If you are wondering how to study Complex Numbers for JEE, start with one number. JEE Main asked 62 questions from this chapter across 2024, 2025 and 2026, and 23 of them were about a locus or a region in the Argand plane. JEE Advanced asked 22 questions from the chapter between 2015 and 2026, at least one in every single year. This guide breaks all of them down by type, based on the papers conducted by the National Testing Agency (jeemain.nta.ac.in) for JEE Main and the JEE Advanced papers from 2015 to 2026.
We went through every one of those questions and tagged it by the idea that solves it: locus and regions in the Argand plane, algebra of complex numbers (modulus, argument, conjugates and equations in z), polar form with cube roots and nth roots of unity, and geometry with rotation. The tables below show exactly where the marks come from, and the rest of the guide shows how to win them.
The chapter has a reputation for being abstract. The papers say otherwise. Most questions reduce to one of two pictures: a circle or a line in the plane, or a point on the unit circle at a known angle. If you can draw the picture, the algebra is usually short.
It is steady, every year. JEE Main had 24 questions from this chapter in 2024, 19 in 2025 and 19 in 2026. That is 4% of all Maths questions in each of those years, or about one question in every paper.
JEE Advanced has never skipped it. Between 2015 and 2026, Advanced asked at least one Complex Numbers question every year, and two or three in every year since 2018. Only a handful of Maths chapters can say that.
Two ideas carry most of the chapter. Locus and region questions (23) and algebra of complex numbers (21) gave 44 of the 62 JEE Main questions, about seven in every ten. Add polar form and roots of unity (10) and you cover 54 of 62.
It feeds other chapters. Roots of unity appear in polynomial and quadratic questions, the geometry of |z − a| = r is the geometry of circles from Coordinate Geometry, and De Moivre's theorem shows up in trigonometric sums. A strong Complex Numbers chapter also helps with Sequences and Series, where sums of powers of ω are a favourite.
"In Complex Numbers, every condition is a picture. |z − a| = r is a circle, |z − a| = |z − b| is a line, arg is an angle. Students who draw first and calculate second solve these questions in half the time." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Real result · JEE 2026
Seven in ten JEE Main Complex Numbers questions are a locus or a piece of algebra done cleanly. Learn both 1-on-1 with an IITian Maths HOD and turn every condition into a picture in seconds.
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Any plan for how to study Complex Numbers for JEE should start from the papers. Here is how many questions JEE Main asked from this chapter in each of the last three years.
| Year | Questions | MCQ + Numerical | Share of Maths Qs |
|---|---|---|---|
| 2024 | 24 | 17 + 7 | 4% |
| 2025 | 19 | 15 + 4 | 4% |
| 2026 | 19 | 17 + 2 | 4% |
| Total | 62 | 49 + 13 | — |
And here are the same 62 questions sorted by the idea that solves them.
| Sub-topic | 2024 | 2025 | 2026 | Total |
|---|---|---|---|---|
| Locus and regions in the Argand plane (circles, lines, ellipses; maximum and minimum of |z| or |z − a|) | 9 | 6 | 8 | 23 |
| Algebra of complex numbers (modulus, argument, conjugate, solving equations in z, powers of i) | 10 | 5 | 6 | 21 |
| Polar form, De Moivre, cube roots of unity (ω) and nth roots | 3 | 2 | 5 | 10 |
| Geometry with complex numbers (rotation, triangles) | 0 | 2 | 0 | 2 |
| Not classified* | 2 | 4 | 0 | 6 |
| Total | 24 | 19 | 19 | 62 |
*The archived text of six questions did not render fully, so we could not tell which idea they need. They are counted in the yearly totals only. Questions that use two ideas are counted under the one that finishes them.
The message of this table is simple. Locus questions gave 23 of 62 and algebra gave 21. Polar form and roots of unity are the part that grew in 2026, with 5 questions against 3 in 2024 and 2 in 2025. Pure geometry with rotation, which many books spend a whole section on, gave only 2.
Pattern 1: Nearest and farthest points on a circle or region. 2024 asked for the minimum of |z + ½(3 + 4i)| when |z| ≤ 1. 2025 asked for the minimum of |z1 − z2| when |z1 − (8 + 2i)| ≤ 1 and |z2 − (2 − 6i)| ≤ 2. 2026 asked for the minimum of |z + ½(5 + 3i)| on the ring 3 ≤ |2z − 3(1 + i)| ≤ 7. Every one of these is "distance between centres, plus or minus the radii".
Pattern 2: Two conditions, one point. 2026 combined |z + 2| = |z − 2| with arg((z + 3)/(z − i)) = π/4 and asked for |z|2. 2024 asked how many z satisfy |z − i| = |z + i| = |z − 1|. Each condition is a line or a circle; the answer is where they meet.
Pattern 3: Ellipses hidden in the modulus. |z − z1| + |z − z2| = k is an ellipse with foci z1 and z2, but only if k is larger than |z1 − z2|. 2026 asked for the largest distance between a point of the disc |z − 2| ≤ 4 and a point of |z − 2| + |z + 2| = 5, and another 2026 question combined a circle with a sum-of-distances condition.
Pattern 4: Large powers. (z201 − i)8 for z = (√3 + i)/2 (2026), (1 + ω)7 written as A + Bω + Cω2 (2024), and the common roots of z1985 + z100 + 1 = 0 and z3 + 2z2 + 2z + 1 = 0 (2024). Write the number as eiθ or use ω3 = 1, reduce the exponent, and the question is over.
Pattern 5: Symmetric functions of two complex numbers. 2024 gave z1 + z2 = 5 and z13 + z23 = 20 + 15i and asked for |z14 + z24|. 2026 gave the roots of a quadratic with β − α = √11 and β2 − α2 = 3i√11. These are quadratic-equation identities used with complex values, and they need careful algebra rather than any new idea.
This is the 12-year view that most guides skip. JEE Advanced asked this chapter in every year from 2015 to 2026.
| Year | Qs | What was asked |
|---|---|---|
| 2026 | 2 | A match-the-columns on roots α, β of given equations and expressions in them · a multi-correct question building complex quantities from two matrices |
| 2025 | 2 | A set defined from z1 = 1 + 2i and z2 = 3i that turns out to be a circle; its centre and radius (multi-correct) · α = arg of Σ(−ω)n for n = 1 to 2025 (numerical) |
| 2024 | 2 | Powers of (−1 + √2) in the set {a + b√2} (multi-correct) · a quartic with real coefficients and the root i√3 (numerical) |
| 2023 | 2 | |z|3 + 2z2 + 4z̄ − 8 = 0 with Im z ≠ 0, matched to values of expressions (match) · the set (1967 + 1686i sin θ)/(7 − 3i cos θ) contains exactly one positive integer (numerical) |
| 2022 | 3 | (2 + 3z + 4z2)/(2 − 3z + 4z2) is real with Im z ≠ 0; find |z|2 (numerical) · the number of distinct roots of an equation in z and z̄ (numerical) · possible |z| when the real and imaginary parts of an expression are integers (multi-correct) |
| 2021 | 2 | Ten angles with sum 2π and a chain of points zk on the unit circle; inequalities on Σ|zk − zk−1| (single correct) · an argument condition that makes a locus a circle (multi-correct) |
| 2020 | 2 | S = {z : |z2 + z + 1| = 1}; true statements (multi-correct) · z4 − |z|4 = 4iz2; minimum of |z1 − z2|2 (numerical) |
| 2019 | 2 | A point z0 of the region |z − 2 + i| ≥ √5 that extremises a modulus; the principal argument of an expression (single correct) · the minimum of |a + bω + cω2|2 over distinct non-zero integers (numerical) |
| 2018 | 2 | False statements about the principal argument (multi-correct) · the solution set of sz + tz̄ + r = 0 and its intersection with a circle (multi-correct) |
| 2017 | 1 | A condition on an expression in z with a − b = 1; possible values of x (multi-correct) |
| 2016 | 1 | z = 1/(a + ibt) for real t ≠ 0; the locus of z (multi-correct) |
| 2015 | 1 | ak = cos(kπ/7) + i sin(kπ/7); the value of a ratio of sums (numerical) |
| Total | 22 | 10 multi-correct · 8 numerical · 2 single correct · 2 match |
Two 2024 questions filed under this chapter in the archive are really about surds and polynomials; they are listed above but counted separately in the breakdown below.
Advanced leans further towards algebra than Main does: 9 of the 22 questions were algebra of z and z̄ (equations, conjugates, "this expression is real"), 5 were locus questions and 4 were polar form or roots of unity. The style is different too. Ten of the 22 were multi-correct, so you have to test every statement, not just find one that works.
The smartest way to decide how to study Complex Numbers for JEE is to match your time to each sub-topic. Use the time column in your mock tests: if you are consistently slower than the target, that sub-topic needs more practice, not more reading.
| Sub-topic | Main Qs (2024–26) | Advanced Qs (2015–26) | Difficulty | Target time |
|---|---|---|---|---|
| Locus and regions | 23 | 5 | Medium | 2–3 min |
| Algebra of complex numbers | 21 | 9 | Easy–Medium | 90 s–2 min |
| Polar form and roots of unity | 10 | 4 | Medium | 2 min |
| Geometry with rotation | 2 | 0 | Medium–Hard | 3 min |
| Other (surds, polynomials, matrices) | — | 4 | Hard | 3–4 min |
Difficulty and target time are our faculty's guidance for exam practice, not exam data. "Other" includes the two 2024 Advanced questions on surds and polynomials and the two 2026 Advanced questions whose archived text did not render fully.
Locus and regions covers every condition that describes a set of points: |z − a| = r (circle), |z − a| = |z − b| (perpendicular bisector), |z − a| + |z − b| = k (ellipse), arg((z − a)/(z − b)) = θ (arc of a circle), Re or Im of a ratio equal to zero, and the nearest and farthest points of such sets.
Algebra of complex numbers covers the arithmetic: modulus, argument, conjugate, powers of i, and solving equations in z by putting z = x + iy or by using z̄ = |z|2/z.
Polar form and roots of unity covers z = r eiθ, De Moivre's theorem, the cube roots of unity 1, ω, ω2, and nth roots of unity.
Geometry with rotation covers (z3 − z1)/(z2 − z1) = (|z3 − z1|/|z2 − z1|) eiθ, equilateral triangles and areas of triangles with complex vertices.
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Nearest-point questions, sum-of-distances traps and principal arguments are where students lose marks in this chapter. Work through them with your own faculty, at your pace, with every doubt answered the same day.
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Knowing how to study Complex Numbers for JEE also means knowing what not to do. These are the mistakes we see most often when students solve this chapter's PYQs.
Keep this table open while you practise. Most of the chapter runs on these results, which is why learning how to study Complex Numbers for JEE is mostly about turning each condition into a picture or a formula from this list.
| Sub-topic | Formula / result |
|---|---|
| Basics | i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i · z z̄ = |z|2 · z + z̄ = 2 Re z · z − z̄ = 2i Im z · 1/z = z̄/|z|2 |
| Modulus | |z1z2| = |z1||z2| · |z1/z2| = |z1|/|z2| · ||z1| − |z2|| ≤ |z1 ± z2| ≤ |z1| + |z2| |
| Modulus identities | |z1 + z2|2 + |z1 − z2|2 = 2(|z1|2 + |z2|2) · |z1 + z2|2 = |z1|2 + |z2|2 + 2 Re(z1z̄2) |
| Argument | arg(z1z2) = arg z1 + arg z2 (mod 2π) · arg(z1/z2) = arg z1 − arg z2 (mod 2π) · arg z̄ = −arg z · principal value in (−π, π] |
| Real and imaginary conditions | z is real ⇔ z = z̄ · z is purely imaginary ⇔ z + z̄ = 0 · for |z| = 1, z̄ = 1/z |
| Polar form | z = r(cos θ + i sin θ) = reiθ · De Moivre: (cos θ + i sin θ)n = cos nθ + i sin nθ |
| Cube roots of unity | ω = e2πi/3 = (−1 + i√3)/2 · 1 + ω + ω2 = 0 · ω3 = 1 · ω̄ = ω2 · 1 + ω = −ω2 · |a + bω + cω2|2 = ½[(a − b)2 + (b − c)2 + (c − a)2] for real a, b, c |
| Factorisations | a2 + ab + b2 = (a − bω)(a − bω2) · a3 + b3 = (a + b)(a + bω)(a + bω2) · z3 + 2z2 + 2z + 1 = (z + 1)(z2 + z + 1) |
| nth roots of unity | e2πik/n, k = 0 … n − 1 · their sum is 0 · their product is (−1)n−1 · they form a regular n-gon on the unit circle |
| Circle | |z − a| = r · |z − a| = k|z − b| with k ≠ 1 is a circle (Apollonius) · Re or Im of (z − a)/(z − b) = 0 gives a circle or line through a and b |
| Line | |z − a| = |z − b| is the perpendicular bisector of a and b · āz + az̄ + c = 0 (c real) is a line |
| Ellipse and hyperbola | |z − z1| + |z − z2| = 2a (2a > |z1 − z2|) ellipse · ||z − z1| − |z − z2|| = 2a (2a < |z1 − z2|) hyperbola branch pair |
| Arc | arg((z − a)/(z − b)) = θ is an arc of a circle through a and b · θ = ±π/2 gives the circle on ab as diameter (one side) |
| Rotation | (z3 − z1)/(z2 − z1) = (|z3 − z1|/|z2 − z1|) eiθ · equilateral triangle: z12 + z22 + z32 = z1z2 + z2z3 + z3z1 |
This is the core of how to study Complex Numbers for JEE: one sub-topic at a time, starting with algebra because every other part uses it. Do not move on until you can solve the PYQ types listed under each heading without notes.
This is the foundation, and it is where speed comes from. Three tools solve almost everything:
Worked example from JEE Main 2026 (numerical): z = (1 + i)(1 + 2i)…(1 + ni) and |z|2 = 44200. Since |z|2 is the product of |1 + ki|2 = 1 + k2, we need 2 × 5 × 10 × 17 × … = 44200. The product 2 × 5 × 10 × 17 × 26 = 44200, so n = 5. No complex arithmetic was needed at all.
Worked example from JEE Main 2024: z1 + z2 = 5 and z13 + z23 = 20 + 15i. With s = 5 and p = z1z2, s3 − 3ps = 125 − 15p = 20 + 15i, so p = 7 − i. Then z12 + z22 = s2 − 2p = 11 + 2i, and z14 + z24 = (11 + 2i)2 − 2p2 = (117 + 44i) − (96 − 28i) = 21 + 72i, whose modulus is 75.
Practice target: 20 algebra PYQs. Write which of the three tools you will use before you start each one.
This is the biggest group in JEE Main, and it rewards drawing. The routine:
Worked example from JEE Main 2026: |z + 2| = |z − 2| and arg((z + 3)/(z − i)) = π/4. The first condition is the perpendicular bisector of −2 and 2, the imaginary axis, so z = iy. Then (3 + iy)/(i(y − 1)) = (y − 3i)/(y − 1), and an argument of π/4 needs equal positive real and imaginary parts: y = −3. So z = −3i and |z|2 = 9.
Worked example from JEE Main 2026: z lies on the ring 3 ≤ |2z − 3(1 + i)| ≤ 7, which is 3/2 ≤ |z − (3/2)(1 + i)| ≤ 7/2. The point −½(5 + 3i) is at distance |4 + 3i| = 5 from the centre, outside the outer circle, so the minimum of |z + ½(5 + 3i)| is 5 − 7/2 = 3/2.
Worked example from JEE Main 2026: the largest distance between a point of the disc |z − 2| ≤ 4 and a point of the ellipse |z − 2| + |z + 2| = 5. The ellipse has a = 5/2 and foci ±2, so the farthest ellipse point from the focus 2 is at distance a + c = 9/2. Adding the disc radius 4 gives 17/2.
Practice target: 25 locus PYQs. Draw every one, even when you think you can do it in your head.
"For a nearest or farthest point question, never parametrise. Join the centres with a straight line. The answer is always on that line, at the distance between centres plus or minus the radii." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Large powers and cyclic sums belong here. Two habits make these quick:
Worked example from JEE Main 2026: z = (√3 + i)/2 = eiπ/6, so z201 = ei·201π/6. Since 201π/6 = 32π + 3π/2, z201 = e3πi/2 = −i. Then z201 − i = −2i and (−2i)8 = 256.
Worked example from JEE Main 2024: the common roots of z1985 + z100 + 1 = 0 and z3 + 2z2 + 2z + 1 = 0. The cubic factors as (z + 1)(z2 + z + 1), with roots −1, ω and ω2. For z = ω, ω1985 = ω2 and ω100 = ω, so the first equation becomes 1 + ω + ω2 = 0, which is true; the same works for ω2. For z = −1 it gives −1 + 1 + 1 = 1, which is not zero. So there are 2 common roots.
Worked example from JEE Main 2024 (numerical): (1 + ω)7 = A + Bω + Cω2. Since 1 + ω = −ω2, (1 + ω)7 = −ω14 = −ω2, so A = 0, B = 0, C = −1 and 5(3A − 2B − C) = 5.
Practice target: 15 PYQs on powers and roots of unity, including at least five with ω.
This is small in recent papers but quick to learn. 2025 asked about an equilateral triangle with complex vertices and about the triangle formed by the origin, √3 + 2√2i and another point. Learn the rotation formula and the equilateral-triangle condition from the formula table, and practise five problems. That is enough for JEE Main.
Practice target: 5 rotation problems, then move on.
That is how to study Complex Numbers for JEE sub-topic by sub-topic. Once you have finished all four, test yourself under exam conditions with our JEE test series, and bring any question that still does not make sense to the JEE Doubt Lab for a 1-on-1 explanation.
Ready for exam-level practice?
JEE Advanced asks this chapter every year, mostly as multi-correct questions mixing z, z̄ and |z|. Practise them 1-on-1 with an IITian Maths HOD who has taught them for years.
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If you are deciding how to study Complex Numbers for JEE Advanced as well as Main, this section is for you. Advanced uses the same ideas but goes further in four places.
Advanced 2022 gave (2 + 3z + 4z2)/(2 − 3z + 4z2) real with Im z ≠ 0 and asked for |z|2; the answer is 0.5. Setting the expression equal to its conjugate and cancelling the factor (z − z̄), which is non-zero because Im z ≠ 0, leaves a condition that forces |z|2 = ½. The skill is spotting which factor to cancel, and using the given condition to justify it.
Advanced 2023 gave |z|3 + 2z2 + 4z̄ − 8 = 0, Advanced 2020 gave z4 − |z|4 = 4iz2, and Advanced 2022 asked for the number of distinct roots of an equation in z and z̄. Put z = x + iy, split into real and imaginary parts, and solve the two real equations together. Use the condition Im z ≠ 0 when it is given; it usually removes a factor.
Advanced 2019 asked for the minimum of |a + bω + cω2|2 over distinct non-zero integers a, b, c. Using |a + bω + cω2|2 = ½[(a − b)2 + (b − c)2 + (c − a)2], the smallest value comes from three consecutive integers, giving ½(1 + 1 + 4) = 3. Advanced 2025 asked for the argument of a long sum of powers of −ω, which needs the period of −ω (it is 6) before anything else.
Advanced 2021 put ten points zk on the unit circle using angles that add to 2π and asked whether Σ|zk − zk−1| ≤ 2π. Each |zk − zk−1| is a chord, and a chord is never longer than its arc, so the total is at most the circumference. Turning a modulus into a chord length is a classic Advanced idea.
"Advanced loves the conjugate. When a question says an expression is real, or gives you z̄ and |z| together, the first move is z z̄ = |z|2 or z = z̄. Students who go straight to x + iy without trying this often drown in algebra." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
The last step in how to study Complex Numbers for JEE is an honest self-check. You should be able to answer "yes" to every line below without notes.
Complex Numbers gave JEE Main 62 questions from 2024 to 2026 and JEE Advanced 22 questions from 2015 to 2026, in every single year. Locus questions (23) and algebra (21) gave 44 of the Main questions, and polar form with roots of unity added 10 more. Advanced leans towards algebra with conjugates, and asks most of its questions as multi-correct. So if you are working out how to study Complex Numbers for JEE, the order is clear: master the algebra tools first, then learn to turn every condition into a picture, then practise powers and roots of unity, and keep rotation as a short final topic.
How we built this data. We went through every JEE Main shift paper from 2024, 2025 and 2026 and every JEE Advanced paper from 2015 to 2026, picked out each question from this chapter and tagged it by the idea that solves it. Where the archived text of a question was too incomplete to classify with confidence, we counted it in the totals and said so instead of guessing. Advanced questions filed here from other topics are listed and footnoted.
If you want a faculty member to check your preparation, every pack starts with a free 30-minute mentorship call. See all Maths chapter packs on our JEE Maths coaching page, and once your exam is done, our JEE counselling team can help you turn your rank into the right college and branch.
JEE Main asked 24 questions from this chapter in 2024, 19 in 2025 and 19 in 2026, counting all shifts. That is 4% of Maths questions each year, or about one question per paper.
JEE Main questions are mostly easy to medium once you can turn each condition into a circle, line or angle. JEE Advanced questions are harder because they mix z, z̄ and |z| in one equation and are usually multi-correct. If you follow this guide on how to study Complex Numbers for JEE, the Main questions become routine with steady practice.
Yes. It was asked 62 times in JEE Main from 2024 to 2026 and in every JEE Advanced exam from 2015 to 2026. Its ideas, especially roots of unity, also appear in questions from other chapters.
No. Skipping it gives up about one question per JEE Main paper and at least one question in every JEE Advanced exam. The chapter is short, and most of its questions use a small set of results, so it is one of the better returns on time in Maths.
Master locus and regions first, then the algebra of z, z̄ and |z|. In JEE Main 2024–2026 these two gave 44 of the 62 questions, about seven in ten. Add polar form and roots of unity and you cover 54 of 62.
Equations and conditions that mix z, z̄ and |z|, such as "this expression is real" or "find the number of roots". They made up 9 of the 22 Advanced questions from 2015 to 2026. Advanced also asks about the minimum of |a + bω + cω2| and inequalities for points on the unit circle. 10 of the 22 Advanced questions were multi-correct.
