JEE Main · Maths · Determinants

Cramer's Rule: JEE Main previous year questions

Name: Cramer's Rule: JEE Main previous year questions
Creator: Lumi

87 questions in the 108 JEE Main shifts Lumi analysed, in 8 of 8 years. Pattern: Asked most years. 24 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 87; in 83 of 108 shifts
Years asked: 8/8; last asked 2026
Chance in a shift: 47%; calibrated model, how we compute it
Weighted estimate: 0.7; model; history: 0.7 a shift on average, typically 0.6–0.9

Key findings

01
Asked in 8 of 8 years with analysed shifts and in 83 of 108 shifts: 87 questions, #11 of 196 topics. It came in every year with analysed shifts.
02
23% of its questions since 2020 tested an idea no earlier year had asked; 61 repeated an idea from the year before.
03
The most repeated idea is Condition for Infinite Solutions, asked 25 times in 25 shifts (2020, 2021, 2022, 2023, 2024, 2025, 2026), most often as “parameters for infinitely many solutions”.
04
JEE Main has asked 19 of the 57 ideas in this topic; 35 have never been asked, directly or inside another question.
05
Mostly asked as single correct (85%); 14% need a numerical answer, 14% are easy and 8% hard.
06
The skill tested most is multi-step (45%), then calculation (32%).

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: 11 in 12 shifts · 2021: 24 in 24 shifts · 2022: 18 in 24 shifts · 2023: 9 in 12 shifts · 2024: 14 in 18 shifts · 2025: 4 in 8 shifts · 2026: 6 in 9 shifts.

Up to 2023: 0.8; a shift, of 25
Since 2024: 0.6; -0.2
Usual gap: 1.3 yrs; between years asking it
Shifts with it: 83/108; 77% of analysed shifts

Recent framings

parameters for infinitely many solutions
non-trivial solution then sum trig roots
parameters for no solution of linear system

Revise first

Condition for Infinite Solutions · asked 2020, 2021, 2022, 2023, 2024, 2025, 2026
Evaluating Parameters for Consistency · asked 2020, 2021, 2022, 2023, 2024, 2025
Condition for No Solution · asked 2017, 2020, 2021, 2022, 2023, 2026

How it is asked

Difficulty, format and skill

Across all 87 questions on Cramer's Rule since 2017.

Difficulty

QUESTIONS

1268

Easy12 · 14%
Medium68 · 78%
Hard7 · 8%

Format

Single correct · 74 Q85%
Numerical value · 12 Q14%
Other · 1 Q1%

Skill tested

Multi-step · 39 Q45%
Calculation · 28 Q32%
Application · 18 Q21%
Conceptual · 2 Q2%

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 19 of the 57 ideas in this topic. 35 have never been asked yet.

1935

Asked directly19 · 33%
Touched inside another question3 · 5%
Never asked35 · 61%

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.

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

Open83 · 95%95%8/811211891445
16 ideas inside
- OpenRepeated24 · 29%29%7/8253382
- OpenRepeated17 · 20%20%6/8455
- OpenRepeated15 · 18%18%6/824323
- OpenRepeated9 · 11%11%5/842
- OpenRepeated4 · 5%5%3/82
- OpenRepeated4 · 5%5%4/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
- Open1 · 1%1%1/8
Open3 · 3%3%2/82
Open1 · 1%1%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

19 different ideas were tested; 6 came more than once.

Condition for Infinite Solutions25×
parameters for infinitely many solutions
Single correct · 25 shifts · 5 numerical
Condition for No Solution17×
parameters for no solution of linear system
Single correct · 17 shifts · 1 numerical
Evaluating Parameters for Consistency15×
parameters for infinitely many solutions
Single correct · 15 shifts
Parameter Evaluation in Homogeneous Systems9×
parameter values for non-trivial homogeneous solution
Single correct · 9 shifts · 2 numerical
Homogeneous System Solutions4×
nature of solutions of homogeneous system
Single correct · 4 shifts
Condition for Unique Solution4×
number of solutions of linear system
Single correct · 4 shifts
Solving Column Vector Matrix Equations1×
parameter interval for negative solution of system
Numerical value · 1 shift · 1 numerical
Consistency of Overdetermined Systems (3 eq, 2 var)1×
unique solution of overdetermined system then distance
Single correct · 1 shift
Complex Coefficients in Cramer's Rule1×
non-trivial solution of homogeneous complex system
Single correct · 1 shift
Parametrizing Infinite Solutions1×
nature of solutions of linear system
Single correct · 1 shift
Quick Identification via Proportionality1×
prime power dividing a factorial determinant
Numerical value · 1 shift · 1 numerical
Trigonometric Coefficients and Solution Counting1×
count parameter values for no solution
Single correct · 1 shift
Arithmetic Progression Coefficients Trick1×
unknown entry from zero determinant with ap columns
Single correct · 1 shift
Conditions for Integer Solutions1×
integer abscissa of line intersection
Single correct · 1 shift
Eigenvalue-Type Homogeneous Systems1×
determinant from ab=b and trace condition
Numerical value · 1 shift · 1 numerical
Probability in Linear Systems1×
probability of unique or no solution of linear system
Single correct · 1 shift
Skew-Symmetric Coefficient Matrices1×
homogeneous system with skew-symmetric coefficient matrix
Single correct · 1 shift
Geometric Meaning of Non-Trivial Homogeneous Solutions1×
count solutions of homogeneous system with norm
Single correct · 1 shift
Variable Ratios in Homogeneous Systems1×
count integer solutions of homogeneous system
Numerical value · 1 shift · 1 numerical

Idea	Times	Shifts	Format	Numerical
Condition for Infinite Solutionsparameters for infinitely many solutions	25	25	Single correctCalculation	5
Condition for No Solutionparameters for no solution of linear system	17	17	Single correctApplication	1
Evaluating Parameters for Consistencyparameters for infinitely many solutions	15	15	Single correctMulti-step	–
Parameter Evaluation in Homogeneous Systemsparameter values for non-trivial homogeneous solution	9	9	Single correctMulti-step	2
Homogeneous System Solutionsnature of solutions of homogeneous system	4	4	Single correctMulti-step	–
Condition for Unique Solutionnumber of solutions of linear system	4	4	Single correctApplication	–
Solving Column Vector Matrix Equationsparameter interval for negative solution of system	1	1	Numerical valueMulti-step	1
Consistency of Overdetermined Systems (3 eq, 2 var)unique solution of overdetermined system then distance	1	1	Single correctMulti-step	–
Complex Coefficients in Cramer's Rulenon-trivial solution of homogeneous complex system	1	1	Single correctMulti-step	–
Parametrizing Infinite Solutionsnature of solutions of linear system	1	1	Single correctMulti-step	–
Quick Identification via Proportionalityprime power dividing a factorial determinant	1	1	Numerical valueMulti-step	1
Trigonometric Coefficients and Solution Countingcount parameter values for no solution	1	1	Single correctMulti-step	–
Arithmetic Progression Coefficients Trickunknown entry from zero determinant with ap columns	1	1	Single correctApplication	–
Conditions for Integer Solutionsinteger abscissa of line intersection	1	1	Single correctMulti-step	–
Eigenvalue-Type Homogeneous Systemsdeterminant from ab=b and trace condition	1	1	Numerical valueMulti-step	1
Probability in Linear Systemsprobability of unique or no solution of linear system	1	1	Single correctMulti-step	–
Skew-Symmetric Coefficient Matriceshomogeneous system with skew-symmetric coefficient matrix	1	1	Single correctMulti-step	–
Geometric Meaning of Non-Trivial Homogeneous Solutionscount solutions of homogeneous system with norm	1	1	Single correctConceptual	–
Variable Ratios in Homogeneous Systemscount integer solutions of homogeneous system	1	1	Numerical valueMulti-step	1

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

Cramer's Rule 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

Chapter, idea, question

Every idea inside this topic

Cramer's Rule questions, year by year

JEE Main 20266 questions in 6 shifts

JEE Main 20254 questions in 4 shifts

JEE Main 202414 questions in 14 shifts

JEE Main 20239 questions in 9 shifts

JEE Main 202218 questions in 17 shifts

JEE Main 202124 questions in 21 shifts

JEE Main 202011 questions in 11 shifts

JEE Main 20171 question in 1 shift

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