CBSE Class 12 Three Dimensional Geometry PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1-2 short answers (2-3 marks), 1-2 long answers on line/plane equations or distance (4-6 marks).
Repeated Topics: Direction cosines/ratios, equation of line (symmetric & vector form), angle between two lines, equation of plane (normal form, intercept form), angle between line & plane, distance of point from plane/line, shortest distance between two skew lines, coplanarity of lines, intersection of line & plane.
Difficulty: Medium; formula application & finding correct parameters (λ, μ) key. Skew lines distance is most scoring & repeated.
Changes: More questions on skew lines distance and angle between line & plane in 2023–2025.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Find the shortest distance between two skew lines: r = (i + 2j + 3k) + λ(2i + 3j + 4k) and r = (4i + 5j + 6k) + μ(i + j + k) (or similar).	Long Answer (4-6 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Use formula d = \|(P2 - P1) · (b1 × b2)\| / \|b1 × b2\|; compute cross product carefully.
Find the equation of plane passing through three points A(1,1,1), B(2,3,4), C(5,6,7) or normal vector given.	Long Answer (4 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Normal n = AB × AC; plane: n · (r - A) = 0.
Find angle between two planes: ax+by+cz+d1=0 and ax+by+cz+d2=0 or line & plane.	Short Answer (2-3 marks)	2022, 2023, 2024, 2025	Repeated 4x; cos θ = \|n1 · n2\| / (\|n1\|\|n2\|); for line & plane sin θ = \|cos φ\| where φ is angle between line & normal.
Assertion: Two lines are coplanar if shortest distance between them is zero. Reason: Skew lines are non-coplanar.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Find direction cosines of line joining points (1,2,3) and (4,5,6) or symmetric form of line.	Short Answer (2 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Direction ratios = differences; cosines = ratios / magnitude.
Find distance of point (1,2,3) from plane 2x - y + 2z - 3 = 0.	Short Answer (2-3 marks)	2022, 2023, 2024	Repeated 3x; Formula \|ax+by+cz+d\| / √(a²+b²+c²).
MCQ: The angle between two planes is equal to the angle between their: (a) normals (b) lines of intersection (c) both	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (a) normals.
Show that line r = a + λ b is perpendicular to plane n · r = d if b · n = 0.	Proof (2-3 marks)	2023, 2025	Repeated 2x; Direction of line parallel to normal of plane → perpendicular.
Case-based: Given two lines (vector or symmetric form), check if parallel/intersecting/skew, find distance if skew.	Case-Based (4-6 marks)	2023, 2025	Repeated 2x; Check if b1 parallel to b2, then coplanar check via triple product.
Find equation of plane through point & perpendicular to given vector or containing two lines.	Short Answer (3 marks)	2021 Term 2, 2024	Repeated 2x; Use normal vector or point-normal form.

Best Format for Preparation and Answering

Study Format: Memorize all formulas (direction cosines, plane forms, distance formulas, skew lines distance). Practice finding intersection/coplanarity daily. Make flowchart for skew lines check (parallel → distance 0 or skew).
Answer Format: Write formula first → substitute values → compute step-by-step (cross product, magnitude). For proofs: Use vector dot/cross properties. Always box final answer (distance, angle, equation).
Tips: Shortest distance between skew lines is highest repeat & marks — master the formula. Practice vector form & symmetric form conversions. Time: 10–12 mins for 4–6 mark questions. Very scoring if no calculation mistakes.

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