JEE Main · Maths · Three Dimensional Geometry

Lines in 3D: JEE Main previous year questions

Name: Lines in 3D: JEE Main previous year questions
Creator: Lumi

73 questions in the 108 JEE Main shifts Lumi analysed, in 7 of 8 years. Pattern: Asked most years. 47 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 73; in 56 of 108 shifts
Years asked: 7/8; last asked 2026
Chance in a shift: 71%; calibrated model, how we compute it
Weighted estimate: 0.9; model; history: 0.6 a shift on average, typically 0.3–1.2

Key findings

01
Asked in 7 of 8 years with analysed shifts and in 56 of 108 shifts: 73 questions, #25 of 196 topics.
02
30% of its questions since 2020 tested an idea no earlier year had asked; 38 repeated an idea from the year before.
03
The most repeated idea is Image of a Point in a Line, asked 16 times in 16 shifts (2020, 2021, 2022, 2024, 2025, 2026), most often as “image of point in a line”.
04
JEE Main has asked 18 of the 65 ideas in this topic; 39 have never been asked, directly or inside another question.
05
Mostly asked as single correct (74%); 26% need a numerical answer, 3% are easy and 18% hard. It went up from 0.3 to 1.3 questions a shift since 2024.
06
The skill tested most is multi-step (66%), then calculation (30%).

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: 3 in 12 shifts · 2021: 10 in 24 shifts · 2022: 8 in 24 shifts · 2023: 5 in 12 shifts · 2024: 25 in 18 shifts · 2025: 9 in 8 shifts · 2026: 13 in 9 shifts.

Up to 2023: 0.3; a shift, of 25
Since 2024: 1.3; +1.0
Usual gap: 1.0 yrs; between years asking it
Shifts with it: 56/108; 52% of analysed shifts

Recent framings

shortest distance between parallel lines
numerical: point on line at given distance
image and foot of point in 3D line

Revise first

Image of a Point in a Line · asked 2020, 2021, 2022, 2024, 2025, 2026
Perpendicular Distance from Point to Line · asked 2021, 2022, 2023, 2024, 2025, 2026
Line Perpendicular to Two Given Lines · asked 2021, 2023, 2024, 2025, 2026

How it is asked

Difficulty, format and skill

Across all 73 questions on Lines in 3D since 2017.

Difficulty

QUESTIONS

5813

Easy2 · 3%
Medium58 · 79%
Hard13 · 18%

Format

Single correct · 54 Q74%
Numerical value · 19 Q26%

Skill tested

Multi-step · 48 Q66%
Calculation · 22 Q30%
Application · 3 Q4%

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 18 of the 65 ideas in this topic. 39 have never been asked yet.

18839

Asked directly18 · 28%
Touched inside another question8 · 12%
Never asked39 · 60%

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

OpenRepeated16 · 22%22%6/8274
- 4 Apr ’26 S2 · Q14
  If $\left(2\alpha+1,\,\alpha^2-3\alpha,\,\frac{\alpha-1}{2}\right)$ is the image of $(\alpha,2\alpha,1)$ in the line…
  mediumSingle correctAsked the year before
- 5 Apr ’26 S2 · Q15
  If the distance of the point $(a, 2, 5)$ from the image of the point $(1, 2, 7)$ in the line $\frac{x}{1}=\frac{y-1}{1}=\frac{z-2}{2}$ is…
  mediumSingle correctAsked the year before
- 6 Apr ’26 S1 · Q15
  Let the image of the point $\mathrm{P}(1,6,a)$ in the line $\mathrm{L}:\frac{x}{1}=\frac{y-1}{2}=\frac{z-a+1}{b}$ , $b>0$ , be…
  hardSingle correctAsked the year before
- 6 Apr ’26 S2 · Q24
  Let the image of the point P $(0,-5,0)$ in the line $\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2}$ be the point R and the image of the point…
  hardNumerical valueAsked the year before
- 2 Apr ’25 S2 · Q14
  If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta, \gamma)$ , then…
  mediumSingle correctAsked the year before
- 4 Apr ’24 S2 · Q29
  Consider a line L passing through the points $P(1,2,1)$ and $Q(2,1,-1)$ . If the mirror image of the point $A(2,2,2)$ in the line L is…
  mediumNumerical value
- 5 Apr ’24 S2 · Q17
  Let $(\alpha, \beta, \gamma)$ be the image of the point $(8, 5, 7)$ in the line $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-2}{5}$ . Then…
  mediumSingle correct
- 6 Apr ’24 S2 · Q17
  Read the question
  hardSingle correct
- 8 Apr ’24 S2 · Q29
  Read the question
  mediumNumerical value
- 9 Apr ’24 S2 · Q29
  Read the question
  mediumNumerical value
- 1 Feb ’24 S2 · Q15
  Read the question
  mediumSingle correct
- 31 Jan ’24 S2 · Q16
  Read the question
  mediumSingle correct
- 27 Jul ’22 S2 · Q14
  Read the question
  mediumSingle correctAsked the year before
- 28 Jun ’22 S2 · Q21
  Read the question
  mediumNumerical valueAsked the year before
- 24 Feb ’21 S2 · Q61
  Read the question
  mediumSingle correctAsked the year before
- 5 Sep ’20 S1 · Q66
  Read the question
  mediumSingle correctNew idea
OpenRepeated12 · 16%16%6/8432
OpenRepeated9 · 12%12%6/8222
OpenRepeated8 · 11%11%6/822
OpenRepeated7 · 10%10%5/822
OpenRepeated3 · 4%4%3/8
OpenRepeated3 · 4%4%2/82
Open3 · 4%4%1/83
OpenRepeated2 · 3%3%2/8
OpenRepeated2 · 3%3%2/8
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

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

18 different ideas were tested; 10 came more than once.

Image of a Point in a Line16×
image of point in a line
Single correct · 16 shifts · 5 numerical
Foot of the Perpendicular12×
foot of perpendicular on line of intersection of planes
Single correct · 12 shifts · 2 numerical
Perpendicular Distance from Point to Line9×
distance of point from line in space
Single correct · 9 shifts · 1 numerical
Point of Intersection Calculation8×
parameters from intersection of lines on coordinate plane
Single correct · 8 shifts · 2 numerical
Line Perpendicular to Two Given Lines7×
point on line at given distance
Single correct · 7 shifts · 3 numerical
Condition for Intersection of Two Lines3×
intersection of two vector lines
Single correct · 3 shifts
General Point Parameterization3×
line with given direction meeting two lines
Numerical value · 3 shifts · 2 numerical
Parametric Points at Distance r3×
point on 3d line at given distance
Single correct · 3 shifts · 1 numerical
Line Parallel to Vector Intersecting Two Lines2×
line with given direction meeting two lines
Numerical value · 2 shifts · 1 numerical
Line Through Point Intersecting Two Skew Lines2×
line through point intersecting two lines
Numerical value · 2 shifts · 1 numerical
Shortest Distance Between Parallel Lines1×
shortest distance between parallel lines
Single correct · 1 shift
Direction Cosines Squares Identity1×
angles of line with axes using direction cosines
Single correct · 1 shift
Distance Measured Parallel to Another Line1×
distance from point to line along given direction
Single correct · 1 shift
Condition for Perpendicular Lines1×
perpendicular lines in asymmetric form
Single correct · 1 shift
Equation of Line of Shortest Distance1×
line of shortest distance between skew lines
Numerical value · 1 shift · 1 numerical
Line Intersecting Two Skew Lines Parallel to a Third1×
line with given direction meeting two lines
Single correct · 1 shift
Shortest Distance Between Skew Lines1×
shortest distance between skew lines
Single correct · 1 shift
Cartesian Symmetric Form1×
line through point perpendicular to given line
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Image of a Point in a Lineimage of point in a line	16	16	Single correctCalculation	5
Foot of the Perpendicularfoot of perpendicular on line of intersection of planes	12	12	Single correctMulti-step	2
Perpendicular Distance from Point to Linedistance of point from line in space	9	9	Single correctMulti-step	1
Point of Intersection Calculationparameters from intersection of lines on coordinate plane	8	8	Single correctMulti-step	2
Line Perpendicular to Two Given Linespoint on line at given distance	7	7	Single correctMulti-step	3
Condition for Intersection of Two Linesintersection of two vector lines	3	3	Single correctMulti-step	–
General Point Parameterizationline with given direction meeting two lines	3	3	Numerical valueMulti-step	2
Parametric Points at Distance rpoint on 3d line at given distance	3	3	Single correctCalculation	1
Line Parallel to Vector Intersecting Two Linesline with given direction meeting two lines	2	2	Numerical valueCalculation	1
Line Through Point Intersecting Two Skew Linesline through point intersecting two lines	2	2	Numerical valueCalculation	1
Shortest Distance Between Parallel Linesshortest distance between parallel lines	1	1	Single correctMulti-step	–
Direction Cosines Squares Identityangles of line with axes using direction cosines	1	1	Single correctMulti-step	–
Distance Measured Parallel to Another Linedistance from point to line along given direction	1	1	Single correctCalculation	–
Condition for Perpendicular Linesperpendicular lines in asymmetric form	1	1	Single correctApplication	–
Equation of Line of Shortest Distanceline of shortest distance between skew lines	1	1	Numerical valueMulti-step	1
Line Intersecting Two Skew Lines Parallel to a Thirdline with given direction meeting two lines	1	1	Single correctMulti-step	–
Shortest Distance Between Skew Linesshortest distance between skew lines	1	1	Single correctCalculation	–
Cartesian Symmetric Formline through point perpendicular to given line	1	1	Single correctApplication	–

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

Lines in 3D 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

Lines in 3D questions, year by year

JEE Main 202613 questions in 9 shifts

JEE Main 20259 questions in 7 shifts

JEE Main 202425 questions in 16 shifts

JEE Main 20235 questions in 5 shifts

JEE Main 20228 questions in 7 shifts

JEE Main 202110 questions in 10 shifts

JEE Main 20203 questions in 2 shifts

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