JEE Main · Maths · Complex Numbers and Quadratic Equations
Roots of Unity: JEE Main previous year questions
18 questions in the 108 JEE Main shifts Lumi analysed, in 6 of 8 years. Pattern: Asked most years. 5 came in 2024–26. Here is how it is asked and which ideas come back.
- Questions
- 18
- in 18 of 108 shifts
- Years asked
- 6/8
- last asked 2025
- Weighted estimate
- 0.2
- model; history: 0.2 a shift on average, typically 0.0–0.4
Key findings
- 01
Asked in 6 of 8 years with analysed shifts and in 18 of 108 shifts: 18 questions, #148 of 196 topics. The years that asked it were 1.6 apart on average. Last asked in 2025.
- 02
59% of its questions since 2020 tested an idea no earlier year had asked; 2 repeated an idea from the year before.
- 03
The most repeated idea is Periodicity of Omega Powers, asked 3 times in 3 shifts (2020, 2024, 2025), most often as “find unknowns from matrix equation then omega powers”.
- 04
JEE Main has asked 11 of the 63 ideas in this topic; 50 have never been asked, directly or inside another question.
- 05
Mostly asked as single correct (56%); 44% need a numerical answer, 0% are easy and 17% hard.
- 06
The skill tested most is multi-step (67%), then calculation (17%).
Questions a shift, year by year
Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.
Raw count: 2017: 1 in 1 shift · 2020: 2 in 12 shifts · 2021: 6 in 24 shifts · 2022: 4 in 24 shifts · 2023: 0 in 12 shifts · 2024: 2 in 18 shifts · 2025: 3 in 8 shifts · 2026: 0 in 9 shifts.
- Up to 2023
- 0.3
- a shift, of 25
- Since 2024
- 0.2
- -0.1
- Usual gap
- 1.6 yrs
- between years asking it
- Shifts with it
- 18/108
- 17% of analysed shifts
How it is asked
Difficulty, format and skill
Across all 18 questions on Roots of Unity since 2017.
Difficulty
QUESTIONS- Easy0 · 0%
- Medium15 · 83%
- Hard3 · 17%
Format
SHARE- Single correct · 10 Q56%
- Numerical value · 8 Q44%
Skill tested
SHARE- Multi-step · 12 Q67%
- Calculation · 3 Q17%
- Application · 2 Q11%
- Conceptual · 1 Q6%
Repeats
New idea, or one asked before?
Each column is a year. Blue questions tested an idea no earlier year had asked; navy ones went back to an idea from an earlier year. 2017 is the first year, so everything in it counts as new. Heights are question counts, and Lumi analysed 1 to 24 shifts a year, so compare the split inside a column rather than heights across years.
- Idea asked for the first time
- Idea asked in an earlier year
Coverage of the topic
JEE Main has asked 11 of the 63 ideas in this topic. 50 have never been asked yet.
- Asked directly11 · 17%
- Touched inside another question2 · 3%
- Never asked50 · 79%
Ideas come from Lumi’s JEE syllabus map. A never-asked idea is not a safe skip: across all of JEE Main, 22% of questions since 2024 tested an idea no earlier year had asked.
Drill down
Chapter, idea, question
Every idea JEE Main tested in this topic, with its eight years. Open one to read the questions.
10 ideas inside
Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.
Ideas
Every idea inside this topic
11 different ideas were tested; 6 came more than once.
find unknowns from matrix equation then omega powers
Single correct · 3 shifts · 1 numericaln from sum over cube root powers
Numerical value · 2 shifts · 2 numericalcomplex numbers as vertices of equilateral triangle
Numerical value · 2 shifts · 1 numericalcount common roots of two polynomials
Numerical value · 2 shifts · 1 numericalpower sum of roots of unity
Numerical value · 2 shifts · 1 numericaldeterminant with cube root of unity entries
Single correct · 2 shiftsvalue of power expression for root of polynomial
Single correct · 1 shiftarea of polygon from roots of complex equation
Single correct · 1 shiftvalue of series in powers of a complex number
Numerical value · 1 shift · 1 numericalpolynomial divisible by x^2+x+1
Numerical value · 1 shift · 1 numericalpower of expression with cube root of unity
Single correct · 1 shift
| Idea | Years asked | Times | Shifts | Format | Numerical |
|---|---|---|---|---|---|
| Periodicity of Omega Powersfind unknowns from matrix equation then omega powers | 3 | 3 | Single correctMulti-step | 1 | |
| Cyclic Powers of Omega Modulo 3n from sum over cube root powers | 2 | 2 | Numerical valueCalculation | 2 | |
| Equilateral Triangle Condition via Omegacomplex numbers as vertices of equilateral triangle | 2 | 2 | Numerical valueConceptual | 1 | |
| Roots of z^2 + z + 1 = 0count common roots of two polynomials | 2 | 2 | Numerical valueMulti-step | 1 | |
| Sum of p-th Powers of Rootspower sum of roots of unity | 2 | 2 | Numerical valueApplication | 1 | |
| Evaluating Determinants with Cube Rootsdeterminant with cube root of unity entries | 2 | 2 | Single correctCalculation | – | |
| Equations Reducible to Roots of Unityvalue of power expression for root of polynomial | 1 | 1 | Single correctApplication | – | |
| Roots of z^n = z_0area of polygon from roots of complex equation | 1 | 1 | Single correctMulti-step | – | |
| De Moivre's Theoremvalue of series in powers of a complex number | 1 | 1 | Numerical valueMulti-step | 1 | |
| Polynomial Remainder via Rootspolynomial divisible by x^2+x+1 | 1 | 1 | Numerical valueMulti-step | 1 | |
| Cube Roots of Unitypower of expression with cube root of unity | 1 | 1 | Single correctCalculation | – |
Times and shifts count every analysed shift. Across JEE Main, 78% of questions since 2024 went back to an idea an earlier year had asked; the rest were new. All repeated ideas
Every question
Roots of Unity questions, year by year
The opening words of each question as it appeared in the official paper, with the idea it tested. Open one for its options and the official answer.
JEE Main 20253 questions in 3 shifts
- 3 Apr, Shift 2Q5If are the vertices of an equilateral triangle, whose centroid is , then…
Idea: Equilateral Triangle Condition via Omega
MediumSingle correct - 4 Apr, Shift 2Q21If is a root of the equation and , then n is equal to…
Idea: Cyclic Powers of Omega Modulo 3
MediumNumerical value - 8 Apr, Shift 2Q5Let be a solution of , and for some and in ,…
Idea: Periodicity of Omega Powers
Asked the year beforeMediumSingle correct
JEE Main 20242 questions in 2 shifts
- 27 Jan, Shift 1Q21If satisfies the equation and , , then is equal to…
Idea: Periodicity of Omega Powers
MediumNumerical value - 30 Jan, Shift 2Q2If is a complex number, then the number of common roots of the equations and ,…
Idea: Roots of z^2 + z + 1 = 0
MediumSingle correct
JEE Main 20224 questions in 4 shifts
- 25 Jul, Shift 1Q2If are the roots of the equation , then…
Idea: Sum of p-th Powers of Roots
Asked the year beforeMediumSingle correct - 26 Jun, Shift 2Q23If , , then is equal to ______.
Idea: Roots of z^2 + z + 1 = 0
New ideaHardNumerical value - 27 Jun, Shift 1Q1The area of the polygon, whose vertices are the non-real roots of the equation is :
Idea: Roots of z^n = z_0
New ideaMediumSingle correct - 29 Jun, Shift 2Q1Let be a root of the equation . Then the value of is equal to :
Idea: Equations Reducible to Roots of Unity
New ideaMediumSingle correct
JEE Main 20216 questions in 6 shifts
- 3 Aug, Shift 1Q81If , then the value of is ________.
Idea: Cyclic Powers of Omega Modulo 3
New ideaMediumNumerical value - 26 Aug, Shift 1Q89Let , . Then the value of…
Idea: De Moivre's Theorem
New ideaHardNumerical value - 31 Aug, Shift 1Q63If , , then the determinant…
Idea: Evaluating Determinants with Cube Roots
HardSingle correct - 26 Feb, Shift 1Q82The sum of power of the roots of the equation is _________.
Idea: Sum of p-th Powers of Roots
New ideaMediumNumerical value - 18 Mar, Shift 1Q81Let be the roots of the equation and form an equilateral triangle with origin. Then, the value of…
Idea: Equilateral Triangle Condition via Omega
New ideaMediumNumerical value - 18 Mar, Shift 2Q81If and are two polynomials such that the polynomial is divisible by , then is equal…
Idea: Polynomial Remainder via Roots
New ideaMediumNumerical value
JEE Main 20202 questions in 2 shifts
- 8 Jan, Shift 2Q52Let . If and , then a…
Idea: Periodicity of Omega Powers
New ideaMediumSingle correct - 4 Sep, Shift 2Q53If a and b are real numbers such that , where , then is equal to :
Idea: Cube Roots of Unity
New ideaMediumSingle correct
JEE Main 20171 question in 1 shift
- 2 Apr (offline)Q63Let be a complex number such that where . If…
Idea: Evaluating Determinants with Cube Roots
MediumSingle correct
Questions from the official JEE Main Paper 1 shifts published by NTA (CBSE for 2017). Questions NTA dropped are left out. Topic and idea tags, counts and notes are Lumi’s.