JEE Main · Maths · Application of Derivatives
Rolle's Theorem: JEE Main previous year questions
9 questions in the 108 JEE Main shifts Lumi analysed, in 4 of 8 years. Pattern: Comes in cycles. 2 came in 2024–26. Here is how it is asked and which ideas come back.
- Questions
- 9
- in 9 of 108 shifts
- Years asked
- 4/8
- last asked 2024
- Weighted estimate
- 0.0
- model; history: 0.0 a shift on average, typically 0.0–0.1
Key findings
- 01
Asked in 4 of 8 years with analysed shifts and in 9 of 108 shifts: 9 questions, #182 of 196 topics. The years that asked it were 1.3 apart on average. Last asked in 2024.
- 02
56% 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 Higher-Order Derivative Roots, asked 4 times in 4 shifts (2021, 2022, 2024), most often as “least number of zeros of higher derivative”.
- 04
JEE Main has asked 5 of the 64 ideas in this topic; 55 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 33% hard.
- 06
The skill tested most is multi-step (56%), then application (33%).
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: 1 in 12 shifts · 2021: 3 in 24 shifts · 2022: 3 in 24 shifts · 2023: 0 in 12 shifts · 2024: 2 in 18 shifts · 2025: 0 in 8 shifts · 2026: 0 in 9 shifts.
- Up to 2023
- 0.1
- a shift, of 25
- Since 2024
- 0.0
- 0.0
- Usual gap
- 1.3 yrs
- between years asking it
- Shifts with it
- 9/108
- 8% of analysed shifts
How it is asked
Difficulty, format and skill
Across all 9 questions on Rolle's Theorem since 2017.
Difficulty
QUESTIONS- Easy0 · 0%
- Medium6 · 67%
- Hard3 · 33%
Format
SHARE- Single correct · 5 Q56%
- Numerical value · 4 Q44%
Skill tested
SHARE- Multi-step · 5 Q56%
- Application · 3 Q33%
- Conceptual · 1 Q11%
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 5 of the 64 ideas in this topic. 55 have never been asked yet.
- Asked directly5 · 8%
- Touched inside another question4 · 6%
- Never asked55 · 86%
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.
4 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
5 different ideas were tested; 2 came more than once.
least number of zeros of higher derivative
Numerical value · 4 shifts · 2 numericalcount real roots of polynomial equation
Numerical value · 2 shifts · 1 numericalminimum number of solutions via rolle on auxiliary function
Numerical value · 1 shift · 1 numericalfind unknown coefficients so theorem holds
Single correct · 1 shiftvalue of second derivative at rolle point
Single correct · 1 shift
| Idea | Years asked | Times | Shifts | Format | Numerical |
|---|---|---|---|---|---|
| Higher-Order Derivative Rootsleast number of zeros of higher derivative | 4 | 4 | Numerical valueApplication | 2 | |
| Polynomial Root Countingcount real roots of polynomial equation | 2 | 2 | Numerical valueApplication | 1 | |
| Auxiliary Functions: Product and Quotient Rulesminimum number of solutions via rolle on auxiliary function | 1 | 1 | Numerical valueMulti-step | 1 | |
| Parameter Evaluationfind unknown coefficients so theorem holds | 1 | 1 | Single correctMulti-step | – | |
| Logarithmic Function Applicationvalue of second derivative at rolle point | 1 | 1 | Single correctMulti-step | – |
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
Rolle's Theorem 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 20242 questions in 2 shifts
- 4 Apr, Shift 2Q25Let be a thrice differentiable function such that and . Then, the…
Idea: Higher-Order Derivative Roots
HardNumerical value - 30 Jan, Shift 1Q9Let be a non constant twice differentiable function such that…
Idea: Higher-Order Derivative Roots
HardSingle correct
JEE Main 20223 questions in 3 shifts
- 28 Jul, Shift 1Q23Let be a twice differentiable function in such that and . If the line intersects…
Idea: Higher-Order Derivative Roots
Asked the year beforeMediumNumerical value - 24 Jun, Shift 2Q11The number of distinct real roots of the equation is
Idea: Polynomial Root Counting
Asked the year beforeMediumSingle correct - 29 Jun, Shift 2Q26Let and be twice differentiable even functions on such that ,…
Idea: Auxiliary Functions: Product and Quotient Rules
New ideaHardNumerical value
JEE Main 20213 questions in 3 shifts
- 31 Aug, Shift 2Q68Let be any continuous function on and twice differentiable on . If , and , then :
Idea: Higher-Order Derivative Roots
New ideaMediumSingle correct - 25 Feb, Shift 1Q65If Rolle's theorem holds for the function , with , then ordered…
Idea: Parameter Evaluation
New ideaMediumSingle correct - 26 Feb, Shift 2Q84Let a be an integer such that all the real roots of the polynomial lie in the interval . Then,…
Idea: Polynomial Root Counting
New ideaMediumNumerical value
JEE Main 20201 question in 1 shift
- 8 Jan, Shift 1Q59If c is a point at which Rolle's theorem holds for the function, in the interval ,…
Idea: Logarithmic Function Application
New ideaMediumSingle 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.