10 What actually comes up
Three years of real questions, part by part, with the grade band each part sat at according to that year's marking schedule. Every question climbs the same ladder: (a) and (b) Achieved, (c) and (d) Merit, (e) Merit–Excellence. Start with the pattern table: it is the fastest way to decide what to revise.
Across 3 years
9 questions · 45 parts 18 Achieved 27 Merit 9 ExcellenceThe pattern — what to prepare
| Topic | Appears in | Grade level | What to practise |
|---|---|---|---|
| The remainder or factor theorem | Every year | Achieved | Remainder r on dividing by x − a: f(a) = r. A factor: f(a) = 0. For x + 3, put x = −3 in. No long division. |
| Surds: squaring a bracket, rationalising, and equations with √x in them | Every year | Achieved–Merit | Square a bracket into three terms. Rationalise with the conjugate. In an equation, get a root alone, square, and put every answer back into the original. |
| A quadratic with a letter in it: solving it, and discriminant conditions | Every year | Achieved–Merit | Treat the letter as a number in the formula. b2 − 4ac = 0 for equal roots, < 0 for none; for “real for every k”, say why it can never be negative. |
| Plotting, dividing, and writing a quotient as a + bi | Every year | Achieved–Merit | Multiply top and bottom by the conjugate of the bottom. If the unknown is inside the fraction, clear the fraction first and collect it on one side. |
| Polar form: a power or a quotient (De Moivre) | Every year | Achieved | Power: raise the modulus, multiply the argument. Quotient: divide the moduli, subtract the arguments. Write it as a + bi if that is the form asked for. |
| Solving zn = a number: all n roots, in polar form | Every year | Merit | Sketch the number to get its argument (a negative real is π, −i is −π/2). Root the modulus, add 2π n to the argument, divide by n, and give all n roots. |
| Given one complex root, find the rest | Every year | Achieved–Merit | Its conjugate is a root too. The two make a real quadratic factor; match the leading and constant terms to find the last factor and any unknown coefficients. |
| A modulus in terms of an unknown | 2023, 2025 | Achieved–Excellence | |a + bi| = √(a2 + b2). Square both sides to clear the roots, solve, and check every answer, including 0 and negative ones. |
| Comparing real and imaginary parts: equal numbers, an argument, a pair of equations | Every year | Merit–Excellence | Get each side to a + bi, then match the real parts and the imaginary parts as two equations. For arg = π/4, both parts are equal AND positive. |
| A locus, converted to Cartesian form | Every year | Merit–Excellence | Put z = x + yi and square. Two roots subtracted: move one across first. A tangent line: substitute it and set the discriminant to zero. |
The 2025 paper ↗ Marking schedule ↗
Question 1 — a remainder, a quadratic in k, a discriminant proof, a cube root, and a hyperbola
Five separate parts: a remainder, a quadratic solved in terms of a constant, a proof that a quadratic always has real roots, the cube roots of an imaginary number, and a locus that turns out to be a hyperbola.
- Not tried yet: a When the polynomial 3x4 + px3 − 4x + 5 is divided by x − 2, the remainder is 21. Find the value of p.Worked solution → Achieved
- Not tried yet: b Solve the equation x2 − 6kx = k2 for x. Simplify your answer as far as possible, giving it in terms of k.Worked solution → Achieved
- Not tried yet: c Prove that the equation (kx)2 = 3 − x/k has real roots for all real values of k, where k ≠ 0.Worked solution → Merit
- Not tried yet: d Solve the equation z3 + 8m27i = 0, where m is a positive real constant. Write your solution(s) in polar form, in terms of m.Worked solution → Merit
- Not tried yet: e The locus of a complex number z is described by |z − 5i| − |z + 5i| = 4. Find the Cartesian equation of the locus of z, giving your answer in the form ay2 − bx2 = k, where a, b and k are constants.Worked solution → Merit to Excellence
Question 2 — two points on an Argand diagram, a fifth power, a modulus, comparing parts, and an argument
Five separate parts: a combination of two points read off an Argand diagram, a power of a number in polar form, an equation with a modulus in it, two unknowns found by comparing real and imaginary parts, and an unknown found from an argument.
- Not tried yet: a On the paper’s Argand diagram, u is the point (4, 2) and w is the point (−2, −3). If z = 2u + 3w, find z, and show it clearly on an Argand diagram.
- u = 4 + 2i
- w = −2 − 3i
- Not tried yet: b If u = m cis(3π/10), write u5 in the form a + bi, where a and b are both real numbers, giving your answer in terms of m.Worked solution → Achieved
- Not tried yet: c Solve the equation |5 − mi| = √5m2 for m, where m is a real number.Worked solution → Merit
- Not tried yet: d If (g + 2i)(3 + hi) = (10 − 4i)(3 − i), find all possible values of g and h.Worked solution → Merit
- Not tried yet: e Given that arg((d + 6i)/(1 − di)) = π/4, find the possible value(s) of d, where d is a real constant.Worked solution → Merit to Excellence
Question 3 — a quadratic from one root, a surd squared, an equation in √x, a cubic with two unknowns, and simultaneous equations
Five separate parts: a quadratic built from one complex root, the square of a complex number written with surds, an equation in √x, a cubic with two unknown coefficients and one root given, and a pair of simultaneous equations in two complex unknowns.
- Not tried yet: a One solution of a quadratic equation, with real coefficients, is x = 2 + √p i. Find the quadratic equation, in terms of p, giving your answer in the form ax2 + bx + c = 0.Worked solution → Achieved
- Not tried yet: b Expand and simplify (√3a − √12a i)2, giving your answer in terms of a, where a is a real number.Worked solution → Achieved
- Not tried yet: c Solve the equation (1 + 2√x)(3 + 2√x) = 5 + 6√x for x.Worked solution → Merit
- Not tried yet: d One solution of the equation 3z3 + pz2 + qz − 8 = 0 is z = 1 + i. If p and q are both real, find the other two solutions of the equation, and the value of both p and q.Worked solution → Merit
- Not tried yet: e u and v are both complex numbers, and ui + 2v = 3 and u + (1 − i)v = 4. Solve this pair of simultaneous equations, giving u and v in the form a + bi. Show your working clearly.Worked solution → Merit to Excellence
The 2024 paper ↗ Marking schedule ↗
Question 1 — a factor, a power in polar form, a surd equation, a locus gradient, and a line a quotient never meets
Five separate parts: a factor of a cubic, a power of a number in polar form, an equation with two square roots, the gradient of a straight-line locus, and a proof that a quotient never lies on y = x.
- Not tried yet: a Given that x + 3 is a factor of x3 + px2 + 5x − 12, find the value of the constant p.Worked solution → Achieved
- Not tried yet: b If z = m cis(nπ/5), where m and n are positive real constants, find z15, giving your answer in polar form, in terms of m and n.Worked solution → Achieved
- Not tried yet: c Solve the equation 4 − √kx = √(kx + 4) for x, in terms of k, where k is a positive real constant.Worked solution → Merit
- Not tried yet: d The locus described by |z − i| = |z + 1| is a straight line. Find the gradient of that line, and justify your answer.Worked solution → Merit
- Not tried yet: e Consider the complex numbers u = 2 + 3ki and v = 4 + 5ki, where k is a real constant. Show that the complex number w = u/v will not lie on the line y = x in the Argand diagram, for any value of k.Worked solution → Merit to Excellence
Question 2 — a quotient in terms of k, equal roots, a modulus from a quotient, a complex cubic, and a circle locus
Five separate parts: a quotient written as a + bi, a quadratic with equal roots, the modulus of an unknown found from a quotient, a cubic with one complex root given, and a locus that is a circle.
- Not tried yet: a Write the complex number i/(2k + i) in the form a + bi, where a, b and k are real numbers, giving your answer in terms of k.Worked solution → Achieved
- Not tried yet: b Find the value(s) of r so that the quadratic equation 2x2 + (3 + 2r)x + 3 − 2r = 0 has equal roots.Worked solution → Achieved
- Not tried yet: c Given that w/(w + i) = 2 − i, find |w|.Worked solution → Merit
- Not tried yet: d One solution of the equation 2z3 + dz2 + 140z − 200 = 0 is z = 6 − 2i. If d is real, find the value of d and the other two solutions of the equation.Worked solution → Merit
- Not tried yet: e The locus of a complex number z is described by |z − 1 − 7i| = 2|z − 4 − 4i|. The complex number u = 3 + di lies on this locus. Find the Cartesian equation of the locus of z, giving your answer in the form (x − a)2 + (y − b)2 = k, and also find the complex number(s) u.Worked solution → Merit to Excellence
Question 3 — a surd fraction, a square on the Argand diagram, a conjugate equation, fourth roots, and a cubic identity
Five separate parts: a fraction of surds simplified, a complex number squared and plotted, an equation in a number and its conjugate, the fourth roots of a negative number, and a cube of x + 1/x.
- Not tried yet: a Write √2p/(√2p − √p) in the form a + √a, where p and a are real constants.Worked solution → Achieved
- Not tried yet: b On an Argand diagram, the point (−2, 3) represents the complex number z. Show clearly, on an Argand diagram, the point representing w = z2.
- z = −2 + 3i
- Not tried yet: c Find the value(s) of the real constant d, given that z = 3 + di and z̄ = 10dz−1.Worked solution → Merit
- Not tried yet: d Solve the equation z4 + 81k8 = 0, where k is a real constant. Give your solution(s) in polar form, in terms of k.Worked solution → Merit
- Not tried yet: e Given that x + 1/x = p, where p is a real constant, find the value of x3 + 1/x3, giving your answers in terms of p.Worked solution → Merit to Excellence
The 2023 paper ↗ Marking schedule ↗
Question 1 — a squared surd, no real roots, a real part of zero, a complex cubic, and a modulus equation
Five separate parts: a bracket with a surd squared, a quadratic with no real roots, a proof from a real part of zero, a cubic with one complex root given, and a constant found from a modulus.
- Not tried yet: a Write (5 − 2√p)2 in the form a + bp + c√p, where a, b and c are integers.Worked solution → Achieved
- Not tried yet: b Find the value(s) of r so that the quadratic equation 4x2 − 4x + 3r − 2 = 0 has no real roots.Worked solution → Achieved
- Not tried yet: c If z = p + qi and w = a + bi and Re(z/w) = 0, show that ap = −bq.Worked solution → Merit
- Not tried yet: d One solution of the equation z3 − 8z2 + 6z + d = 0 is z = 5 − i. If d is real, find the value of d and the other two solutions of the equation.Worked solution → Merit
- Not tried yet: e The complex numbers u and v are u = 3 + i and v = 1 + 2i. Determine the possible value(s) of the real constant k if |u/v + k| = √(k + 2).Worked solution → Merit to Excellence
Question 2 — dividing in polar form, a modulus in k, an argument, cube roots, and a tangent to a circle
Five separate parts: a quotient of two numbers in polar form, the modulus of a difference, an argument found from an equation, the cube roots of a negative number, and a line that touches a circular locus.
- Not tried yet: a If u = q6 cis(5π/8) and v = q2 cis(2π/5), write u/v in the form r cisθ.Worked solution → Achieved
- Not tried yet: b If z = 1 + ki and w = 7 − ki, find |z − w|, giving your answer in terms of k.Worked solution → Achieved
- Not tried yet: c Find Arg(z) if 13z/(z + 1) = 11 − 3i.Worked solution → Merit
- Not tried yet: d Solve the equation z3 + 64m12 = 0, where m is a real constant. Write your solution(s) in polar form, in terms of m.Worked solution → Merit
- Not tried yet: e The straight line with equation y = mx − 1, where m is a real constant and m > 0, is a tangent to the locus described by |z − 2 + i| = √3. Find the Cartesian equation of the locus AND the value of m.Worked solution → Merit to Excellence
Question 3 — a remainder, an unknown from a quotient, a surd equation, comparing parts, and a pair of conditions
Five separate parts: a remainder, an unknown found from a quotient, an equation with two square roots, two unknowns found from a reciprocal, and a constant pinned down by two conditions at once.
- Not tried yet: a When the polynomial 2x3 + px2 + 7x − 3 is divided by x + 3, the remainder is 30. Find the value of p.Worked solution → Achieved
- Not tried yet: b The complex numbers u and v are u = n − i and v = 2 − 3i. Given that u/v = 3 + 4i, find the value of n.Worked solution → Achieved
- Not tried yet: c Solve the equation 4√(4x − w) = 5 − 8√x for x, in terms of w.Worked solution → Merit
- Not tried yet: d Find the values of x and y, given that x and y are real and 1/(x + yi) = 1 − 1/(1 + pi), giving your answers in terms of p, where p is a positive real constant.Worked solution → Merit
- Not tried yet: e If z + 2i = iz + k, where k is a real number, and w/z = 2 + 2i, where Im(w) = 8, find the value of k.Worked solution → Merit to Excellence
Read the last column as a to-do list. In every question of 2023–2025, (a) and (b) are Achieved, (c) and (d) Merit, and (e) Merit–Excellence, the same ladder as 91578 and 91579. The cubic with one complex root given is in every paper, and it is a Merit part, not the hardest one.
Practice for this standard covers the whole topic — flashcards and quiz.