an Argand diagram. 2 Re Im – 3 O Which of the equations below represents the locus shown above? Circle your answer. [1 mark]
- June 2023, Paper 2, Q6 (5 marks): (a) Express −5 − 5i in the form reiy, where −p < y × p [2 marks] 6 (b) The point on an Argand diagram that represents −5 − 5i is one of the vertices of an equil
- June 2023, Paper 2, Q9 (7 marks): The complex number z is such that z = 1 + i 1 − ki where k is a real number. 9 (a) Find the real part of z and the imaginary part of z, giving your answers in t
- June 2023, Paper 2, Q10 (8 marks): The region R on an Argand diagram satisfies both jz + 2ij × 3 and − p 6 × arg (z) × p 2 10 (a) Sketch R on the Argand diagram below. [3 marks] 1 1 – 1 – 2 – 3 –
- June 2023, Paper 2, Q15 (10 marks): (a) Given that z = cos y + i siny , use de Moivre’s theorem to show that zn − z−n = 2i sin ny [2 marks] 15 (b) The series S is defined as S = siny + sin 3y + ::
- June 2023, AS Paper 1, Q8 (4 marks): Abdoallah wants to write the complex number –1 + i√ 3 in the form r (cos θ + i sin θ) where r ≥ 0 and – π < θ ≤ π Here is his method: r = √ (–1)2 + (√ 3)2 = √ 1
- June 2023, AS Paper 1, Q12 (13 marks): (a) Show that (1 + i)൯ = – 4 [3 marks] 12 (b) The function f is defined by f (z) = z4 + 3z2 – 6z + 10 z ℂ 12 (b) (i) Show that (1 + i) is a root of f (z) = 0 [2
- June 2022, Paper 1, Q2 (1 mark): Simplify cos 6p 13 + i sin 6p 13 cos 2p 13 − i sin 2p 13 Tick (3) one box. [1 mark] cos 8p 13 + i sin 8p 13 cos 8p 13 − i sin 8p 13 cos 4p 13 + i sin 4p 13 cos
- June 2022, Paper 1, Q5 (6 marks): It is given that z = − 3 2 + i 11 p 2 is a root of the equation z4 − 3z3 − 5z2 + kz + 40 = 0 where k is a real number. 5 (a) Find the other three roots. [5 mark
- June 2022, Paper 1, Q8 (11 marks): (a) The complex number w is such that arg (w + 2i) = tan−1 1 2 It is given that w = x + iy , where x and y are real and x > 0 Find an equation for y in terms of
- June 2022, Paper 1, Q12 (17 marks): The Argand diagram shows the solutions to the equation z5 = 1 Im Re z5 z2 z3 z4 z1 12 (a) Solve the equation z5 = 1 giving your answers in the form z = cosy + i
- June 2022, Paper 2, Q1 (1 mark): Find the imaginary part of 5 + i 1 − i Circle your answer. [1 mark] −3 −2 2 3
- June 2022, AS Paper 1, Q4 (1 mark): The complex numbers w and z are defined as w = 2( cos a + i sin a) z = 3( cos b + i sin b) Find the product wz Tick (3) one box. [1 mark] 5(cos (ab) + i sin (ab
- June 2022, AS Paper 1, Q5 (3 marks): Show that (2 + i)3 is 2 + 11i [3 marks]
- June 2022, AS Paper 1, Q12 (6 marks): (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation jz − 2ij = 2 [2 marks] Re Im O – 1 – 1 – 2 – 3 – 4 – 5 5 4 3 2 1 – 2 – 3 –
- November 2021, Paper 1, Q2 (1 mark): Given that z = 1 − 3i is one root of the equation z2 + pz + r = 0 , where p and r are real, find the value of r. Circle your answer. [1 mark] −8 −2 6 10
- November 2021, Paper 1, Q6 (10 marks): (a) Show that the equation (2z − z≤)≤ = z2 has exactly four solutions and state these solutions. [7 marks] 6 (b) (i) Plot the four solutions to the equation in
- November 2021, Paper 2, Q2 (1 mark): Find arg (−4 − 7i) to the nearest degree. Circle your answer. [1 mark] −120° −60° 30° 60°
- November 2021, Paper 2, Q8 (6 marks): The complex number z satisfies the equations jz× − 1 − 2ij = jz − 3j and jz − aj = 3 where a is real. Show that a must lie in the interval 1 − s t p , 1 + s t p
- November 2021, Paper 2, Q13 (16 marks): (a) Two of the solutions to the equation cos 6y = 0 are y = p 4 and y = 3p 4 Find the other solutions to the equation cos 6y = 0 for 0 × y × p [2 marks] 13 (b)
- November 2021, AS Paper 1, Q1 (1 mark): The complex number o is shown below on the Argand diagram. Im ω O Re Which of the following complex numbers could be o? Tick (3) one box. [1 mark] cos (−2) + i
- November 2021, AS Paper 1, Q8 (6 marks): Stephen is correctly told that (1 + i) and −1 are two roots of the polynomial equation z3 − 2iz2 + pz + q = 0 where p and q are complex numbers. 8 (a) Stephen s
- November 2021, AS Paper 1, Q10 (8 marks): Matrix A is given by A = 3 i − 1 i 2 10 (a) Show that det A = a + i where a is an integer to be determined. [2 marks] 10 (b) Matrix B is given by B = 14 − 2i b
- November 2020, Paper 1, Q4 (6 marks): It is given that 1 − 3i is one root of the quartic equation z4 − 2z3 + pz2 + rz + 80 = 0 where p and r are real numbers. 4 (a) Express z4 − 2z3 + pz2 + rz + 80
- November 2020, Paper 1, Q6 (9 marks): Let w be the root of the equation z7 = 1 that has the smallest argument a in the interval 0 < a < p 6 (a) Prove that wn is also a root of the equation z7 = 1 fo
- November 2020, Paper 2, Q2 (1 mark): Given that arg (a + bi) = j , where a and b are positive real numbers and 0 < j < p 2 , three of the following four statements are correct. Which statement is n
- November 2020, AS Paper 1, Q1 (1 mark): Express the complex number 1 − i 3 p in modulus-argument form. Tick (3) one box. [1 mark] 2 cosp 3 + i sinp 3 2 cos 2p 3 + i sin 2p 3 2 cos − p 3 + i sin − p 3
- November 2020, AS Paper 1, Q2 (1 mark): Given that 1 − i is a root of the equation z3 − 3z2 + 4z − 2 = 0 , find the other two roots. Tick (3) one box. [1 mark] −1 + i and −1 1 + i and 1 −1 + i and 1 1
- November 2020, AS Paper 1, Q18 (5 marks): The locus of points L1 satisfies the equation jzj = 2 The locus of points L2 satisfies the equation arg (z + 4) = p 4 18 (a) Sketch L1 on the Argand diagram bel
- June 2019, Paper 1, Q4 (4 marks): Solve the equation 2z − 5iz* = 12 [4 marks]
- June 2019, Paper 1, Q8 (10 marks): (a) If z = cos y + i sin y , use de Moivre’s theorem to prove that zn − 1 zn = 2i sinny [3 marks] 8 (b) Express sin5 y in terms of sin 5y, sin 3y and siny [4 ma
- June 2019, Paper 1, Q9 (9 marks): (a) Solve the equation z3 = 2 p − 6 p i , giving your answers in the form reiy where r > 0 and 0 × y < 2p [5 marks] 9 (b) The transformation represented by the
- June 2019, Paper 2, Q1 (1 mark): Given that z is a complex number, and that z* is the complex conjugate of z, which of the following statements is not always true? Circle your answer. [1 mark]
- June 2019, Paper 2, Q6 (6 marks): A circle C in the complex plane has equation jz − 2 − 5ij = a The point z1 on C has the least argument of any point on C, and arg (z1) = p 4 Prove that a = 3p2
- June 2019, Paper 2, Q12 (5 marks): Abel and Bonnie are trying to solve this mathematical problem: z = 2 − 3i is a root of the equation 2z3 + mz2 + pz + 91 = 0 Find the value of m and the value of
- June 2019, AS Paper 1, Q8 (7 marks): Given that z1 = 2 cosp 6 + i sin p 6 and z2 = 2 cos 3p 4 + i sin 3p 4 8 (a) Find the value of jz1z2j [1 mark] 8 (b) Find the value of arg z1 z2 [1 mark] 8 (c) S
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