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AS & A Level · AS/A Level

Further Mathematics

Complex geometry and roots

Name: ____________________Date: October 10, 2026
  1. 1.

    Express (3 + 4i)/(1 - 2i) as a + bi and find its modulus.

    [3 marks] · no calculator
  2. 2.

    For z = x + iy, identify and sketch the locus |z - (2 + i)| = 3, labelling its real-axis intersections.

    [3 marks] · no calculator
  3. 3.

    Find all cube roots of -8i in modulus-argument form with arguments in (-pi, pi].

    [3 marks] · no calculator
  4. 4.

    A real-coefficient cubic x^3 + ax^2 + bx - 10 has root 1 + 2i. Find a, b and the remaining roots.

    [3 marks] · no calculator
  5. 5.

    Find the points satisfying both |z - 1| = |z + i| and |z| = sqrt(2). Give exact answers and explain the geometry.

    [3 marks] · no calculator
  6. 6.

    Use De Moivre's theorem to derive cos(3theta) = 4cos^3(theta) - 3cos(theta). Then solve 4u^3 - 3u = 0 for real u.

    [3 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.

Marking points are indicative, not an official mark scheme. Accept equivalent valid methods and supported interpretations that address the task; award each mark once without requiring the model wording.