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Three Dimensional Geometry

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 33 times: 2 in 2021, 4 in 2022, 10 in 2023, 8 in 2024, 5 in 2025, 4 in 2026. This one: 6 Apr 2026 S2 See all 33 of this type

    64
    The 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:

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  2. Question 2

    This type asked 19 times: 2 in 2021, 2 in 2022, 7 in 2024, 2 in 2025, 6 in 2026. This one: 6 Apr 2026 S1 See all 19 of this type

    65
    Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1 b, b>0L:\frac{x}{1}=\frac{y - 1}{2}=\frac{z - a + 1}{\text{ }b},\text{ }b > 0, be (a3,0,a+c)\left( \frac{a}{3},0,a + c \right). If S(α,β,γ),α>0S(\alpha,\beta,\gamma),\alpha> 0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to :

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  3. Question 3

    This type asked 15 times: 2021 Paper 2, 2021 Paper 7, 2021 Paper 18, 2021 Paper 19, 2021 Paper 20, 2021 Paper 21, 2021 Paper 22, 28 Jun 2022, 26 Jul 2022, 24 Jan 2023, 25 Jan 2023, 1 Feb 2023, 6 Apr 2023, 8 Apr 2023, 13 Apr 2023. This one: 13 Apr 2023 See all 15 of this type

    80
    Let the equation of plane passing through the line of intersection of the planes x+2y+az=2x + 2y + az = 2 and x−y+z=3x - y + z = 3 be 5x−11y+bz=6a−15x - 11y + bz = 6a - 1. For c∈Zc \in Z, if the distance of this plane from the point (a,−c,c)(a, - c,c) is 2a\frac{2}{\sqrt{a}}, then a+bc\frac{a + b}{c} is equal to

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  4. Question 4

    This type asked 11 times: 2021 Paper 10, 2021 Paper 21, 2021 Paper 26, 25 Jun 2022, 27 Jun 2022, 29 Jun 2022, 1 Feb 2023, 8 Apr 2023, 10 Apr 2023, 11 Apr 2023, 13 Apr 2023. This one: 13 Apr 2023 See all 11 of this type

    88
    Let the image of the point (53,53,83)\left( \frac{5}{3},\frac{5}{3},\frac{8}{3} \right) in the plane x−2y+z−2=0x - 2y + z - 2 = 0 be PP. If the distance of the point Q(6,−2,α),α>0Q(6, - 2,\alpha),\alpha > 0, from PP is 13 , then α\alpha is equal to

  5. Question 5

    This type asked 10 times: 2021 Paper 4, 11 Apr 2023, 30 Jan 2024, 31 Jan 2024, 29 Jan 2025, 7 Apr 2025, 24 Jan 2026 S1, 2 Apr 2026 S1, 6 Apr 2026 S1, 8 Apr 2026 S2. This one: 8 Apr 2026 S2 See all 10 of this type

    73
    Let a line L1L_{1} pass through the origin and be perpendicular to the lines L2:r→=(3+t)i^+(2t−1)j^+(2t+4)k^L_{2}:\overrightarrow{r} = (3 + t)\widehat{i} + (2t - 1)\widehat{j} + (2t + 4)\widehat{k} and L3:r→=(3+2s)i^+(3+2s)j^+(2+s)k^,tL_{3}:\overrightarrow{r} = (3 + 2s)\widehat{i} + (3 + 2s)\widehat{j} + (2 + s)\widehat{k},t, s∈Rs \in R. If (a,b,c),a∈Z(a,b,c),a \in Z, is the point on L3L_{3} at a distance of 17\sqrt{17} from the point of intersection of L1L_{1} and L2L_{2}, then (a+b+c)2(a + b + c)^{2} is equal to ____\_\_\_\_.

  6. Question 6

    This type asked 9 times: 27 Jan 2024, 9 Apr 2024, 23 Jan 2025, 28 Jan 2025, 3 Apr 2025, 22 Jan 2026 S2, 28 Jan 2026 S1, 28 Jan 2026 S2, 4 Apr 2026 S1. This one: 4 Apr 2026 S1 See all 9 of this type

    65
    The 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 x+51=y+5−1=z2\frac{x+ 5}{1}=\frac{y+ 5}{- 1}=\frac{z}{2} is equal to:

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  7. Question 7

    This type asked 9 times: 27 Jun 2022, 25 Jan 2023, 13 Apr 2023, 30 Jan 2024, 6 Apr 2024, 22 Jan 2025, 29 Jan 2025, 21 Jan 2026 S1, 2 Apr 2026 S2. This one: 2 Apr 2026 S2 See all 9 of this type

    65
    Let the point A be the foot of perpendicular drawn from the point P(a,b,0)P(a,b,0) on the line x−12=y−21=z−α3\frac{x- 1}{2}=\frac{y- 2}{1}=\frac{z-\alpha}{3}. If the midpoint of the line segment PA is (0,34,−14)\left( 0,\frac{3}{4},\frac{- 1}{4} \right), then the value of a2+b2+α2a^{2}+b^{2}+\alpha^{2} is equal to :

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  8. Question 8

    This type asked 8 times: 2021 Paper 12, 2021 Paper 17, 28 Jul 2022, 29 Jan 2023, 6 Apr 2023, 12 Apr 2023, 13 Apr 2023, 2 Apr 2026 S1. This one: 2 Apr 2026 S1 See all 8 of this type

    63
    If 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} lies on xyxy-plane, then the value of a+ba+b is :

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  9. Question 9

    This type asked 7 times: 14 Jun 2022, 25 Jan 2023, 10 Apr 2023, 29 Jan 2024, 9 Apr 2024, 21 Jan 2026 S2, 4 Apr 2026 S1. This one: 4 Apr 2026 S1 See all 7 of this type

    64
    A 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 x+1−1=y−21=z4\frac{x + 1}{- 1}=\frac{y - 2}{1}=\frac{z}{4} at the points P and Q , respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^{2} is equal to:

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  10. Question 10

    This type asked 7 times: 2021 Paper 6, 2021 Paper 20, 29 Jul 2022, 24 Jan 2023, 11 Apr 2023, 13 Apr 2023, 15 Apr 2023. This one: 15 Apr 2023 See all 7 of this type

    83
    Let the plane PP contain the line 2x+y−z−3=0=5x−3y+4z+92x + y - z - 3 = 0 = 5x - 3y + 4z + 9 and be parallel to the line x+22=3−y−4=z−75\frac{x + 2}{2}=\frac{3 - y}{- 4}=\frac{z - 7}{5}. Then the distance of the point A(8,−1,−19)A(8, - 1, - 19) from the plane P measured parallel to the line x−3=y−54=2−z−12\frac{x}{- 3}=\frac{y - 5}{4}=\frac{2 - z}{- 12} is equal to