New Zealand Scholarship · Calculus and Advanced Mathematics

Complex Numbers, De Moivre's Theorem and Polynomial Theory

40 practice questions, each with the answer and an explanation.

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All 40 questions

  1. What is the modulus and principal argument of the complex number z = -1 + i*sqrt(3)?
  2. Using De Moivre's theorem, evaluate (1 + i)^8.
  3. If 2 - 3i is a root of the cubic equation x^3 - 3x^2 + 9x + 13 = 0 with real coefficients, what are the other two roots?
  4. What geometric locus is described by the equation |z - (2 + 3i)| = 5 in the complex Argand plane?
  5. What geometric locus is described by |z - 4| = |z + 2i| in the complex plane?
  6. What are the three cube roots of unity (solutions to z^3 = 1) in exponential form?
  7. What is the remainder when the polynomial P(x) = 2x^4 - 3x^3 + 5x - 7 is divided by (x - 2)?
  8. Express (cos(theta) + isin(theta))^3 in terms of cos(3theta) and sin(3*theta) using De Moivre's theorem.
  9. If z = a + bi, what is z * conjugate(z)?
  10. What is the value of i^(2026)?
  11. Using De Moivre's theorem, express cos(3*theta) entirely in terms of powers of cos(theta).
  12. Find all complex solutions to the polynomial equation z^4 + 16 = 0.
  13. If w is a non-real cube root of unity satisfying w^3 = 1 and w != 1, evaluate the algebraic expression (1 + w)^5.
  14. If alpha, beta, gamma are roots of x^3 - 4x^2 + 6x - 2 = 0, what is the value of alpha^2 + beta^2 + gamma^2?
  15. What is the Cartesian equation of the locus defined by arg((z - 2) / (z + 2)) = pi/2?
  16. Evaluate the product of the four fourth roots of unity: 1 i (-1) * (-i).
  17. If z + 1/z = 2*cos(theta), what is z^n + 1/z^n for any positive integer n?
  18. Find the cubic polynomial with integer coefficients whose roots are twice the roots of x^3 - 2x^2 + 3x - 1 = 0.
  19. What is the modulus of (3 + 4i) / (1 - 2i)?
  20. If P(x) = x^4 - 2x^3 + 3x^2 - 4x + 2, what is the value of P(1) and what does it indicate about (x - 1)?
  21. What is the principal argument of z = -5i in radians?
  22. If 1 + i is a root of the quadratic equation x^2 + bx + c = 0 with real coefficients, find b and c.
  23. Express sin(3*theta) in terms of sin(theta) using de Moivre's theorem.
  24. What geometric locus is described by |z - 3| = 2*|z + 3| in the complex plane?
  25. What is the value of (1 + i*sqrt(3))^6?
  26. If polynomial P(x) = x^3 + px + q has roots alpha, beta, gamma, what is the sum alpha + beta + gamma?
  27. Simplify the complex fraction (2 + 3i) / (3 - 2i).
  28. What are the solutions to the equation z^2 + 2z + 5 = 0?
  29. Find the sum of the infinite series 1 + cos(theta)/2 + cos(2theta)/4 + cos(3theta)/8 + ... for any real theta.
  30. What is the condition on real coefficients a, b for x^2 + ax + b to have two distinct purely imaginary roots?
  31. If w is a complex root of z^5 = 1 with w != 1, evaluate 1 + w + w^2 + w^3 + w^4.
  32. What is the remainder when the polynomial P(x) = x^100 is divided by (x^2 - 1)?
  33. If z = re^(itheta), what is the conjugate of z in exponential polar form?
  34. Find all complex roots of the polynomial equation z^3 + 8 = 0.
  35. How many distinct roots does the polynomial P(x) = (x^2 + 1)^3 * (x - 2)^4 have in the complex field C?
  36. What is the modulus of the complex number 3 - 4i?
  37. Use De Moivre's theorem to write (1 + i)^6 in the form a + bi.
  38. What is the sum of the three complex cube roots of 8?
  39. A cubic equation with real coefficients has 2 + 3i as one root. Which of these must also be a root?
  40. Find the two square roots of the complex number 3 + 4i.

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