JEE Main Maths · 3D Geometry PYQ
JEE Main 3D Geometry PYQ (2002–2025)
3D Geometry is a P1 chapter — ~5% JEE Main weightage with 1–2 questions per session. Heavily vector-driven, with template-repetitive question patterns (shortest distance, angle between line and plane, image of point in plane). Quick to learn once you master Vectors.
3D Geometry PYQs from 2002 to 2025, tagged by sub-topic (direction cosines, lines, planes, shortest distance) and difficulty. Each KaTeX solution shows both vector and Cartesian forms where applicable.
3D Geometry at a Glance
Key Sub-Topics & What's Tested
Direction Cosines & Ratios
l² + m² + n² = 1 for direction cosines, relationship with direction ratios (a, b, c), angle between two lines using d.c.s.
Equation of Line in 3D
Vector form r = a + λb, Cartesian form (x-x₁)/a = (y-y₁)/b = (z-z₁)/c, symmetric form, line through two points.
Equation of Plane
Vector form r·n̂ = d, Cartesian form ax + by + cz + d = 0, plane through three points, plane through line and point.
Angle Between Line and Plane
sin θ = |a·n|/(|a||n|) where a is direction vector of line and n is normal to plane; (90° - angle between line and normal).
Shortest Distance Between Lines
Skew lines: d = |[(a₂ - a₁) · (b₁ × b₂)]|/|b₁ × b₂|. Parallel lines: use perpendicular distance formula from point to line.
Distance of Point from Plane
d = |ax + by + cz + d|/√(a² + b² + c²), sign indicates which side of plane, projection formulation.
Image of Point in Plane
Reflection formula: 2× perpendicular from point to plane, image = point - 2·perpendicular. Key PYQ template.
Foot of Perpendicular
From point to line: parametrise line, find λ minimising distance. From point to plane: drop normal from point to plane.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Line-Plane Angle | 20% | Angle between line (x-1)/2 = y/3 = (z-2)/-1 and plane x + 2y - z = 5. |
| Shortest Distance (Skew Lines) | 20% | Find shortest distance between two given skew lines using the scalar triple product formula. |
| Plane Through Points/Line | 20% | Find equation of plane passing through (1,0,1), (2,1,2) and perpendicular to plane x + y - 2z = 4. |
| Distance from Point to Plane | 15% | Find distance from (1, 2, 3) to plane 2x - y + z - 5 = 0. |
| Image of Point in Plane/Line | 15% | Find reflection of point (1, 2, 3) in the plane 2x + y - z = 7. |
| Intersection / Coplanarity | 10% | Find point of intersection of two lines, or determine coplanarity of two given lines. |
How to Solve 3D Geometry PYQs
- 1Master vector vs Cartesian form switching. Line: r = a + λb ↔ (x-x₀)/d₁ = (y-y₀)/d₂ = (z-z₀)/d₃. Plane: r·n̂ = p ↔ ax + by + cz = d. Both are standard.
- 2For shortest distance, use formula directly. d = |[(a₂ - a₁) · (b₁ × b₂)]|/|b₁ × b₂| for skew lines. Set up vector difference, cross-product, dot-product — three steps.
- 3Angle between line and plane involves normal. If θ is angle between line direction and normal, then angle between line and plane is (90° - θ). sin(line-plane) = cos(line-normal).
- 4Image of point in plane: add 2× perpendicular. Perpendicular vector from P to plane = (normal) × (signed distance). Image = P - 2·perpendicular.
- 5For plane through line and point: use cross-product. Normal to plane = (direction of line) × (vector from any point on line to external point).
Common Mistakes That Cost Marks
- Missing direction cosine constraint. l² + m² + n² = 1 always. If working with direction ratios (a, b, c), normalize: l = a/√(a²+b²+c²), etc.
- Applying shortest distance formula to intersecting lines. Intersecting lines have shortest distance = 0. Formula works only for skew lines (non-intersecting, non-parallel).
- Confusing angle between planes vs angle between normals. Angle between two planes = angle between their normals (can be acute or obtuse; usually take acute).
- Using |n| without normalisation in distance formula. Distance from point to plane = |ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²). Don't forget the denominator.
- Wrong sign in image of point reflection. Image = 2F - P where F is foot of perpendicular. Don't calculate image by flipping sign of only one coordinate.
Related JEE Main Practice
Frequently asked questions
How many 3D Geometry PYQs should I solve?
Target 50–70 PYQs across 2010–2025. Given high template repeat (shortest distance, image, angle), practice builds instant pattern recognition.
Is 3D Geometry harder than Coordinate Geometry (2D)?
Computationally yes — more variables, more vector algebra. But templates are fewer. Once you master 5–6 problem types, 3D Geometry PYQs become formulaic.
Do I need Vectors before 3D Geometry?
Yes — Vectors is foundational. Study Vectors (dot/cross products, scalar triple) first, then 3D Geometry becomes natural. Attempting 3D without Vectors is painful.
What's the most-tested 3D Geometry template?
Image of point in plane (or shortest distance between skew lines). Both appear in 4-5 PYQs per 10-year window with slight variations.
Do I need parametric form for both lines and planes?
Line: parametric (r = a + λb) is almost always useful. Plane: parametric is rarely needed; Cartesian form (ax + by + cz = d) suffices for nearly all PYQs.
Does 3D Geometry connect to other chapters?
Yes — Vectors (direct prerequisite), Matrices & Determinants (scalar triple as determinant), Limits/Calculus (rare). Strong Vectors + Matrices fluency makes 3D trivial.
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