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Friday, 9 October

JEE Main Maths · Class 12

Three Dimensional Geometry: JEE Main previous year questions

Three Dimensional Geometry had 84 questions in the 42 JEE Main shifts Lumi analysed, about 1.8 a shift, asked in 42 of 42 shifts. It ranks 2 of 22 Maths chapters by questions; 18% of its questions were hard.

Questions
84
in 42 of 42 shifts
A shift, on average
1.8
of 25 Maths questions
Since 2024
+0.2
1.7 a shift in 2017–23, 1.9 in 2024–26
Numerical answer
21
25% of its questions

Year by year

How often it came

Questions in the analysed shifts of each year. Years differ in how many shifts Lumi analysed, so read the bars with the shift count.

Questions each year
09182’174’208’218’2212’2317’2416’2517’26

2017: 1 shifts · 2020: 4 shifts · 2021: 4 shifts · 2022: 4 shifts · 2023: 4 shifts · 2024: 8 shifts · 2025: 8 shifts · 2026: 9 shifts

Difficulty by year
  • 20172
  • 20204
  • 20218
  • 20228
  • 202312
  • 202417
  • 202516
  • 202617
  • Easy
  • Medium
  • Hard

Inside the chapter

Topics and the ideas that repeat

7 topics and 40 distinct ideas; 17 ideas were asked more than once. Most questions are single correct and multi-step.

Topics, by questions

3 questions had no close topic in our taxonomy and are counted for the chapter only.

Ideas asked most
17 repeat

Study it on Lumi

Three Dimensional Geometry: the interactive lesson

Every question

All 84 Three Dimensional Geometry questions

Newest first. Each opens with its options and the official answer.

2026 17 questions

  1. 2 Apr 2026, Shift 1 · Q13If the point of intersection of the lines x+13=y+a5=z+b+17\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} and x−21=y−b4=z−2a7\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7}…MediumSingle correct
  2. 2 Apr 2026, Shift 1 · Q15Let a line L passing through the point (1,1,1)(1, 1, 1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and…MediumSingle correct
  3. 2 Apr 2026, Shift 2 · Q15Let the point AA be the foot of perpendicular drawn from the point P(a,b,0)P(a, b, 0) on the line…MediumSingle correct
  4. 4 Apr 2026, Shift 1 · Q14A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3} and…MediumSingle correct
  5. 4 Apr 2026, Shift 1 · Q15The square of the distance of the point (−2,−8,6)(-2,-8,6) from the line x−11=y−12=z−1\frac{x-1}{1}=\frac{y-1}{2}=\frac{z}{-1} along the line…MediumSingle correct
  6. 4 Apr 2026, Shift 2 · Q13The shortest distance between the lines…EasySingle correct
  7. 4 Apr 2026, Shift 2 · Q14If (2α+1, α2−3α, α−12)\left(2\alpha+1,\,\alpha^2-3\alpha,\,\frac{\alpha-1}{2}\right) is the image of (α,2α,1)(\alpha,2\alpha,1) in the line…MediumSingle correct
  8. 5 Apr 2026, Shift 1 · Q13The square of the distance of the point P(5,6,7)P(5, 6, 7) from the line x−22=y−53=z−24\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4} is equal to:EasySingle correct
  9. 5 Apr 2026, Shift 1 · Q15The square of the distance of the point of intersection of the lines…MediumSingle correct
  10. 5 Apr 2026, Shift 2 · Q14Let a triangle PQR be such that P and Q lie on the line x+38=y−42=z+12\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2} and are at a distance of 6 units from…MediumSingle correct
  11. 5 Apr 2026, Shift 2 · Q15If the distance of the point (a,2,5)(a, 2, 5) from the image of the point (1,2,7)(1, 2, 7) in the line x1=y−11=z−22\frac{x}{1}=\frac{y-1}{1}=\frac{z-2}{2} is…MediumSingle correct
  12. 6 Apr 2026, Shift 1 · Q15Let the image of the point P(1,6,a)\mathrm{P}(1,6,a) in the line L:x1=y−12=z−a+1b\mathrm{L}:\frac{x}{1}=\frac{y-1}{2}=\frac{z-a+1}{b}, b>0b>0, be…HardSingle correct
  13. 6 Apr 2026, Shift 1 · Q16Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57\mathrm{L}_1:\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} and…MediumSingle correct
  14. 6 Apr 2026, Shift 2 · Q14The shortest distance between the lines x−41=y−32=z−2−3\frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} and x+22=y−64=z−5−5\frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} is :EasySingle correct
  15. 6 Apr 2026, Shift 2 · Q24Let the image of the point P(0,−5,0)(0,-5,0) in the line x−12=y1=z+1−2\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2} be the point R and the image of the point…HardNumerical value
  16. 8 Apr 2026, Shift 2 · Q15Let the foot of perpendicular from the point (λ,2,3)(\lambda, 2, 3) on the line x−41=y−92=z−51\frac{x-4}{1} = \frac{y-9}{2} = \frac{z-5}{1} be the point…MediumSingle correct
  17. 8 Apr 2026, Shift 2 · Q23Let a line L1L_1 pass through the origin and be perpendicular to the lines…MediumNumerical value

2025 16 questions

  1. 2 Apr 2025, Shift 1 · Q5Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be…MediumSingle correct
  2. 2 Apr 2025, Shift 1 · Q13Let the vertices Q and R of the triangle PQR lie on the line x+35=y−12=z+43\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}, QR=5\mathrm{QR}=5 and the…MediumSingle correct
  3. 2 Apr 2025, Shift 2 · Q14If the image of the point P(1,0,3)P(1, 0, 3) in the line joining the points A(4,7,1)A(4, 7, 1) and B(3,5,3)B(3, 5, 3) is Q(α,β,γ)Q(\alpha, \beta, \gamma), then…MediumSingle correct
  4. 2 Apr 2025, Shift 2 · Q15The line L1L_1 is parallel to the vector a⃗=−3i^+2j^+4k^\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} and passes through the point (7,6,2)(7, 6, 2) and the line…EasySingle correct
  5. 3 Apr 2025, Shift 1 · Q14Line L1\mathrm{L}_1 passes through the point (1,2,3)(1, 2, 3) and is parallel to zz-axis. Line L2\mathrm{L}_2 passes through the point…MediumSingle correct
  6. 3 Apr 2025, Shift 1 · Q15Let a line passing through the point (4,1,0)(4, 1, 0) intersect the line L1:x−12=y−23=z−34\mathrm{L}_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} at the…HardSingle correct
  7. 3 Apr 2025, Shift 2 · Q14Each of the angles β\beta and γ\gamma that a given line makes with the positive yy- and zz-axes, respectively, is half of the angle…MediumSingle correct
  8. 3 Apr 2025, Shift 2 · Q15The distance of the point (7,10,11)(7, 10, 11) from the line x−41=y−40=z−23\frac{x-4}{1} = \frac{y-4}{0} = \frac{z-2}{3} along the line…MediumSingle correct
  9. 4 Apr 2025, Shift 2 · Q14Let the values of p, for which the shortest distance between the lines x+13=y4=z5\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} and…MediumSingle correct
  10. 4 Apr 2025, Shift 2 · Q15Let A be the point of intersection of the lines L1:x−71=y−50=z−3−1L_1:\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1} and…MediumSingle correct
  11. 7 Apr 2025, Shift 1 · Q13Let the line L pass through (1,1,1)(1, 1, 1) and intersect the lines x−12=y+13=z−14\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} and…MediumSingle correct
  12. 7 Apr 2025, Shift 1 · Q14If the shortest distance between the lines x−12=y−23=z−34\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and…MediumSingle correct
  13. 7 Apr 2025, Shift 2 · Q16If the equation of the line passing through the point (0,−12,0)\left(0,-\frac{1}{2},0\right) and perpendicular to the lines…MediumSingle correct
  14. 7 Apr 2025, Shift 2 · Q17Consider the lines L1:x−1=y−2=zL_1: x-1=y-2=z and L2:x−2=y=z−1L_2: x-2=y=z-1. Let the feet of the perpendiculars from the point P(5,1,−3)P(5,1,-3) on the lines L1L_1…MediumSingle correct
  15. 8 Apr 2025, Shift 2 · Q15Let the values of λ\lambda for which the shortest distance between the lines x−12=y−23=z−34\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and…HardSingle correct
  16. 8 Apr 2025, Shift 2 · Q24Let the area of the triangle formed by the lines x+2=y−1=zx + 2 = y - 1 = z, x−35=y−1=z−11\frac{x-3}{5} = \frac{y}{-1} = \frac{z-1}{1} and…HardNumerical value

2024 17 questions

  1. 6 Apr 2024, Shift 2 · Q17Let P(α,β,γ)P(\alpha,\beta,\gamma) be the image of the point Q(3,−3,1)Q(3,-3,1) in the line x−01=y−31=z−1−1\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1} and RR be the…HardSingle correct
  2. 6 Apr 2024, Shift 2 · Q28If the shortest distance between the lines x−λ3=y−2−1=z−11\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1} and…MediumNumerical value
  3. 8 Apr 2024, Shift 1 · Q14If the shortest distance between the lines L1:r⃗=(2+λ)i^+(1−3λ)j^+(3+4λ)k^L_1:\vec{r}=(2+\lambda)\hat{i}+(1-3\lambda)\hat{j}+(3+4\lambda)\hat{k},…MediumSingle correct
  4. 8 Apr 2024, Shift 1 · Q17Let P(x,y,z)P(x,y,z) be a point in the first octant, whose projection in the xyxy-plane is the point QQ. Let OP=γOP=\gamma; the angle between…MediumSingle correct
  5. 9 Apr 2024, Shift 1 · Q15Let the line L intersect the lines x−2=−y=z−1x-2=-y=z-1, 2(x+1)=2(y−1)=z+12(x+1)=2(y-1)=z+1 and be parallel to the line…MediumSingle correct
  6. 9 Apr 2024, Shift 1 · Q16The shortest distance between the lines x−34=y+7−11=z−15\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} and x−53=y−9−6=z+21\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1} is :EasySingle correct
  7. 9 Apr 2024, Shift 2 · Q16Consider the line L passing through the points (1,2,3)(1, 2, 3) and (2,3,5)(2, 3, 5). The distance of the point…HardSingle correct
  8. 9 Apr 2024, Shift 2 · Q29The square of the distance of the image of the point (6,1,5)(6, 1, 5) in the line x−13=y2=z−24\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}, from the origin is…MediumNumerical value
  9. 29 Jan 2024, Shift 1 · Q16Let PQR be a triangle with R(−1,4,2)R(-1,4,2). Suppose M(2,1,2)M(2,1,2) is the mid point of PQ. The distance of the centroid of ΔPQR\Delta PQR from the…MediumSingle correct
  10. 29 Jan 2024, Shift 1 · Q29A line with direction ratios 2,1,22,1,2 meets the lines x=y+2=zx=y+2=z and x+2=2y=2zx+2=2y=2z respectively at the points P and Q. If the length of the…HardNumerical value
  11. 30 Jan 2024, Shift 2 · Q17Let L1:r⃗=(i^−j^+2k^)+λ(i^−j^+2k^),λ∈RL_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k}), \lambda \in \mathbb{R},…MediumSingle correct
  12. 30 Jan 2024, Shift 2 · Q29Let a line passing through the point (−1,2,3)(-1, 2, 3) intersect the lines L1:x−13=y−22=z+1−2L_1: \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{-2} at…MediumNumerical value
  13. 31 Jan 2024, Shift 1 · Q15The distance of the point Q(0,2,−2)Q(0,2,-2) form the line passing through the point P(5,−4,3)P(5,-4,3) and perpendicular to the lines…MediumSingle correct
  14. 31 Jan 2024, Shift 1 · Q29Let Q and R be the feet of perpendiculars from the point P(a,a,a)P(a,a,a) on the lines x=y, z=1x=y,\ z=1 and x=−y, z=−1x=-y,\ z=-1 respectively. If…MediumNumerical value
  15. 31 Jan 2024, Shift 2 · Q16Let (α,β,γ)(\alpha,\beta,\gamma) be the mirror image of the point (2,3,5)(2,3,5) in the line x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}. Then,…MediumSingle correct
  16. 31 Jan 2024, Shift 2 · Q17The shortest distance, between lines L1L_1 and L2L_2, where L1:x−12=y+1−3=z+42L_1:\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2} and L2L_2 is the line,…MediumSingle correct
  17. 31 Jan 2024, Shift 2 · Q29A line passes through A(4,−6,−2)A(4,-6,-2) and B(16,−2,4)B(16,-2,4). The point P(a,b,c)P(a,b,c), where a,b,ca, b, c are non-negative integers, on the line ABAB…MediumNumerical value

2023 12 questions

  1. 6 Apr 2023, Shift 1 · Q13If the equation of the plane passing through the line of intersection of the planes 2x−y+z=32x-y+z=3, 4x−3y+5z+9=04x-3y+5z+9=0 and parallel to the line…MediumSingle correct
  2. 6 Apr 2023, Shift 1 · Q14One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along xx, yy and zz axes are 3, 4 and 5…MediumSingle correct
  3. 6 Apr 2023, Shift 1 · Q30Let the image of the point (1,2,3)(1,2,3) in the plane 2x−y+z=92x-y+z=9 be Q. If the coordinates of the point R are (6,10,7)(6,10,7), then the square of…MediumNumerical value
  4. 8 Apr 2023, Shift 2 · Q13Let PP be the plane passing through the line x−11=y−2−3=z+57\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+5}{7} and the point (2,4,−3)(2, 4, -3). If the image of…MediumSingle correct
  5. 8 Apr 2023, Shift 2 · Q14For a,b∈Za, b \in \mathbb{Z} and ∣a−b∣≤10|a - b| \leq 10, let the angle between the plane P:ax+y−z=bP: ax + y - z = b and the line…HardSingle correct
  6. 8 Apr 2023, Shift 2 · Q30Let P1P_1 be the plane 3x−y−7z=113x - y - 7z = 11 and P2P_2 be the plane passing through the points (2,−1,0)(2, -1, 0), (2,0,−1)(2, 0, -1), and (5,1,1)(5, 1, 1).…HardNumerical value
  7. 11 Apr 2023, Shift 1 · Q2Let (α,β,γ)(\alpha, \beta, \gamma) be the image of the point P(2,3,5)P(2, 3, 5) in the plane 2x+y−3z=62x + y - 3z = 6. Then α+β+γ\alpha + \beta + \gamma is…MediumSingle correct
  8. 11 Apr 2023, Shift 1 · Q8If equation of the plane that contains the point (−2,3,5)(-2, 3, 5) and is perpendicular to each of the planes 2x+4y+5z=82x + 4y + 5z = 8 and…MediumSingle correct
  9. 11 Apr 2023, Shift 1 · Q22Let a line ll pass through the origin and be perpendicular to the lines…HardNumerical value
  10. 13 Apr 2023, Shift 2 · Q14The plane, passing through the points (0,−1,2)(0,-1,2) and (−1,2,1)(-1,2,1) and parallel to the line passing through (5,1,−7)(5,1,-7) and (1,−1,−1)(1,-1,-1), also…MediumSingle correct
  11. 13 Apr 2023, Shift 2 · Q15The line, that is coplanar to the line x+3−3=y−11=z−55\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}, isMediumSingle correct
  12. 13 Apr 2023, Shift 2 · Q16Let N be the foot of perpendicular from the point P (1,−2,3)(1,-2,3) on the line passing through the points (4,5,8)(4,5,8) and (1,−7,5)(1,-7,5). Then the…MediumSingle correct

2022 8 questions

  1. 25 Jul 2022, Shift 1 · Q14Let PP be the plane containing the straight line x−39=y+4−1=z−7−5\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5} and perpendicular to the plane containing…HardSingle correct
  2. 28 Jul 2022, Shift 1 · Q16The foot of the perpendicular from a point on the circle x2+y2=1, z=0x^2+y^2=1,\ z=0 to the plane 2x+3y+z=62x+3y+z=6 lies on which one of the following…HardSingle correct
  3. 28 Jul 2022, Shift 1 · Q22Let P(−2,−1,1)P(-2,-1,1) and Q(5617,4317,11117)Q\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right) be the vertices of the rhombus PRQS. If the direction…MediumNumerical value
  4. 24 Jun 2022, Shift 1 · Q24Let a line having direction ratios 1,−4,21, -4, 2 intersect the lines x−73=y−1−1=z+21\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1} and…MediumNumerical value
  5. 24 Jun 2022, Shift 1 · Q30If the shortest distance between the lines r⃗=(−i^+3k^)+λ(i^−aj^)\vec{r}=(-\hat{i}+3\hat{k})+\lambda(\hat{i}-a\hat{j}) and…MediumNumerical value
  6. 29 Jun 2022, Shift 1 · Q3If the mirror image of the point (2,4,7)(2, 4, 7) in the plane 3x−y+4z=23x - y + 4z = 2 is (a,b,c)(a, b, c), then 2a+b+2c2a + b + 2c is equal to :EasySingle correct
  7. 29 Jun 2022, Shift 1 · Q23Let dd be the distance between the foot of perpendiculars of the points P(1,2,−1)P(1, 2, -1) and Q(2,−1,3)Q(2, -1, 3) on the plane −x+y+z=1-x + y + z = 1.…MediumNumerical value
  8. 29 Jun 2022, Shift 1 · Q29Let P1:r⃗⋅(2i^+j^−3k^)=4P_1: \vec{r}\cdot(2\hat{i} + \hat{j} - 3\hat{k}) = 4 be a plane. Let P2P_2 be another plane which passes through the points…MediumNumerical value

2021 8 questions

  1. 3 Aug 2021, Shift 2 · Q76Let L1L_1 and L2L_2 be two lines passing through the points P(b−c,c−a,a−b)P(b-c, c-a, a-b) and Q(1l,1m,1n)Q\left(\frac{1}{l}, \frac{1}{m}, \frac{1}{n}\right)…HardSingle correct
  2. 3 Aug 2021, Shift 2 · Q77A plane intersects the yzyz-plane at x=0, 2y−3z=5x = 0,\ 2y - 3z = 5 and it intersects the xyxy-plane at z=0, 7x+4y=10z = 0,\ 7x + 4y = 10. Then the distance…MediumSingle correct
  3. 25 Jul 2021, Shift 1 · Q69Let the foot of perpendicular from a point P(1, 2, −1-1) to the straight line L:x1=y0=z−1L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1} be N. Let a line be…MediumSingle correct
  4. 16 Mar 2021, Shift 2 · Q61If the foot of the perpendicular from point (4, 3, 8) on the line L1:x−al=y−23=z−b4L_1:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4}, l≠0l\neq0 is (3, 5, 7),…MediumSingle correct
  5. 16 Mar 2021, Shift 2 · Q70If (x,y,z)(x,y,z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the…MediumSingle correct
  6. 16 Mar 2021, Shift 2 · Q83If the distance of the point (1, −2-2, 3) from the plane x+2y−3z+10=0x+2y-3z+10=0 measured parallel to the line,…MediumNumerical value
  7. 18 Mar 2021, Shift 1 · Q85Let the plane ax+by+cz+d=0ax+by+cz+d=0 bisect the line joining the points (4,−3,1)(4,-3,1) and (2,3,−5)(2,3,-5) at the right angles. If a, b, c, d are integers,…EasyNumerical value
  8. 18 Mar 2021, Shift 1 · Q86The equation of the planes parallel to the plane x−2y+2z−3=0x-2y+2z-3=0 which are at unit distance from the point (1,2,3)(1,2,3) is ax+by+cz+d=0ax+by+cz+d=0. If…MediumNumerical value

2020 4 questions

  1. 8 Jan 2020, Shift 1 · Q65The shortest distance between the lines x−33=y−8−1=z−31\dfrac{x-3}{3}=\dfrac{y-8}{-1}=\dfrac{z-3}{1} and…MediumSingle correct
  2. 9 Jan 2020, Shift 2 · Q74If the distance between the plane, 23x−10y−2z+48=023x-10y-2z+48=0 and the plane containing the lines x+12=y−34=z+13\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3} and…HardNumerical value
  3. 2 Sep 2020, Shift 1 · Q66The plane passing through the points (1,2,1)(1, 2, 1), (2,1,2)(2, 1, 2) and parallel to the line, 2x=3y2x=3y, z=1z=1 also passes through the point :MediumSingle correct
  4. 6 Sep 2020, Shift 2 · Q67A plane P meets the coordinate axes at A, B and C respectively. The centroid of Δ\DeltaABC is given to be (1,1,2)(1,1,2). Then the equation of…MediumSingle correct

2017 2 questions

  1. 2 Apr 2017 (offline) · Q82The distance of the point (1,3,−7)(1,3,-7) from the plane passing through the point (1,−1,−1)(1,-1,-1), having normal perpendicular to both the lines…MediumSingle correct
  2. 2 Apr 2017 (offline) · Q83If the image of the point P(1,−2,3)P(1,-2,3) in the plane, 2x+3y−4z+22=02x+3y-4z+22=0 measured parallel to the line, x1=y4=z5\frac{x}{1}=\frac{y}{4}=\frac{z}{5}…MediumSingle correct

Pattern: Asked most years.