plumiComplex numbers and Argand diagrams: WJEC A‑Level Further Maths questions
Every WJEC A-Level Further Maths exam question on 1.2 Complex numbers and Argand diagrams, 2018–2025, with its mark scheme and examiner’s report.
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All 20 questions on Complex numbers and Argand diagrams, as a list
- Summer 2025, AS Unit 1, Q1 (13 marks): 2 f x ( ) x 8 x 25 x 26 1. The function f is defined by . Show that f 3 2i . You must show all your working. 0 (a) [3] Solve the equation ( ) f
- Summer 2025, AS Unit 1, Q4 (5 marks): z 7 24i z a i b 4. The complex number z is given by . By writing , where a, b are real, find the two values of z . [5]
- Summer 2025, AS Unit 1, Q6 (6 marks): 6. The complex number z is represented by the point P(x, y) in an Argand diagram and z 2 5i 3 z q . i 5 Given that the locus of P is a circle, centre
- Summer 2024, AS Unit 1, Q1 (5 marks): 1. The complex numbers z, v and w are related by the equation v z . w , find z in the form 16 11i 5 2i cos isin v w r . Given that and [5]
- Summer 2024, AS Unit 1, Q2 (3 marks): 3 2 x 4 x 5 x x 7 x 15 2. Given that is a factor of , solve the equation 3 2 x x 7 x 15 0 . [3]
- Summer 2024, AS Unit 1, Q4 (10 marks): 4. The complex numbers z and w are represented, respectively, by the points P(x, y) and Q(u, v) in Argand diagrams and z w . 1 z y v (a) Show that and obta
- Summer 2024, AS Unit 1, Q6 (12 marks): 6. The complex number z is represented by the point P(x, y) in an Argand diagram. Two loci, L1 and L2, are given by: : z 2 i z 2 3i , L 1 : z 2 i
- Summer 2023, AS Unit 1, Q1 (4 marks): The complex number z is given by z = 3 + λi, where λ is a positive constant. The complex 1. conjugate of z is denoted by z . Given that z2 + z 2 = 2 , find the
- Summer 2023, AS Unit 1, Q3 (6 marks): Given that 5 – i is a root of the equation x4 −10x3 +10x2 +160x − 416 = 0 , 3. (a) write down another root of the equation, [1] (b) find the remaining roots. [5
- Summer 2023, AS Unit 1, Q6 (6 marks): The complex number z is represented by the point P(x, y) in an Argand diagram. 6. Given that z − 3 + i = 2 z − 5 − 2i , show that the locus of P is a circle and
- Summer 2023, AS Unit 1, Q9 (12 marks): 9. The complex numbers z and w are represented by the points P(x, y) and Q(u, v) respectively, in Argand diagrams, and w = 1− z2. (a) Find expressions for u and
- Summer 2022, AS Unit 1, Q1 (12 marks): z = 3 − 4i, w = 2 − i. 1. The complex numbers z, w are given by (a) (i) Find the modulus and argument of zw. r(cosθ + isinθ). (ii) Express zw in the form [5] v
- Summer 2022, AS Unit 1, Q5 (4 marks): 5. (a) The complex number z is represented by the point P(x, y) in an Argand diagram. Given that 3 + 2i = z 3 , z find the equation of the locus of P. [3] (b) G
- Summer 2019, AS Unit 1, Q3 (7 marks): The complex numbers z and w are represented by the points Z and W in an Argand diagram. 3. π The complex number z is such that |z| = 6 and arg z = . 3 The point
- Summer 2019, AS Unit 1, Q5 (6 marks): x = − 1 Given that and x = – 3 are two roots of the equation 5. 2 2x4 − x3 −15x2 + 23x +15 = 0 , find the remaining roots. [6]
- Summer 2019, AS Unit 1, Q6 (3 marks): The complex number z is represented by the point P(x, y) in an Argand diagram. 6. Given that |z – 1| = |z – 2i|, show that the locus of P is a straight line. [3
- Summer 2019, AS Unit 1, Q9 (8 marks): 9. The complex numbers z and w are represented by the points P(x, y) and Q(u, v) respectively in Argand diagrams and w = z2 – 1. (a) Show that v = 2xy and obtai
- Summer 2018, AS Unit 1, Q4 (7 marks): A complex number is defined by z = –3 + 4i. 4. Express z in the form r (cosθ + i sinθ ), where –πX θX π. (a) (i) Express z, the complex conjugate of z, in the f
- Summer 2018, AS Unit 1, Q6 (7 marks): Show that 1 – 2i is a root of the cubic equation x3 + 5x2 – 9x + 35 = 0. 6. (a) [3] (b) Find the other two roots of the equation. [4]
- Summer 2018, AS Unit 1, Q7 (5 marks): The complex number z is represented by the point P(x, y) in the Argand diagram and 7. |z – 4 – i| = |z + 2|. (a) Find the equation of the locus of P. [4] (b) Gi
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