JEE (Main) · Mathematics, Class 12
Three Dimensional Geometry: JEE Main PYQ, 2020–2025
Three Dimensional Geometry (Mathematics, Class 12) has been asked 224 times in the 111 JEE (Main) shift papers from 2020–2025: about 2 questions a paper, 7% of all Mathematics questions. It is the 3rd most-asked of 19 Mathematics chapters. The 2025 shifts had 30 from it in all. Last checked 28 October 2026.
Questions per year, all shifts
Year by year.
| Year | Questions |
|---|---|
| 2020 | 16in 14 papers |
| 2021 | 56in 26 papers |
| 2022 | 49in 24 papers |
| 2023 | 36in 12 papers |
| 2024 | 37in 18 papers |
| 2025 | 30in 17 papers |
JEE (Main) is sat in many shifts a year, each with its own paper; the counts add up every shift. 2021–2024 papers had 30 questions a subject, the others 25.
Inside the chapter
Topics asked most.
- Equation of a line127
- Skew lines and shortest distance41
- Coordinates of a point in space22
- Direction ratios and direction cosines21
- Angle between two lines13
Try it
10 Three Dimensional Geometry questions to practise.
Our own questions, written in the JEE Main pattern and checked, each with an explanation. Pick an option to see the answer.
Q1 · Mathematics · Three Dimensional Geometry
The vector equation of a line is r = (î + 2ĵ − 3k̂) + λ(2î − ĵ + k̂). Its Cartesian equation is:
Answer: (A)
Compare r = a + λb with the position vector a = x₁î + y₁ĵ + z₁k̂ and b = aî + bĵ + ck̂. The Cartesian form is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. Here (x₁, y₁, z₁) = (1, 2, −3) and (a, b, c) = (2, −1, 1), so the equation is (x − 1)/2 = (y − 2)/(−1) = (z + 3)/1.
Q2 · Mathematics · Three Dimensional Geometry
A cube has side a and one vertex at the origin O, with the opposite vertex at G(a, a, a). Find the shortest distance between the body diagonal OG and the diagonal of the face x = a that joins (a, a, 0) and (a, 0, a).
Answer: (D)
The body diagonal has direction b1 = (1,1,1) and passes through O. The face diagonal has direction b2 = (0,−1,1) and passes through (a, a, 0). Then b1 × b2 = (2, −1, −1) and |b1 × b2| = √6. With a2 − a1 = (a, a, 0), the dot product (a2 − a1)·(b1 × b2) = 2a − a = a. The distance is a/√6.
Q3 · Mathematics · Three Dimensional Geometry
The direction cosines (l, m, n) of two distinct lines both satisfy l + m + n = 0 and l² = m² + n². What is the acute angle between these two lines?
Answer: (A)
From l + m + n = 0, l = −(m + n), so l² = m² + 2mn + n². Setting l² = m² + n² gives mn = 0. If m = 0, then l = −n, and l² + n² = 1 gives (1/√2, 0, −1/√2). If n = 0, then l = −m, giving (1/√2, −1/√2, 0). Then cosθ = (1/√2)(1/√2) + 0 + 0 = 1/2, so θ = 60°.
Q4 · Mathematics · Three Dimensional Geometry
Two straight lines in three-dimensional space are neither parallel nor do they ever meet. Such a pair of lines is called
Answer: (A)
Skew lines are lines in space that are non-parallel and non-intersecting. They do not lie in a single plane. Two coplanar lines are either parallel or intersecting, so skew lines cannot be coplanar. Concurrent lines meet at one point. So the answer is skew lines.
Q5 · Mathematics · Three Dimensional Geometry
Line L is the intersection of the planes 2x − y + z = 3 and x + y − z = 1. Line M passes through the points (1, 0, −1) and (3, 1, 1). What is the acute angle between L and M?
Answer: (C)
The line of intersection of two planes is perpendicular to both normals, so its direction is n₁ × n₂ = (2, −1, 1) × (1, 1, −1) = (0, 3, 3) ∝ (0, 1, 1). Line M has direction (3 − 1, 1 − 0, 1 + 1) = (2, 1, 2). Then cos θ = |0 + 1 + 2| / (√2 · 3) = 1/√2, so θ = 45°.
Q6 · Mathematics · Three Dimensional Geometry
A rectangular box has edges 2, 3 and 6 units along the x, y and z axes, with one vertex at O(0, 0, 0) and the opposite vertex at G(2, 3, 6). Line m is the space diagonal OG. Line n is the diagonal of the base face (z = 0) joining the points (2, 0, 0) and (0, 3, 0). What is the cosine of the acute angle between m and n?
Answer: (A)
Write both lines as direction vectors. m = OG = (2, 3, 6) with |m| = √(4 + 9 + 36) = 7. n goes from (2, 0, 0) to (0, 3, 0), so n = (−2, 3, 0) with |n| = √13. The dot product is m·n = −4 + 9 + 0 = 5. Then cos θ = |m·n| / (|m||n|) = 5/(7√13).
Q7 · Mathematics · Three Dimensional Geometry
What are the coordinates of the foot of the perpendicular drawn from the point P(4, −3, 6) to the y-axis?
Answer: (A)
A point on the y-axis has the form (0, y, 0). The perpendicular from P(4, −3, 6) to the y-axis is parallel to the zx-plane and meets the axis at the point with the same y-coordinate as P. So the foot is (0, −3, 0). The other options set only one coordinate to 0, which gives feet on coordinate planes rather than on the axis.
Q8 · Mathematics · Three Dimensional Geometry
On the line (x − 3)/2 = (y + 2)/3 = (z − 1)/6, a point P lies at a distance 14 units from A(3, −2, 1) and has z-coordinate greater than 1. The coordinates of P are:
Answer: (D)
A general point on the line is (3 + 2t, −2 + 3t, 1 + 6t). Its distance from A is |t|√(2² + 3² + 6²) = 7|t|. Setting 7|t| = 14 gives t = ±2. Since z = 1 + 6t must exceed 1, t = 2, which gives P = (7, 4, 13).
Q9 · Mathematics · Three Dimensional Geometry
A cube has one vertex at the origin O and edges of length a along the positive coordinate axes. What is the shortest distance between the body diagonal through O and the diagonal of the face z = 0 that does not pass through O?
Answer: (A)
Body diagonal: through O(0,0,0) with direction b₁ = (1,1,1). Face diagonal: through A(a,0,0) and C(0,a,0), with direction b₂ = (−1,1,0). The two lines are skew, because the body diagonal meets the plane z = 0 only at O and O is not on AC. b₁ × b₂ = (1·0 − 1·1, 1·(−1) − 1·0, 1·1 − 1·(−1)) = (−1, −1, 2), with magnitude √6. a₂ − a₁ = (a,0,0), so the triple product is −a. d = a/√6.
Q10 · Mathematics · Three Dimensional Geometry
The segment joining A(1, −2, 3) and B(4, 4, 9) crosses the zx-plane at the point P. What is the z-coordinate of P?
Answer: (D)
Let P divide AB in the ratio k : 1. Its y-coordinate is (4k − 2)/(k + 1) = 0, so k = 1/2, a ratio of 1 : 2. Then z = (1 × 9 + 2 × 3)/3 = 5, and P is (2, 0, 5).
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All 224 real JEE Main questions from Three Dimensional Geometry.
The questions from the actual papers, with the official answer and an explanation for each — chapter by chapter, or as full timed papers.
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About Three Dimensional Geometry in JEE Main.
How many questions come from Three Dimensional Geometry in JEE Main?
Three Dimensional Geometry (Mathematics, Class 12) has been asked 224 times in the 111 JEE (Main) shift papers from 2020–2025: about 2 questions a paper, 7% of all Mathematics questions. It is the 3rd most-asked of 19 Mathematics chapters. The 2025 shifts had 30 from it in all.
Which topics of Three Dimensional Geometry are asked most in JEE Main?
Equation of a line (127), Skew lines and shortest distance (41), Coordinates of a point in space (22), Direction ratios and direction cosines (21), counted over the papers from 2020–2025.
Is Three Dimensional Geometry important for JEE Main 2027?
It is in the syllabus in force, so it can be asked. By past papers it ranks 3 of 19 in Mathematics; use that to decide how much time it gets, not whether to study it.
Where can I solve Three Dimensional Geometry JEE Main previous year questions?
On Nikaalle, free: sign up and practise every past JEE Main question from Three Dimensional Geometry with an explanation for each, or sit the full papers timed. The ten questions on this page are our own, written in the same pattern.
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