plumiFurther matrices and complex numbers: Edexcel A‑Level Further Maths questions
Every Edexcel A-Level Further Maths exam question on FP.8 Further matrices and complex numbers, 2018–2025, with its mark scheme and examiner’s report.
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All 32 questions on Further matrices and complex numbers, as a list
- June 2025, Further Pure 2, Q2 (8 marks): A 1 2 2 4 (a) Determine the eigenvalues of matrix A (3) (b) Hence determine an orthogonal matrix P and a diagonal matrix D such that D = P TAP (5)
- 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, Further Pure 2, Q9 (8 marks): The transformation T from the z‑plane to the w‑plane, where z = x + iy and w = u + iv is given by w z z 2i z ≠ –2i The circle with equation ½ z ½ = 4 is mapped
- June 2025, AS Further Pure 2, Q4 (7 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A = 6 1 2 3 Determine a
- 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, Further Pure 2, Q4 (12 marks): A = 4 2 0 2 2 0 2 2 p where p is a constant Given that 2 1 2 is an eigenvector of A, (a) determine the eigenvalue corresponding to this eigenv
- 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 Further Pure 2, Q3 (7 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A 3 5 2 k where k is a constant
- June 2024, AS Further Pure 2, Q4 (9 marks): A circle C in the complex plane has equation i) (1 3 3 ( 3 i) z α z where α is a real constant with α > 1 Given that the imaginary axis is a tangent to C (a) sk
- June 2023, Further Pure 2, Q1 (6 marks): A 1 3 8 a where a is a constant. (a) Determine, in expanded form in terms of a, the characteristic equation for A. (2) (b) Hence use the Cayley-Hamilton
- 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 Further Pure 2, Q2 (8 marks): A linear transformation T : 2 → 2 is represented by the matrix M 5 1 3 k where k is a constant. Given that matrix M has a repeated eigenvalue, (a) de
- 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, Further Pure 2, Q2 (8 marks): Matrix M is given by M = 1 3 0 0 1 1 b a a where a and b are integers, such that a < b Given that the characteristic equation for M is λ3 – 7λ2 + 13λ +
- 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 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
- June 2022, AS Further Pure 2, Q2 (7 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. M 4 3 2 1 Find a matrix P and a
- 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 2021, Further Pure 2, Q8 (17 marks): A = 5 2 5 0 3 6 6 4 − − − p where p is a constant Given that 2 1 2 − is an eigenvector for A (a) (i) determine the eigenvalue co
- November 2020, Further Pure 2, Q3 (10 marks): M = 1 2 2 4 1 1 2 3 k − − where k is a constant. (a) Show that, in terms of k, a characteristic equation for M is given by λ3 – (2k + 13)λ + 5(k
- November 2020, Further Pure 2, Q5 (10 marks): A transformation T from the z‑plane to the w‑plane is given by w = 1 3 2 − + z z i z ≠ –2i The circle with equation ½ z + i ½ = 3 is mapped by T onto the circle
- November 2020, AS Further Pure 2, Q3 (10 marks): (i) A = 1 2 1 4 − (a) Show that the characteristic equation for A is λ2 – 5λ + 6 = 0 (2) (b) Use the Cayley‑Hamilton theorem to find integers p and q
- 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, Further Pure 2, Q2 (11 marks): The matrix A is given by A = − − − − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 6 2 2 1 2 1 (a) Show that 2 is a repeated eigenvalue of A and find the other eigenvalue. (5) (b) Henc
- June 2019, Further Pure 2, Q7 (6 marks): A transformation from the z-plane to the w-plane is given by w z z z = − + ≠− i i i (a) Show that the circle C with equation | z + i | = 1 in the z-plane is
- June 2019, AS Further Pure 2, Q1 (5 marks): Given that A = ⎛ ⎝⎜ ⎞ ⎠⎟ 3 2 2 2 (a) find the characteristic equation for the matrix A, simplifying your answer. (2) (b) Hence find an expression for the matrix
- 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 Further Pure 2, Q4 (7 marks): A = 1 1 2 4 − Find a matrix P and a diagonal matrix D such that D = P–1AP (7)
- 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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