JEE Main · Maths · Complex Numbers and Quadratic Equations

Complex Numbers - Polar Form: JEE Main previous year questions

Name: Complex Numbers - Polar Form: JEE Main previous year questions
Creator: Lumi

57 questions in the 108 JEE Main shifts Lumi analysed, in 7 of 8 years. Pattern: Asked most years. 16 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 57; in 54 of 108 shifts
Years asked: 7/8; last asked 2026
Chance in a shift: 47%; calibrated model, how we compute it
Weighted estimate: 0.4; model; history: 0.4 a shift on average, typically 0.2–0.6

Key findings

01
Asked in 7 of 8 years with analysed shifts and in 54 of 108 shifts: 57 questions, #43 of 196 topics.
02
51% of its questions since 2020 tested an idea no earlier year had asked; 21 repeated an idea from the year before.
03
The most repeated idea is Modulus and Argument, asked 13 times in 13 shifts (2020, 2021, 2022, 2023, 2024), most often as “distance of point from complex modulus and argument”.
04
JEE Main has asked 20 of the 59 ideas in this topic; 30 have never been asked, directly or inside another question.
05
Mostly asked as single correct (72%); 23% need a numerical answer, 7% are easy and 28% hard.
06
The skill tested most is multi-step (70%), then calculation (12%).

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: 0 in 1 shift · 2020: 8 in 12 shifts · 2021: 13 in 24 shifts · 2022: 15 in 24 shifts · 2023: 5 in 12 shifts · 2024: 12 in 18 shifts · 2025: 2 in 8 shifts · 2026: 2 in 9 shifts.

Up to 2023: 0.4; a shift, of 25
Since 2024: 0.3; -0.1
Usual gap: 1.0 yrs; between years asking it
Shifts with it: 54/108; 50% of analysed shifts

How it is asked

Difficulty, format and skill

Across all 57 questions on Complex Numbers - Polar Form since 2017.

Difficulty

QUESTIONS

3716

Easy4 · 7%
Medium37 · 65%
Hard16 · 28%

Format

Single correct · 41 Q72%
Numerical value · 13 Q23%
Two statements · 3 Q5%

Skill tested

Multi-step · 40 Q70%
Calculation · 7 Q12%
Conceptual · 5 Q9%
Application · 5 Q9%

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.

’17’20’21’22’23’24’25’26

Idea asked for the first time
Idea asked in an earlier year

Coverage of the topic

JEE Main has asked 20 of the 59 ideas in this topic. 30 have never been asked yet.

20930

Asked directly20 · 34%
Touched inside another question9 · 15%
Never asked30 · 51%

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

Idea, year, question

Every idea JEE Main tested in this topic, with its eight years. Open one to read the questions.

ideaAskedShareYears’17’20’21’22’23’24’25’26

OpenRepeated13 · 23%23%5/8326
- 5 Apr ’24 S1 · Q2
  Consider the following two statements : Statement I : For any two non-zero complex numbers $z_1, z_2$ ,…
  hardTwo statementsAsked the year before
- 6 Apr ’24 S2 · Q3
  If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z_2}}\right|=2$ , then
  hardSingle correctAsked the year before
- 1 Feb ’24 S1 · Q1
  Let $S=\{z\in\mathbb{C}:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\}$ . Let $z_1, z_2\in S$ be such that…
  hardSingle correctAsked the year before
- 29 Jan ’24 S1 · Q3
  If $z=\frac{1}{2}-2i$ is such that $|z+1|=\alpha z+\beta(1+i)$ , $i=\sqrt{-1}$ and $\alpha,\beta\in\mathbb{R}$ , then $\alpha+\beta$ is…
  easySingle correctAsked the year before
- 30 Jan ’24 S1 · Q2
  If $z=x+iy,\ xy\neq0$ , satisfies the equation $z^2+i\bar{z}=0$ , then $|z^2|$ is equal to :
  mediumSingle correctAsked the year before
- 31 Jan ’24 S1 · Q22
  If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$ , where…
  mediumNumerical valueAsked the year before
- 8 Apr ’23 S1 · Q2
  If for $z=\alpha+i\beta$ , $|z+2|=z+4(1+i)$ , then $\alpha+\beta$ and $\alpha\beta$ are the roots of the equation
  mediumSingle correctAsked the year before
- 28 Jul ’22 S2 · Q88
  Let $z=a+ib,\ b\neq 0$ be complex numbers satisfying $z^2=\bar{z}\cdot 2^{1-|z|}$ . Then the least value of $n\in\mathbb{N}$ , such that…
  hardNumerical valueAsked the year before
- 30 Jul ’22 S2 · Q62
  Let $z_1 = 1 + 2i$ , $z_2 = 2 + i$ , $\frac{1}{z_1}+\frac{1}{z_2}=\frac{6}{w}$ , and $z = \frac{iw}{2-\overline{w}}$ . Consider the statements…
  mediumTwo statementsAsked the year before
- 4 Aug ’21 S1 · Q62
  Read the question
  mediumSingle correctAsked the year before
- 9 Jan ’20 S1 · Q53
  Read the question
  mediumSingle correctNew idea
- 9 Jan ’20 S2 · Q53
  Read the question
  mediumSingle correctNew idea
- 6 Sep ’20 S2 · Q53
  Read the question
  mediumSingle correctNew idea
OpenRepeated8 · 14%14%5/832
OpenRepeated6 · 11%11%3/832
OpenRepeated5 · 9%9%3/822
Open4 · 7%7%1/84
OpenRepeated4 · 7%7%3/82
Open2 · 4%4%1/82
OpenRepeated2 · 4%4%2/8
OpenRepeated2 · 4%4%2/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8
Open1 · 2%2%1/8

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

20 different ideas were tested; 9 came more than once.

Modulus and Argument13×
distance of point from complex modulus and argument
Single correct · 13 shifts · 2 numerical
Polar Triangle Inequality Extrema8×
max/min distance between points on circles
Single correct · 8 shifts · 2 numerical
Locus of Argument Ratio (Circular Arc)6×
modulus from locus and argument conditions
Single correct · 6 shifts · 1 numerical
Polar Conjugate and Inverse5×
real part from modulus and imaginary condition
Single correct · 5 shifts · 2 numerical
Purely Real and Imaginary Euler Conditions4×
statement i/ii on locus of complex numbers
Single correct · 4 shifts · 1 numerical
De Moivre's Theorem for Integer Powers4×
form quadratic from real and imaginary parts of power
Numerical value · 3 shifts · 3 numerical
Geometric Rotation2×
complex plane circle and argument of point
Single correct · 2 shifts
Principal Argument Arithmetic2×
argument of complex numbers from conjugate relation
Single correct · 2 shifts
Area of Rotated Complex Triangles2×
area of triangle from modulus and argument
Single correct · 2 shifts
Locus of Argument (Ray)1×
area of region defined by inequalities
Single correct · 1 shift
90-Degree Rotation via 'i'1×
principal argument after rotation of complex numbers
Single correct · 1 shift
Distance Formula in Polar Coordinates1×
radius of circle locus of complex numbers
Numerical value · 1 shift · 1 numerical
Coni Method (Rotation Theorem)1×
identify false statement on complex number geometry
Single correct · 1 shift
Locus of Full Line via Argument1×
range of parameter from complex locus conditions
Single correct · 1 shift
Polar Form for Binomial Powers1×
real part of complex expression
Single correct · 1 shift
Simplifying Conjugate Binomial Sums1×
value of expression with complex number powers and conjugate
Single correct · 1 shift
Collinearity via Argument1×
locus of complex number with real ratio
Single correct · 1 shift
Conversion of sin(θ) + i cos(θ)1×
power of complex ratio with trig terms
Single correct · 1 shift
Smallest Integer Power for Real/Imaginary1×
gcd of least exponents making power equal one
Numerical value · 1 shift · 1 numerical
Square Root via Polar Form1×
imaginary part of complex square roots
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Modulus and Argumentdistance of point from complex modulus and argument	13	13	Single correctMulti-step	2
Polar Triangle Inequality Extremamax/min distance between points on circles	8	8	Single correctMulti-step	2
Locus of Argument Ratio (Circular Arc)modulus from locus and argument conditions	6	6	Single correctMulti-step	1
Polar Conjugate and Inversereal part from modulus and imaginary condition	5	5	Single correctMulti-step	2
Purely Real and Imaginary Euler Conditionsstatement i/ii on locus of complex numbers	4	4	Single correctMulti-step	1
De Moivre's Theorem for Integer Powersform quadratic from real and imaginary parts of power	4	3	Numerical valueMulti-step	3
Geometric Rotationcomplex plane circle and argument of point	2	2	Single correctMulti-step	–
Principal Argument Arithmeticargument of complex numbers from conjugate relation	2	2	Single correctMulti-step	–
Area of Rotated Complex Trianglesarea of triangle from modulus and argument	2	2	Single correctApplication	–
Locus of Argument (Ray)area of region defined by inequalities	1	1	Single correctMulti-step	–
90-Degree Rotation via 'i'principal argument after rotation of complex numbers	1	1	Single correctMulti-step	–
Distance Formula in Polar Coordinatesradius of circle locus of complex numbers	1	1	Numerical valueMulti-step	1
Coni Method (Rotation Theorem)identify false statement on complex number geometry	1	1	Single correctMulti-step	–
Locus of Full Line via Argumentrange of parameter from complex locus conditions	1	1	Single correctMulti-step	–
Polar Form for Binomial Powersreal part of complex expression	1	1	Single correctCalculation	–
Simplifying Conjugate Binomial Sumsvalue of expression with complex number powers and conjugate	1	1	Single correctCalculation	–
Collinearity via Argumentlocus of complex number with real ratio	1	1	Single correctConceptual	–
Conversion of sin(θ) + i cos(θ)power of complex ratio with trig terms	1	1	Single correctMulti-step	–
Smallest Integer Power for Real/Imaginarygcd of least exponents making power equal one	1	1	Numerical valueApplication	1
Square Root via Polar Formimaginary part of complex square roots	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

Complex Numbers - Polar Form questions, year by year

The newest 40 show their opening words as they appeared in the official paper; older ones name the idea they tested. Open one for its options and the official answer.

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.

Key findings

Questions a shift, year by year

Difficulty, format and skill

Difficulty

Format

Skill tested

New idea, or one asked before?

Coverage of the topic

Idea, year, question

Every idea inside this topic

Complex Numbers - Polar Form questions, year by year

JEE Main 20262 questions in 2 shifts

JEE Main 20252 questions in 2 shifts

JEE Main 202412 questions in 11 shifts

JEE Main 20235 questions in 5 shifts

JEE Main 202215 questions in 14 shifts

JEE Main 202113 questions in 12 shifts

JEE Main 20208 questions in 8 shifts

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