Complex numbers

Practice

Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.

Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.

Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.

Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.

What actually comes up

Three years of real questions, with the grade band each part sat at according to that year's marking schedule. The remainder theorem opens almost every paper — the least revision time for the most reliable marks — and a complex cubic with one root given closes almost every one, worth over-practising because it recurs in exactly the same shape.

Across 3 years

The pattern — what to prepare

The 2025 paper ↗ Marking schedule ↗

Question 1 — Remainder theorem, a parametric quadratic, a proof, a cube root, and a locus

The polynomial 3x⁴ + px³ − 4x + 5 is divided by x − 2, leaving remainder 21.

• f(2) = 21

• a When the polynomial 3x⁴ + px³ − 4x + 5 is divided by x − 2, the remainder is 21. Find the value of p. Achieved
• b x² − 6kx = k², where k is a real constant. Solve the equation for x, simplifying your answer as far as possible in terms of k. Merit
• c Prove that the equation (kx)² = 3 − kx has real roots for all real values of k, where k ≠ 0. Merit
• d m is a positive real constant. Solve z³ + 8m³i = 0. Write your solution(s) in polar form, in terms of m. Excellence
• e The locus of z is described by |z − 5i| − |z + 5i| = 4. Find the Cartesian equation of the locus, in the form ay² − bx² = k. Excellence

Question 3 — A quadratic from one complex root

One solution of a quadratic equation with real coefficients is x = p + 2i.

• a Find the quadratic equation, in terms of p, giving your answer in the form ax² + bx + c = 0. Merit

The 2024 paper ↗ Marking schedule ↗

Question 1 — Remainder theorem, a surd equation, a locus gradient, and a symmetry proof

x + 3 is a factor of x³ + px² + 5x − 12.

• a Given that x + 3 is a factor of x³ + px² + 5x − 12, find the value of the constant p. Achieved
• c k is a positive real constant. Solve the equation (4 − √(kx))² = kx + 4 for x, in terms of k. Merit
• d The locus described by |z − i| = |z + 1| is a straight line. Find the gradient of that line, justifying your answer. Merit
• e u = 2 + 3ki and v = 4 + 5ki, where k is real. Show that w = u/v never lies on the line y = x in the Argand diagram, for any value of k. Excellence

Question 2 — A quotient in terms of k, equal roots, a modulus equation, and a complex cubic

Four unrelated parts of the same question.

• a k is a real constant. Write the complex number i/(2k + i) in the form a + bi, in terms of k. Achieved
• b 2x² + (3 + 2r)x + 3 − 2r = 0. Find the value(s) of r so that the quadratic equation has equal roots. Merit
• c Given that w/(w + i) = 2 − i, find w. Merit
• d One solution of 2z³ + dz² + 140z − 200 = 0 is z = 6 − 2i, where d is real. Find the value of d and the other two solutions of the equation. Excellence
• e The locus of z is described by |z − 1 − 7i| = 2|z − 4 − 4i|, and u = 3 + di lies on this locus. Find the Cartesian equation of the locus in the form (x−a)² + (y−b)² = k, and find the complex number(s) u. Excellence

Question 3 — An Argand-diagram square, a modulus condition, a quartic root, and a cubic identity

Five unrelated parts of the same question.

• b On the Argand diagram, the point (−2, 3) represents z. Show clearly, by calculation, the point representing w = z². Achieved
• c z = 3 + di, and z̄ = 10d/z, where d is real. Find the value(s) of d, given z = 3 + di and z̄ = 10d/z. Merit
• d k is a real constant. Solve z⁴ + 81k⁸ = 0. Give your solution(s) in polar form, in terms of k. Excellence
• e x + 1/x = p, where p is a real constant. Find the value of x³ + 1/x³, in terms of p. Excellence

The 2023 paper ↗ Marking schedule ↗

Question 1 — A binomial surd, no real roots, a general proof, a complex cubic, and a scaled sum

Five unrelated parts of the same question.

• a p is a positive real constant. Write (5 − 2√p)² in the form a + bp + c√p, where a, b, c are integers. Achieved
• b 4x² − 4x + 3r − 2 = 0. Find the value(s) of r so that the quadratic equation has no real roots. Merit
• c z = p + qi and w = a + bi. Given that Re(z/w) = 0, show that ap = −bq. Merit
• d One solution of z³ − 8z² + 6z + d = 0 is z = 5 − i. Find the value of d and the other two solutions of the equation. Excellence
• e u = 3 + i and v = 1 + 2i, where k is real. Find the value(s) of k satisfying |u/(v + k)|² = k + 2. Excellence

Question 2 — Dividing in polar form, a modulus difference, an argument, a cube root, and a tangent line

Five unrelated parts of the same question.

• a u = q⁶ cis(5π/8) and v = q² cis(2π/5). Write u/v in the form r cis θ. Achieved
• b z = 1 + ki and w = 7 − ki, where k is real. Find |z − w|, giving your answer in terms of k. Merit
• c Find Arg(z) if 13z/(z + 1) = 11 − 3i. Merit
• d m is a real constant. Solve z³ + 64m¹² = 0. Write your solution(s) in polar form, in terms of m. Merit
• e The line y = mx − 1 (m > 0, real) is a tangent to the locus |z − 2 + i| = √3. Find the Cartesian equation of the locus AND the value of m. Excellence

Question 3 — Remainder theorem, a quotient constant, a surd equation, a reciprocal equation, and a modulus condition

2x³ + px² + 7x − 3 is divided by x + 3, leaving remainder 30.

• f(−3) = 30

• a When the polynomial 2x³ + px² + 7x − 3 is divided by x + 3, the remainder is 30. Find the value of p. Achieved
• b u = n − i and v = 2 − 3i, where u/v = 3 + 4i. Find the value of n, given u = n − i, v = 2 − 3i, and u/v = 3 + 4i. Achieved
• c w is a real constant. Solve 4√(4x − w) = 5 − 8√x for x, in terms of w. Merit
• d 1/(x + yi) = 1 − 1/(1 + pi), where p is a positive real constant and x, y are real. Find the values of x and y, in terms of p. Merit
• e z + 2i = iz + k, and w/z = 2 + 2i, where Im(w) = 8. Find the value of k, given z + 2i = iz + k, w/z = 2 + 2i, and Im(w) = 8. Excellence

Read the last column as a to-do list. The remainder theorem is worth the least revision time for the most reliable marks — it opens every paper and is pure real algebra. The complex-cubic question is the opposite: it is the hardest reliably-appearing part, and it always follows the same shape (conjugate → real quadratic factor → divide), which is worth over-practising precisely because it recurs.

Past exams & resources

⏱ Timed opens the paper and starts a countdown matching the exam length. Official NZQA PDFs. If a link 404s, that year’s paper may sit under a slightly different code: browse the NZQA subject page.