2022 May/June · Paper 31 · Q7 (a, b, c)8 marksComplex arithmetic & equations · Modulus-argument / exponential form · Square roots of complex numbers
2022 May/June · Paper 32 · Q10 (a, b, c, d)11 marksArgand loci · Polynomial with complex roots
2022 May/June · Paper 33 · Q5 (a, b, c)8 marksArgand loci · Complex arithmetic & equations · Modulus-argument / exponential form
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Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Argand loci · Core · Paper 3 style · 4 marks
(a)Describe fully the locus of points in an Argand diagram representing complex numbers z that satisfy |z − 3 + 2i| = 4.[2]
(b)Find the greatest value of |z| for points on this locus. Give your answer correct to 3 significant figures.[2]
Show answer
(a) A circle with centre (3, −2), i.e. the point 3 − 2i, and radius 4.
B1 circle centre 3 − 2i; B1 radius 4.
(b) √13 + 4 ≈ 7.61
M1 distance from origin to centre = √(3² + 2²) = √13; A1 √13 + 4 = 7.61.
Q2Argand loci · Core · Paper 3 style · 4 marks
(a)The complex number z = x + iy satisfies |z − 1| = |z + 3i|. Describe the locus of z geometrically.[1]
(b)Find the Cartesian equation of the locus in the form ax + by + c = 0.[3]
Show answer
(a) The perpendicular bisector of the line joining the points 1 and −3i (i.e. (1, 0) and (0, −3)).
(a)The locus of z is given by arg(z + 2) = π/3. Find a Cartesian equation for the locus, stating any restriction on x.[2]
(b)Determine whether the point representing 1 + 3√3 i lies on the locus.[2]
Show answer
(a) y = √3(x + 2), x > −2
M1 tan(π/3) = y/(x + 2); A1 with x > −2.
(b) Yes: arg(3 + 3√3 i) = tan⁻¹(√3) = π/3.
M1 z + 2 = 3 + 3√3 i; A1 argument is π/3, so yes.
Q4Argand loci · Core · Paper 3 style · 3 marks
(a)The complex number z satisfies both |z| = 5 and |z − 6| = 5. Explain why Re z = 3.[1]
(b)Find the two possible values of z.[2]
Show answer
(a) z is equidistant from 0 and 6, so it lies on the perpendicular bisector x = 3.
B1.
(b) z = 3 + 4i or z = 3 − 4i
M1 3² + y² = 25; A1 y = ±4.
Q5Argand loci · Standard · Paper 3 style · 5 marks
(a)The complex number z satisfies |z − 4 − 3i| = 2. Find the least value of |z|.[2]
(b)Find the greatest value of arg z for points on this locus. Give your answer in radians correct to 3 significant figures.[3]
Show answer
(a) 3
M1 |4 + 3i| = 5; A1 5 − 2 = 3.
(b) 1.06 rad
B1 arg(4 + 3i) = tan⁻¹(3/4) = 0.6435; M1 angle between the line to the centre and the tangent = sin⁻¹(2/5) = 0.4115; A1 0.6435 + 0.4115 = 1.055 ≈ 1.06.
Q6Argand loci · Standard · Paper 3 style · 5 marks
(a)The region R in an Argand diagram consists of points z satisfying both |z − 2 − 3i| ≤ 3 and |z| ≥ |z − 4|. Describe R in words.[2]
(b)Find the greatest value of |z| for points in R. Give your answer correct to 3 significant figures.[2]
(c)State the least value of Re z for points in R.[1]
Show answer
(a) The part of the disc with centre 2 + 3i and radius 3 that lies on or to the right of the line Re z = 2 (the perpendicular bisector of 0 and 4).
B1 disc centre (2, 3), radius 3; B1 |z| ≥ |z − 4| ⇔ Re z ≥ 2, so R is the right-hand half of the disc.
(b) √13 + 3 ≈ 6.61
M1 farthest point lies on the line through O and the centre (it has Re z = 2 + 6/√13 > 2, so it is in R); A1 √13 + 3 = 6.61.
(c) 2
B1.
Q7Argand loci · Standard · Paper 3 style · 5 marks
(a)The complex number z = x + iy satisfies |z − 4i| = |z − 6|. Find the Cartesian equation of the locus of z.[3]
(b)Find the least value of |z| for points on this locus, giving your answer correct to 3 significant figures.[2]
M1 points on L₁: (1 + t) + (1 + t)i, t > 0; M1 (t − 4)² + t² = 9 ⇒ 2t² − 8t + 7 = 0; A1 t = 2 ± √2/2 (both positive, so both points lie on the half-line); A1 both complex numbers.
Q9Argand loci · Challenge · Paper 3 style · 6 marks
(a)The complex number z satisfies |z + 2 − i| = 1. Show that the locus of z touches the real axis, and state the point of contact.[2]
(b)Find the least and greatest values of arg z for points on the locus, where −π < arg z ≤ π. Give the least value in radians correct to 3 significant figures.[4]
Show answer
(a) Centre −2 + i is 1 unit from the real axis, equal to the radius; contact at z = −2.
B1 centre (−2, 1), radius 1; B1 distance to real axis = 1, contact at (−2, 0).
(b) Least ≈ 2.21 rad (exactly π − 2 tan⁻¹(½)); greatest = π
B1 greatest is π, at the point of contact −2; M1 arg(−2 + i) = π − tan⁻¹(½) = 2.678; M1 subtract sin⁻¹(1/√5) = 0.4636 for the other tangent; A1 2.214 ≈ 2.21.
Q10Argand loci · Challenge · Paper 3 style · 7 marks
(a)The region R consists of points z satisfying |z − 3i| ≤ 2 and Re z ≥ 1. Find the least value of arg z for points in R. Give your answer in radians correct to 3 significant figures, and show that the point where it occurs is in R.[4]
(b)Find the greatest value of |z| for points in R. Give your answer correct to 3 significant figures.[3]
Show answer
(a) ≈ 0.841 rad (π/2 − sin⁻¹(2/3)); at the tangent point Re z = 2√5/3 ≈ 1.49 ≥ 1
M1 tangent from O to circle centre 3i, radius 2; M1 angle = π/2 − sin⁻¹(2/3); A1 0.841; B1 tangent length √(9 − 4) = √5, so Re z = √5 cos(π/2 − sin⁻¹(2/3)) = √5 × 2/3 = 2√5/3 ≈ 1.49 ≥ 1.
(b) √(13 + 6√3) ≈ 4.84
M1 on the circle x² + (y − 3)² = 4 we have |z|² = 6y − 5, so maximise y subject to x ≥ 1; M1 x = 1 gives y = 3 + √3; A1 |z| = √(13 + 6√3) = 4.84.
Q11Complex arithmetic & equations · Core · Paper 3 style · 5 marks
(a)The complex numbers z and w are given by z = 3 − 4i and w = 1 + 2i. Find zw in the form x + iy.[2]
(b)Find z/w in the form x + iy, showing all necessary working.[3]
Show answer
(a) 11 + 2i
M1 expand, using i² = −1; A1.
(b) −1 − 2i
M1 multiply numerator and denominator by 1 − 2i; M1 denominator 5; A1 (−5 − 10i)/5 = −1 − 2i.
Q12Complex arithmetic & equations · Core · Paper 3 style · 4 marks
(a)Solve the equation z² − 6z + 13 = 0, giving your answers in the form x + iy.[3]
(b)State the sum and the product of the two roots.[1]
Show answer
(a) z = 3 + 2i, z = 3 − 2i
M1 use the formula or complete the square; M1 √(36 − 52) = √(−16) = 4i; A1 3 ± 2i.
(b) Sum 6, product 13
B1.
Q13Complex arithmetic & equations · Core · Paper 3 style · 5 marks
(a)Express (3 + 2i)/(4 − i) in the form x + iy, where x and y are exact fractions.[3]
(b)Find the modulus of (3 + 2i)/(4 − i), giving your answer in exact form.[2]
Q30Polynomial with complex roots · Challenge · Paper 3 style · 7 marks
(a)The equation z³ + az² + bz + 10 = 0, where a and b are real, has a root 1 + 3i. Write down a second root and find the quadratic factor of z³ + az² + bz + 10 with these two roots.[3]
(b)Find the values of a and b and the third root.[4]
