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CBSE Class 12 Mathematics Chapter 11 Three Dimensional Geometry Worksheet with Answers

Three Dimensional Geometry forms a critical component of CBSE Class 12 Mathematics, bridging algebraic concepts with spatial reasoning. This chapter typically carries 10-13 marks in the board examination through a mix of objective and subjective questions. This worksheet provides structured practice across all difficulty levels, from basic direction cosine problems to advanced applications involving skew lines and planes, ensuring comprehensive preparation for the 2025 board exams.

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Key takeaways

  • ✓Direction cosines satisfy the fundamental relation l² + m² + n² = 1 for any line in three-dimensional space
  • ✓Vector and Cartesian forms of line equations are interchangeable; mastering both forms is crucial for board exam success
  • ✓The equation of a plane can be expressed in normal form, intercept form, or general form ax + by + cz + d = 0
  • ✓Shortest distance between two skew lines uses the formula involving cross product and requires careful vector manipulation
  • ✓Angle between two planes equals the angle between their normal vectors, calculated using dot product formula
  • ✓Distance of a point from a plane and from a line are high-weightage topics carrying 4-6 marks in CBSE board exams
  • ✓Case-study questions on 3D geometry often integrate real-world applications like architecture, navigation, and engineering design

Chapter 11 Quick Recap: Three Dimensional Geometry Essentials

Three Dimensional Geometry extends coordinate geometry into three-dimensional space, introducing the z-axis perpendicular to the familiar x-y plane. Direction cosines (l, m, n) are the cosines of angles a line makes with the positive x, y, and z axes respectively, always satisfying l² + m² + n² = 1. Direction ratios are any three numbers proportional to direction cosines, commonly denoted as a, b, c. The chapter covers equations of lines in both vector form (r⃗ = a⃗ + λb⃗) and Cartesian form ((x-x₁)/a = (y-y₁)/b = (z-z₁)/c). Plane equations appear in normal form (r⃗·n⃗ = d), general form (ax + by + cz + d = 0), and intercept form (x/a + y/b + z/c = 1). Key applications include finding angles between lines and planes, calculating distances from points to lines and planes, determining shortest distance between skew lines, and solving problems involving coplanar lines. Understanding vector operations—dot product for angles and projections, cross product for perpendicularity and area—is fundamental to mastering this chapter.
  • Direction cosines: l = cos α, m = cos β, n = cos γ where α, β, γ are angles with axes
  • Angle between two lines: cos θ = |l₁l₂ + m₁m₂ + n₁n₂|
  • Perpendicular lines condition: a₁a₂ + b₁b₂ + c₁c₂ = 0
  • Parallel lines condition: a₁/a₂ = b₁/b₂ = c₁/c₂
  • Distance from point (x₁, y₁, z₁) to plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²)
  • Shortest distance between skew lines uses formula: |((b⃗₁ × b⃗₂)·(a⃗₂ - a⃗₁))|/|b⃗₁ × b⃗₂|

Section A: Multiple Choice Questions (1 mark each)

This section tests your conceptual understanding and quick problem-solving skills across all topics in Three Dimensional Geometry. Each question has been carefully designed to reflect the pattern seen in CBSE board exams and previous years' question papers. Focus on eliminating obviously wrong options first, then apply formulas accurately. Pay special attention to sign conventions, particularly when dealing with direction cosines and angles. Remember that MCQs in the board exam often test common misconceptions—read each option carefully before marking your answer. Time management is crucial; allocate no more than 60-90 seconds per MCQ to ensure you complete the entire worksheet within the suggested time frame. These questions cover direction cosines and ratios, line equations, plane equations, angles, and distance formulas, providing comprehensive coverage of the chapter for your board exam preparation.
  • Q1. If a line makes angles 90°, 60°, and 30° with the positive x, y, and z axes respectively, then its direction cosines are: (a) (0, 1/2, √3/2) (b) (0, √3/2, 1/2) (c) (1, 1/2, √3/2) (d) (0, 1/√2, 1/√2)
  • Q2. The equation of the line passing through (1, 2, 3) and parallel to the vector 3î + 2ĵ - k̂ is: (a) (x-1)/3 = (y-2)/2 = (z-3)/(-1) (b) (x+1)/3 = (y+2)/2 = (z+3)/(-1) (c) x/3 = y/2 = z/(-1) (d) (x-3)/1 = (y-2)/2 = (z+1)/3
  • Q3. The distance of the point (2, 3, 4) from the plane 3x - 6y + 2z + 11 = 0 is: (a) 1 unit (b) 2 units (c) 3 units (d) 4 units
  • Q4. Two lines with direction ratios (1, -1, 2) and (k, 3, 5) are perpendicular if k equals: (a) -11 (b) 11 (c) 13 (d) -13
  • Q5. The angle between the planes r⃗·(2î - ĵ + 2k̂) = 5 and r⃗·(3î + 6ĵ - 2k̂) = 9 is: (a) 30° (b) 45° (c) 60° (d) 90°
  • Q6. The equation of the plane with intercepts 2, 3, and 4 on the x, y, and z axes respectively is: (a) 6x + 4y + 3z = 12 (b) x/2 + y/3 + z/4 = 1 (c) 2x + 3y + 4z = 1 (d) Both (a) and (b)

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions demand precise knowledge of formulas, definitions, and key results from Three Dimensional Geometry. Unlike MCQs where you can use elimination, these require direct recall and accurate computation. Write your answers clearly in the space provided, ensuring proper mathematical notation. Pay attention to units where applicable and maintain consistency in vector notation throughout. These questions test fundamental concepts such as the relationship between direction cosines, conditions for perpendicularity and parallelism, standard forms of equations, and special properties of planes and lines. Review the basic formulas before attempting this section, and double-check calculations involving square roots and fractions. Marks are awarded only for completely correct answers with proper notation, so be meticulous. This section is particularly important as similar questions frequently appear in CBSE board exams as part of the objective section introduced in recent years.
  • Q7. If the direction ratios of a line are 2, -3, 6, then its direction cosines are ________, ________, ________.
  • Q8. The Cartesian equation of the line r⃗ = (2î - 3ĵ + 5k̂) + λ(î + 2ĵ + 3k̂) is ________.
  • Q9. A plane passes through the point (1, 1, 1) and is perpendicular to the vector î + 2ĵ + 3k̂. Its equation in vector form is ________.
  • Q10. Two lines are coplanar if the shortest distance between them is ________.
  • Q11. The angle between the line (x-1)/2 = (y+1)/(-1) = z/3 and the plane 2x + y - z = 4 is given by sin θ = ________.
  • Q12. If a line makes equal angles with the coordinate axes, then each angle is ________ and the direction cosines are ________.

Section C: True or False (1 mark each)

Determine whether each statement is true or false, providing a brief justification in one line. This section evaluates your deep conceptual understanding and ability to identify subtle errors in mathematical statements. Read each statement carefully—sometimes a single word can change the truth value. Think about counterexamples for statements you suspect are false, and recall standard theorems for those you believe are true. The justification is as important as the answer itself; in board exams, partially correct reasoning may earn partial credit even if the final answer is wrong. Focus on the logical structure of each statement and test it against standard results from NCERT Class 12 Mathematics Chapter 11. Pay special attention to statements involving necessary versus sufficient conditions, and universal versus existential quantifiers. These subtle logical distinctions are often tested in CBSE examinations. Review definitions of parallel lines, perpendicular lines, skew lines, and different forms of plane equations before attempting this section.
  • Q13. Any line perpendicular to the z-axis has direction cosines of the form (l, m, 0) where l² + m² = 1. True/False: ________. Justification: ________.
  • Q14. If two lines are parallel, then the angle between them is 0° or 180°. True/False: ________. Justification: ________.
  • Q15. The shortest distance between two parallel lines is unique. True/False: ________. Justification: ________.
  • Q16. If two planes are perpendicular, the dot product of their normal vectors is zero. True/False: ________. Justification: ________.
  • Q17. Every line in 3D space can be represented uniquely in Cartesian form. True/False: ________. Justification: ________.
  • Q18. The distance from the origin to the plane x + y + z = 1 is 1/√3. True/False: ________. Justification: ________.

Section D: Short Answer Questions (2-3 marks each)

Short-answer questions require you to show complete working within 4-6 lines, demonstrating logical progression from given data to the final answer. Each question typically carries 2 or 3 marks in CBSE board exams and tests your ability to apply formulas correctly, perform algebraic manipulations, and present solutions clearly. Write the relevant formula first, substitute the given values accurately, perform step-by-step calculations, and box or underline your final answer. Marks are distributed between method and accuracy—even if your final answer is incorrect due to a calculation error, you can earn method marks for correct approach. These questions cover standard problem types: finding direction cosines from direction ratios, writing equations of lines and planes in different forms, calculating angles between geometric objects, and computing distances. Practice writing solutions neatly, as presentation matters in board exams. Show all intermediate steps; do not skip to the answer directly even if you can calculate mentally, as examiners award marks based on visible working.
  • Q19. Find the equation of the line passing through the points (1, 2, 3) and (4, 5, 6) in both vector and Cartesian forms. (3 marks)
  • Q20. Find the angle between the line (x+1)/2 = (y-2)/3 = (z+3)/6 and the plane 2x + y - 2z = 5. (2 marks)
  • Q21. Find the perpendicular distance from the point (3, -2, 1) to the plane 2x - y + 2z + 3 = 0. (2 marks)
  • Q22. Show that the lines (x-1)/3 = (y-2)/1 = (z-3)/5 and (x+2)/4 = (y-1)/3 = (z+3)/(-2) are perpendicular. (2 marks)
  • Q23. Find the equation of the plane passing through the intersection of planes x + y + z = 6 and 2x + 3y + 4z = 5, and passing through the point (1, 1, 1). (3 marks)

Section E: Long Answer and HOTS Questions (4-6 marks each)

Long-answer questions demand comprehensive solutions with detailed reasoning, often involving multiple steps and integration of several concepts from Three Dimensional Geometry. These questions carry 4 to 6 marks and test your ability to tackle complex problems systematically. Begin by identifying what is given and what needs to be found, then plan your approach before writing. Break complex problems into manageable parts, solving each step completely before moving to the next. Use diagrams where appropriate to visualize the geometric situation—this not only helps you solve the problem but also earns presentation marks. HOTS (Higher Order Thinking Skills) questions require you to apply concepts in unfamiliar contexts, combine multiple topics, or provide rigorous proofs. They may involve proving geometric properties, finding loci, or solving optimization problems in 3D space. Allocate 8-12 minutes per question in this section. These questions closely mirror the 5-mark and 6-mark questions that appear in Section C of the CBSE Class 12 Mathematics board exam, so practice them thoroughly. Clear communication of mathematical reasoning is as important as getting the correct numerical answer.
  • Q24. Find the shortest distance between the lines r⃗ = (î + 2ĵ + 3k̂) + λ(î - 3ĵ + 2k̂) and r⃗ = (4î + 5ĵ + 6k̂) + μ(2î + 3ĵ + k̂). Also, determine whether the lines are skew, parallel, or intersecting. (6 marks)
  • Q25. A plane passes through the point (1, -2, 3) and is perpendicular to the line joining the points (3, 4, -1) and (2, -1, 5). Find the equation of the plane and calculate the distance of this plane from the origin. (5 marks)
  • Q26. Find the coordinates of the foot of the perpendicular drawn from the point (2, 3, 4) to the line (x-1)/5 = (y-2)/3 = (z-3)/(-1). Also, find the length of the perpendicular. (6 marks)

Case Study Question: Architectural Design Application

Case-study questions, introduced in recent CBSE board exams, present a real-world scenario followed by 4-5 sub-questions of varying difficulty. These questions test your ability to extract mathematical information from contextual descriptions and apply Three Dimensional Geometry concepts to practical situations. Read the case study paragraph carefully, identifying all numerical data and geometric relationships described. Sketch a rough diagram if needed to visualize the situation. The sub-questions typically progress from easy (1 mark) to moderate (2-3 marks), allowing students of all abilities to attempt them. Such questions integrate mathematics with fields like architecture, engineering, physics, and navigation, demonstrating the practical relevance of 3D geometry. In the board exam, case studies usually carry 4-5 marks total and appear in Section C. Practice reading comprehension is as important as mathematical skills for these questions—underline key information as you read. The scenario below involves an architectural application, a common theme in CBSE case-study questions for this chapter.

Complete Answer Key with Explanations

This comprehensive answer key provides correct answers along with brief explanations and relevant formulas for every question in the worksheet. Use this section for self-assessment after attempting all questions on your own—resist the temptation to look at answers while solving. For MCQs and fill-in-the-blanks, verify that your answers match exactly. For numerical problems, check both your method and final answer; if they differ, trace back through your working to identify errors. The explanations highlight common mistakes students make, helping you avoid similar errors in the actual exam. When practicing, maintain an error log noting which types of questions you consistently get wrong, then revisit those concepts in your NCERT Class 12 Mathematics textbook and solved examples. In board exams, even if you make a calculation error, correct method can fetch you most of the marks, so pay attention to the solution approaches demonstrated here. For case-study questions, notice how information from the passage is translated into mathematical expressions. Regular practice with such worksheets significantly improves speed and accuracy, both crucial for scoring well in the CBSE Class 12 Mathematics board examination.

How CBSETUTOR.ai Helps Master Three Dimensional Geometry

Three Dimensional Geometry challenges many Class 12 students because it requires strong spatial visualization skills alongside algebraic manipulation—abilities that develop through extensive practice with immediate feedback. CBSETUTOR.ai provides exactly this combination through its AI-powered learning platform. Students can upload photos of any 3D geometry problem—whether from this worksheet, NCERT exercises, or previous years' board papers—and receive instant step-by-step solutions with detailed explanations. The AI tutor identifies conceptual gaps, such as confusion between direction cosines and direction ratios, or errors in applying the shortest distance formula, and provides targeted practice problems to strengthen weak areas. Available 24×7, the platform supports learning at any hour, crucial during board exam preparation when doubts arise during late-night study sessions. Unlike coaching classes with fixed schedules, CBSETUTOR.ai adapts to each student's pace, spending more time on challenging topics like skew lines while quickly reviewing mastered concepts like basic direction cosines. The platform covers the entire CBSE Class 12 Mathematics syllabus at a flat fee of just ₹999 per month—one price for all classes from 6 to 12—making quality education accessible without the ₹15,000-25,000 monthly fees typical of metro-city coaching centres. Parents can try the platform risk-free with a 3-day free trial, experiencing firsthand how AI-powered personalized learning transforms their child's understanding of complex topics. For Three Dimensional Geometry specifically, the visual problem-solving approach helps students develop the spatial intuition essential for both board exam success and future engineering or architecture studies.
  • Upload photo of any 3D geometry problem from worksheet or textbook for instant detailed solution
  • AI identifies whether errors stem from conceptual misunderstanding or calculation mistakes
  • Generates unlimited practice problems on specific weak topics like angle between line and plane
  • Explains geometric visualizations with diagrams for better spatial understanding
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  • Covers all of CBSE Class 12 Mathematics including integration with other chapters at ₹999/month flat fee
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Worksheet Difficulty Level and Time Guidelines

This comprehensive worksheet is designed at the CBSE board exam standard difficulty level, with questions ranging from basic recall (Section A, B, C) to application and analysis (Section D, E) to integration and evaluation (Case Study). It aligns with the cognitive levels specified in the CBSE Class 12 Mathematics curriculum: remembering and understanding (40%), applying and analysing (40%), and evaluating and creating (20%). Students should attempt this worksheet under timed conditions to simulate board exam pressure and develop speed alongside accuracy. Suggested time allocation: Section A (MCQs) - 6-8 minutes for 6 questions; Section B (Fill-ins) - 6-8 minutes; Section C (True/False) - 8-10 minutes including justifications; Section D (Short answer) - 15-18 minutes for 5 questions; Section E (Long answer) - 25-30 minutes for 3 questions; Case Study - 8-10 minutes. Total suggested time: 75-90 minutes, matching the time proportion for this chapter in the actual 3-hour board exam. First-time attempts may take longer; aim to complete within 90 minutes initially, then work down to 75 minutes with practice. After completing the worksheet, spend 20-25 minutes reviewing answers using the answer key, focusing especially on questions you got wrong or found difficult. Mark these for revision and reattempt them after a few days. Regular practice with such worksheets across all chapters is the most effective strategy for scoring 90+ in CBSE Class 12 Mathematics board exams, as confirmed by toppers year after year.
  • Difficulty: CBSE Board Exam Standard (mix of easy, moderate, and difficult questions)
  • Total Questions: 32 across multiple formats testing all cognitive levels
  • Suggested Time: 75-90 minutes under exam conditions without interruptions
  • Marking Scheme: Total 50 marks distributed as per typical board exam weightage
  • Recommended Approach: Attempt all sections sequentially, marking difficult questions for later review
  • Success Target: Score 40+ marks (80%) to indicate strong chapter preparation for boards
  • Follow-up: Reattempt incorrect questions after reviewing concepts from NCERT textbook

Frequently asked questions

What is the weightage of Three Dimensional Geometry in CBSE Class 12 Mathematics board exam?+
Three Dimensional Geometry typically carries 10-13 marks in the CBSE Class 12 Mathematics board exam, appearing as 2-3 questions including one long-answer question of 5-6 marks, one or two short-answer questions of 2-3 marks, and possibly MCQs or case-study sub-parts worth 1-2 marks each.
What is the difference between direction cosines and direction ratios?+
Direction cosines (l, m, n) are the cosines of angles a line makes with the positive x, y, and z axes, always satisfying l² + m² + n² = 1. Direction ratios (a, b, c) are any three numbers proportional to direction cosines; they give the same direction but need not satisfy any specific relation. To convert direction ratios to direction cosines, divide each by √(a² + b² + c²).
How do I determine if two lines in 3D space are parallel, intersecting, or skew?+
Two lines are parallel if their direction ratios are proportional (a₁/a₂ = b₁/b₂ = c₁/c₂). They intersect if they are coplanar and not parallel; check by setting the Cartesian equations equal and solving for a consistent parameter value. They are skew if the shortest distance between them is non-zero, calculated using the cross-product formula for lines in vector form.
What are the most common mistakes students make in Three Dimensional Geometry questions?+
Common errors include: forgetting the ± sign when converting direction ratios to direction cosines; mixing up the formulas for angle between lines versus angle between line and plane; sign errors when calculating distances; not checking whether lines are skew before applying shortest distance formula; and incorrectly writing vector equations by confusing position and direction vectors.
How should I prepare Three Dimensional Geometry for scoring high marks in boards?+
Master all standard formulas and their derivations from NCERT. Practice converting between vector and Cartesian forms fluently. Solve all NCERT examples and exercises thoroughly, then attempt previous years' board papers (at least 10 years). Focus on high-weightage topics: shortest distance between skew lines, angle calculations, and distance from point to plane. Regular timed practice with worksheets like this one builds speed and accuracy essential for board exam success.
Are case-study questions difficult in Three Dimensional Geometry?+
Case-study questions test the same concepts as regular questions but present them in a real-world context requiring careful reading. They are generally moderate in difficulty, with sub-parts ranging from easy (1 mark) to moderate (2 marks). The key is extracting mathematical information from the contextual description. Practice 15-20 case studies from sample papers to become comfortable with this format introduced in recent CBSE exams.
What is the formula for shortest distance between two skew lines and when should I use it?+
For lines r⃗ = a⃗₁ + λb⃗₁ and r⃗ = a⃗₂ + μb⃗₂, shortest distance = |((b⃗₁ × b⃗₂)·(a⃗₂ - a⃗₁))|/|b⃗₁ × b⃗₂|. Use this formula only after confirming the lines are skew (neither parallel nor intersecting). If lines intersect, shortest distance is zero. If parallel, use the perpendicular distance formula between a point on one line and the other line.
How do I convert between vector and Cartesian forms of line equations?+
Vector form r⃗ = (x₁î + y₁ĵ + z₁k̂) + λ(aî + bĵ + ck̂) converts to Cartesian form (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Conversely, from Cartesian form, read off the point (x₁, y₁, z₁) from the numerators and direction ratios (a, b, c) from denominators. Practice both conversions until they become automatic, as board questions may ask for either form.
What is the best way to remember all the formulas in this chapter?+
Create a formula sheet organized by topic: direction cosines, line equations, plane equations, angles, distances. Write each formula, its conditions of applicability, and one example. Review this sheet daily for a week before exams. More effective than rote memorization is understanding derivations from NCERT textbook—when you know why a formula works, you can reconstruct it even if you forget the exact form during the exam.
How much time should I spend on Three Dimensional Geometry during board exam preparation?+
Allocate 8-10 days across your preparation schedule: 3-4 days for initial concept learning and NCERT exercise solving, 2-3 days for worksheet practice and previous years' questions, 2 days for revision and doubt clearing, and 1 day for final formula review. Given its 10-13 mark weightage (about 13% of the paper), it deserves proportional attention. Combine it with Vector Algebra study as many concepts overlap, making learning more efficient.

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