JEE Main · Maths · Integrals

Integral as Limit of Sum: JEE Main previous year questions

Name: Integral as Limit of Sum: JEE Main previous year questions
Creator: Lumi

13 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: 13; in 13 of 108 shifts
Years asked: 4/8; last asked 2024
Chance in a shift: 13%; calibrated model, how we compute it
Weighted estimate: 0.1; model; history: 0.1 a shift on average, typically 0.0–0.1

Key findings

01
Asked in 4 of 8 years with analysed shifts and in 13 of 108 shifts: 13 questions, #166 of 196 topics. Last asked in 2024.
02
38% of its questions since 2020 tested an idea no earlier year had asked; 8 repeated an idea from the year before.
03
The most repeated idea is Summation to Riemann Form, asked 9 times in 9 shifts (2021, 2022, 2023, 2024), most often as “limit of sum as definite integral”.
04
JEE Main has asked 5 of the 65 ideas in this topic; 55 have never been asked, directly or inside another question.
05
Mostly asked as single correct (77%); 15% need a numerical answer, 0% are easy and 31% hard.
06
The skill tested most is calculation (46%), then multi-step (38%).

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: 0 in 12 shifts · 2021: 4 in 24 shifts · 2022: 6 in 24 shifts · 2023: 1 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; -0.1
Usual gap: 1.0 yrs; between years asking it
Shifts with it: 13/108; 12% of analysed shifts

How it is asked

Difficulty, format and skill

Across all 13 questions on Integral as Limit of Sum since 2017.

Difficulty

QUESTIONS

Easy0 · 0%
Medium9 · 69%
Hard4 · 31%

Format

Single correct · 10 Q77%
Numerical value · 2 Q15%
Two statements · 1 Q8%

Skill tested

Calculation · 6 Q46%
Multi-step · 5 Q38%
Application · 2 Q15%

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 5 of the 65 ideas in this topic. 55 have never been asked yet.

Asked directly5 · 8%
Touched inside another question5 · 8%
Never asked55 · 85%

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

Open11 · 85%85%3/8362
4 ideas inside
- OpenRepeated8 · 73%73%3/852
- Open1 · 9%9%1/8
- Open1 · 9%9%1/8
- Open1 · 9%9%1/8
Open2 · 15%15%2/8

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

5 different ideas were tested; 1 came more than once.

Summation to Riemann Form9×
limit of sum as definite integral
Single correct · 9 shifts · 2 numerical
Algebraic Manipulation to Extract 1/n1×
limit of sum as definite integral
Single correct · 1 shift
Fractional or Multiple Upper Bounds1×
limit of sum as definite integral
Single correct · 1 shift
Greatest Integer Function Sums1×
limit of sum with greatest integer function
Single correct · 1 shift
Product Series with Variable Exponents1×
limit of product series via integral
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Summation to Riemann Formlimit of sum as definite integral	9	9	Single correctCalculation	2
Algebraic Manipulation to Extract 1/nlimit of sum as definite integral	1	1	Single correctMulti-step	–
Fractional or Multiple Upper Boundslimit of sum as definite integral	1	1	Single correctCalculation	–
Greatest Integer Function Sumslimit of sum with greatest integer function	1	1	Single correctApplication	–
Product Series with Variable Exponentslimit of product series via integral	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

Integral as Limit of Sum 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

9 Apr, Shift 1Q28
Let…
Idea: Summation to Riemann Form
Asked the year beforeHardNumerical value
30 Jan, Shift 1Q10
The value of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :
Idea: Summation to Riemann Form
Asked the year beforeMediumSingle correct

JEE Main 20231 question in 1 shift

13 Apr, Shift 1Q7
Among (S1) : $\lim_{n\to\infty}\frac{1}{n^2}(2+4+6+\ldots\ldots+2n)=1$ (S2) :…
Idea: Summation to Riemann Form
Asked the year beforeMediumTwo statements

JEE Main 20226 questions in 6 shifts

25 Jul, Shift 1Q27
If…
Idea: Summation to Riemann Form
Asked the year beforeMediumNumerical value
25 Jul, Shift 2Q7
…
Idea: Summation to Riemann Form
Asked the year beforeMediumSingle correct
26 Jul, Shift 1Q10
If $\mathrm{a} = \lim_{n \to \infty} \sum_{k=1}^{n} \dfrac{2n}{n^2 + k^2}$ and $f(x) = \sqrt{\dfrac{1 - \cos x}{1 + \cos x}}$ ,…
Idea: Summation to Riemann Form
Asked the year beforeMediumSingle correct
30 Jul, Shift 2Q72
$\lim_{n\to\infty} \sum_{k=1}^{n} \frac{n(n^2+k^2)}{(n^4+n^2k^2+k^4)}$ is equal to
Idea: Algebraic Manipulation to Extract 1/n
New ideaHardSingle correct
24 Jun, Shift 2Q7
…
Idea: Summation to Riemann Form
Asked the year beforeHardSingle correct
30 Jun, Shift 1Q11
$\lim_{n\to\infty} \sum_{r=1}^{n} \frac{r}{2r^2 - 7rn + 6n^2}$ is equal to :
Idea: Summation to Riemann Form
Asked the year beforeMediumSingle correct

JEE Main 20214 questions in 4 shifts

26 Aug, Shift 1Q70
The value of $\lim_{n\to\infty} \frac{1}{n}\sum_{r=0}^{2n-1} \frac{n^2}{n^2+4r^2}$ is :
Idea: Fractional or Multiple Upper Bounds
New ideaMediumSingle correct
27 Aug, Shift 1Q68
If $U_n=\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)^2 \dots \left(1+\frac{n^2}{n^2}\right)^n$ , then…
Idea: Product Series with Variable Exponents
New ideaHardSingle correct
25 Feb, Shift 2Q74
$\lim_{n\to\infty}\left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots\ldots\ldots + \frac{n}{(2n-1)^2}\right]$ is equal to :
Idea: Summation to Riemann Form
New ideaMediumSingle correct
17 Mar, Shift 2Q70
The value of $\lim_{n\to\infty}\frac{[r]+[2r]+\ldots+[nr]}{n^2}$ , where r is a non-zero real number and $[r]$ denotes the greatest integer…
Idea: Greatest Integer Function Sums
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.