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CBSE Class 11 Mathematics Chapter 11 Introduction to Three Dimensional Geometry Worksheet with Answers

Q: What is the difficulty level and suggested time for this CBSE Class 11 Mathematics Chapter 11 worksheet?

The worksheet is Moderate to Advanced difficulty, suitable for comprehensive exam preparation. The suggested time is 90 minutes under exam-like conditions, though initial attempts may take longer as you familiarize yourself with 3D coordinate problems.

Q: Which topics from NCERT Class 11 Mathematics Chapter 11 are covered in this worksheet?

This worksheet covers all major topics: coordinate systems in space, identification of octants, distance formula in three dimensions, section formula (internal and external division), midpoint formula, and coordinates of points on axes and planes.

Q: How should I use the answer key provided with this worksheet?

Attempt the entire worksheet independently first without referring to solutions. After completion, check your answers against the detailed answer key, award marks honestly, and identify weak areas. Revisit those concepts in your NCERT textbook before reattempting similar questions.

Q: What is the sign convention for octants in three-dimensional geometry?

Octant I: (+,+,+), II: (−,+,+), III: (−,−,+), IV: (+,−,+), V: (+,+,−), VI: (−,+,−), VII: (−,−,−), VIII: (+,−,−). Remember these sign patterns as they are frequently tested in CBSE Class 11 Mathematics exams.

Q: How is the distance formula different in 3D compared to 2D coordinate geometry?

The 3D distance formula includes the z-coordinate: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. This is an extension of the 2D formula with an additional term for the third dimension, following the Pythagorean theorem in three dimensions.

Q: What is the section formula for internal division in three dimensions?

If point P divides the line joining A(x₁,y₁,z₁) and B(x₂,y₂,z₂) internally in ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)). Each coordinate is calculated independently using the same ratio.

Q: How can I identify which coordinate plane a point lies on?

A point lies on the xy-plane if z=0, on the yz-plane if x=0, and on the zx-plane if y=0. A point on the x-axis has y=0 and z=0; similarly for other axes. Remember: the plane name tells you which coordinates are non-zero.

Q: Are case-study questions important for CBSE Class 11 Mathematics term exams?

Yes, case-study questions carry 4-5 marks and are a regular feature in recent CBSE exam patterns. They test your ability to apply mathematical concepts to real-world scenarios. Practice these thoroughly as they are scoring opportunities if you read carefully and extract data accurately.

Q: What are common mistakes students make in Introduction to Three Dimensional Geometry?

Common errors include: confusing octant sign conventions, forgetting to include the z-term in distance formula, mixing up which coordinate is zero on which plane, calculation errors in section formula, and not showing sufficient working in exam answers.

Q: How does CBSETUTOR.ai help with Class 11 Mathematics Chapter 11 doubts?

CBSETUTOR.ai provides 24×7 AI tutoring where you can upload photos of any 3D Geometry problem and receive instant step-by-step solutions with explanations. It covers all CBSE Class 11 Mathematics topics at ₹999/month for Classes 6-12, with a 3-day free trial available to start learning interactively.

Introduction to Three Dimensional Geometry marks a crucial transition from plane geometry to spatial reasoning in CBSE Class 11 Mathematics. This worksheet offers a structured practice tool covering coordinate systems in space, distance formula, and section formula as per NCERT Class 11 Mathematics Chapter 11. Whether you are preparing for school term exams or strengthening foundational concepts for JEE, this worksheet provides varied question types to test comprehension and application skills.

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Key takeaways

✓Worksheet difficulty: Moderate to Advanced; suitable for comprehensive revision before term exams.
✓Suggested time: 90 minutes for completion under exam-like conditions.
✓Covers all NCERT topics: coordinate systems in space, distance formula, and section formula in 3D.
✓Includes 6 MCQs, 5 fill-in-the-blanks, 5 true/false, 5 short-answer, 3 long-answer, and 1 case-study question.
✓Complete answer key provided with step-by-step explanations for self-assessment.
✓Aligned with latest CBSE Class 11 Mathematics examination pattern and NCERT solutions.
✓Printable PDF-ready format for classroom or home practice sessions.

Quick Chapter Recap: Introduction to Three Dimensional Geometry

Chapter 11 of NCERT Class 11 Mathematics introduces students to the cartesian coordinate system extended into three dimensions. The chapter builds on Class 10 coordinate geometry by adding the z-axis perpendicular to the xy-plane. Key concepts include representing a point P as (x, y, z) where x, y, z are distances from the yz-plane, zx-plane, and xy-plane respectively. The distance formula in 3D space between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. The section formula helps find coordinates of a point dividing a line segment internally or externally in a given ratio m:n. Understanding octants (eight regions formed by the three coordinate planes) and sign conventions is fundamental. These concepts form the foundation for vectors, planes, and straight lines in 3D geometry covered in Class 12.

Coordinate axes: X-axis, Y-axis, Z-axis mutually perpendicular at origin O(0,0,0)
Coordinate planes: XY-plane (z=0), YZ-plane (x=0), ZX-plane (y=0)
Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Section formula (internal): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))
Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Eight octants determined by signs of x, y, z coordinates

Worksheet Information and Instructions

This worksheet is designed for CBSE Class 11 students preparing for Mathematics term examinations. The difficulty level is Moderate to Advanced, suitable for students who have completed NCERT Class 11 Mathematics Chapter 11 and want rigorous practice. Suggested completion time is 90 minutes under exam conditions, though students may take longer during initial practice. The worksheet contains 25+ questions across six sections (A to F) covering all topics: coordinate systems in space, distance formula applications, section formula problems, and octant identification. Each section tests different skills – from quick recall (MCQs) to conceptual understanding (short answers) to problem-solving ability (long answers and case study). Attempt all questions in sequence. For best results, avoid referring to notes or solutions initially. After completion, check your answers against the detailed answer key provided at the end. Mark incorrect answers and revisit those concepts in your NCERT textbook or Class 11 Mathematics notes.

Total marks: 50 (indicative weightage for practice)
No calculators allowed; all calculations must be shown
Draw diagrams wherever applicable, especially for 3D problems
Write assumptions clearly when solving section formula problems
Check units and sign conventions carefully in distance problems

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing quick recall and conceptual clarity on Introduction to Three Dimensional Geometry. Each question carries 1 mark. Select the most appropriate option from the four given choices. These MCQs mirror the pattern seen in CBSE Class 11 Mathematics term exams and cover octants, coordinate identification, distance formula basics, and section formula applications. Attempt all six questions within 10-12 minutes to simulate exam time pressure. Remember that negative marking is not typically applied in school exams, so make educated guesses if unsure. Pay special attention to sign conventions when identifying octants and coordinate planes. Many students lose marks by confusing the plane equations (e.g., xy-plane has z=0, not x=0). Review your NCERT Class 11 Mathematics textbook examples before attempting if this is your first practice session with 3D coordinate geometry.

Q1. The point (−2, 3, −5) lies in which octant? (a) I (b) IV (c) VI (d) VIII
Q2. The distance of point P(3, 4, 5) from the origin is: (a) 5√2 (b) 5 (c) 12 (d) √50
Q3. If a point lies on the xy-plane, which coordinate is zero? (a) x (b) y (c) z (d) x and y both
Q4. The coordinates of the midpoint of the segment joining A(1, 2, 3) and B(3, 4, 5) are: (a) (2, 3, 4) (b) (4, 6, 8) (c) (1, 1, 1) (d) (2, 2, 2)
Q5. In which octant do all coordinates have negative values? (a) I (b) IV (c) VII (d) VIII
Q6. The point equidistant from (0,0,0), (2,0,0), and (0,2,0) lies on: (a) x-axis (b) y-axis (c) z-axis (d) a plane x+y=2

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

This section tests your ability to recall formulas, definitions, and key terminology from Chapter 11 Introduction to Three Dimensional Geometry. Each blank carries 1 mark and should be filled with precise mathematical expressions, numbers, or terms as per NCERT Class 11 Mathematics standard notation. Avoid writing long sentences; concise answers are expected. These questions cover coordinate plane equations, octant properties, special point locations, and formula components. Students often make errors in remembering which coordinate is zero on which plane – the yz-plane has x=0, the zx-plane has y=0, and the xy-plane has z=0. Similarly, remember that in octant notation, octant I has all positive coordinates (+,+,+) while octant VIII has all negative coordinates (−,−,−). Review section formulas carefully; the denominator is (m+n) for internal division and (m−n) for external division. Allocate about 8-10 minutes for this section during your 90-minute practice window.

Q7. The equation of the yz-plane is ______.
Q8. The distance between points A(x, y, z) and B(x, y, 0) is ______.
Q9. If a point divides the join of A and B in ratio 2:3 internally, the section formula denominator is ______.
Q10. The coordinates of a point on the z-axis are of the form ______.
Q11. The octant in which x<0, y>0, z<0 is octant number ______.

Section C: True or False (1 mark each)

This section contains five statements related to the coordinate system in space, distance formulas, and section formulas from CBSE Class 11 Mathematics Chapter 11. Mark each statement as True (T) or False (F). Each correct answer carries 1 mark. Be cautious with statements involving special cases, such as points on coordinate axes or planes. A common pitfall is assuming that the distance formula in 3D is identical to the 2D version – it includes an additional z-component term. Similarly, students sometimes incorrectly believe that the midpoint formula changes in 3D; it remains an average of corresponding coordinates. Statements may also test understanding of octant sign conventions and symmetry properties. Read each statement twice before marking your answer. If a statement is partially correct but contains one incorrect clause, it is considered False. Attempt this section within 6-8 minutes. Cross-verify your answers with worked examples from your NCERT Class 11 Mathematics textbook or Class 11 Mathematics solutions after completing the worksheet independently.

Q12. The coordinates of the origin are (1, 1, 1). (T / F)
Q13. The distance formula in 3D is √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. (T / F)
Q14. A point on the y-axis has coordinates (0, y, 0). (T / F)
Q15. The xy-plane divides space into two octants. (T / F)
Q16. If a point P divides AB externally in ratio m:n, the section formula uses denominator (m−n). (T / F)

Section D: Short Answer Questions (2 marks each)

This section comprises five short-answer questions, each worth 2 marks, requiring concise working and final answers. These questions test your ability to apply the distance formula, section formula, and coordinate geometry concepts in straightforward problem contexts. Show all essential steps; marks are awarded for method even if the final answer is incorrect due to a calculation slip. Typical questions include finding distances between given points, identifying coordinates of points satisfying certain conditions, locating midpoints, and determining which octant a point lies in based on given information. Each question should take about 3-4 minutes. Use proper mathematical notation and write coordinates in standard (x, y, z) format. Draw a rough 3D sketch if it helps visualize the problem, especially when dealing with perpendicular distances to planes or symmetry. These questions mirror the 2-mark questions seen in CBSE Class 11 term exams and are excellent practice for building speed and accuracy. Refer to NCERT Class 11 Mathematics exemplar problems for additional practice after completing this worksheet.

Q17. Find the distance between the points A(2, 3, 5) and B(4, 3, 1). (2 marks)
Q18. Find the coordinates of the point which divides the line segment joining A(1, −2, 3) and B(3, 4, −5) in the ratio 2:3 internally. (2 marks)
Q19. Show that the points (0, 7, 10), (−1, 6, 6), and (−4, 9, 6) form a right-angled triangle. (2 marks)
Q20. If the distance of point P(x, 2, 3) from the origin is 7, find the possible values of x. (2 marks)
Q21. In which octant does the point (−3, 2, −4) lie? Justify your answer. (2 marks)

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

This section contains three long-answer or Higher Order Thinking Skills (HOTS) questions, each carrying 4 marks. These problems require multi-step reasoning, integration of multiple concepts, and sometimes algebraic manipulation. Expect questions involving verification of geometric properties (collinearity, forming specific triangles), coordinate calculations with parameters, or application of section formula in both internal and external division contexts. Allocate 10-12 minutes per question. Write clear, step-by-step solutions with logical flow. State any formula you use before applying it. For geometry verification problems, compute all necessary distances or use section formula to show required conditions. HOTS questions may present unfamiliar contexts or require you to derive results rather than simply apply formulas. These questions test depth of understanding and are similar to 4-mark questions in CBSE board exams. After checking your solutions against the answer key, revisit any mistakes and understand where your reasoning diverged. Practice similar questions from Class 11 Mathematics solutions or reference books to master this question type.

Q22. Find the ratio in which the yz-plane divides the line segment joining points A(2, 4, 5) and B(3, 5, −4). Also find the coordinates of the point of division. (4 marks)
Q23. Show that the points A(1, 2, 3), B(−1, −1, −1), and C(3, 5, 7) are collinear. (4 marks)
Q24. The vertices of a triangle are A(1, 1, 2), B(4, 3, 1), and C(2, 3, 5). Find the coordinates of the centroid and verify that the centroid divides each median in the ratio 2:1. (4 marks)

Section F: Case Study Question (4 marks)

Case-study questions are now a regular feature in CBSE Class 11 Mathematics examinations and test your ability to extract mathematical information from real-world or descriptive contexts. This question presents a scenario involving 3D coordinates, followed by 3-4 sub-questions totaling 4 marks. Read the case study carefully, identify given data, and note what is being asked in each sub-part. The scenario below involves a telecommunications tower location problem, a practical application of coordinate geometry in space. Sub-questions typically include straightforward applications (1 mark), formula-based calculations (2 marks), and a slightly analytical part (1 mark). Attempt this section after completing Sections A-E, as it integrates multiple concepts. Write answers to each sub-question clearly labeled (i), (ii), (iii). Show essential working for calculation-based parts. Allocate about 12-15 minutes. Case studies are scoring opportunities if you read carefully and avoid misinterpreting the given information. Practice similar case-based questions from CBSE sample papers or Class 11 Mathematics notes to build familiarity with this format.

Complete Answer Key with Explanations

This section provides detailed solutions to all questions in the worksheet. Use this answer key only after attempting the entire worksheet independently. For MCQs and fill-in-the-blanks, correct answers are stated first, followed by brief explanations. For numerical and derivation-based questions, step-by-step working is shown to help you understand the method. Compare your solutions with these model answers. If your final answer matches but the method differs, verify that your approach is mathematically sound. Award yourself marks honestly based on the marking scheme indicated. Identify topics where you lost marks and revise those sections from your NCERT Class 11 Mathematics textbook. If you score below 60%, revisit the chapter thoroughly and reattempt the worksheet after a few days. For additional practice, explore Class 11 Mathematics solutions or use CBSETUTOR.ai where you can upload photos of 3D geometry problems and get instant step-by-step solutions at ₹999/month for all subjects across Classes 6-12, with a 3-day free trial to start mastering concepts interactively.

Section A Answers: 1(c), 2(d), 3(c), 4(a), 5(d), 6(d)
Section B Answers: 7. x=0 | 8. |z| | 9. 5 | 10. (0, 0, z) | 11. VI
Section C Answers: 12. F | 13. T | 14. T | 15. F (divides into 4 regions, not 2 octants) | 16. T
Section D Answers: See detailed solutions below
Section E Answers: See detailed solutions below
Section F Answers: See detailed solutions below

How CBSETUTOR.ai Supports Your 3D Geometry Mastery

Mastering Introduction to Three Dimensional Geometry requires visualizing 3D space, remembering multiple formulas, and applying them correctly under time pressure. While worksheets like this one provide excellent practice, students often need personalized guidance when stuck on specific problems. CBSETUTOR.ai offers an AI-powered tutor available 24×7 to help CBSE Class 11 students with Mathematics and all other subjects. Simply take a photo of any problem from this worksheet or your NCERT textbook and upload it to receive step-by-step solutions instantly. The platform explains not just the 'what' but the 'why' behind each step, helping you build conceptual clarity. Whether you are struggling with octant identification, section formula applications, or complex distance calculations, the AI tutor adapts to your learning pace. All of this is available at a flat ₹999 per month for Classes 6-12, covering every subject with unlimited queries. Start with a 3-day free trial to experience how interactive, on-demand help can transform your preparation. Especially valuable for Chapter 11, where spatial reasoning can be challenging without immediate feedback, CBSETUTOR.ai acts as your personal Mathematics coach, available anytime you need it.

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Frequently asked questions

What is the difficulty level and suggested time for this CBSE Class 11 Mathematics Chapter 11 worksheet?+

The worksheet is Moderate to Advanced difficulty, suitable for comprehensive exam preparation. The suggested time is 90 minutes under exam-like conditions, though initial attempts may take longer as you familiarize yourself with 3D coordinate problems.

Which topics from NCERT Class 11 Mathematics Chapter 11 are covered in this worksheet?+

This worksheet covers all major topics: coordinate systems in space, identification of octants, distance formula in three dimensions, section formula (internal and external division), midpoint formula, and coordinates of points on axes and planes.

How should I use the answer key provided with this worksheet?+

Attempt the entire worksheet independently first without referring to solutions. After completion, check your answers against the detailed answer key, award marks honestly, and identify weak areas. Revisit those concepts in your NCERT textbook before reattempting similar questions.

What is the sign convention for octants in three-dimensional geometry?+

Octant I: (+,+,+), II: (−,+,+), III: (−,−,+), IV: (+,−,+), V: (+,+,−), VI: (−,+,−), VII: (−,−,−), VIII: (+,−,−). Remember these sign patterns as they are frequently tested in CBSE Class 11 Mathematics exams.

How is the distance formula different in 3D compared to 2D coordinate geometry?+

The 3D distance formula includes the z-coordinate: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. This is an extension of the 2D formula with an additional term for the third dimension, following the Pythagorean theorem in three dimensions.

What is the section formula for internal division in three dimensions?+

If point P divides the line joining A(x₁,y₁,z₁) and B(x₂,y₂,z₂) internally in ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)). Each coordinate is calculated independently using the same ratio.

How can I identify which coordinate plane a point lies on?+

A point lies on the xy-plane if z=0, on the yz-plane if x=0, and on the zx-plane if y=0. A point on the x-axis has y=0 and z=0; similarly for other axes. Remember: the plane name tells you which coordinates are non-zero.

Are case-study questions important for CBSE Class 11 Mathematics term exams?+

Yes, case-study questions carry 4-5 marks and are a regular feature in recent CBSE exam patterns. They test your ability to apply mathematical concepts to real-world scenarios. Practice these thoroughly as they are scoring opportunities if you read carefully and extract data accurately.

What are common mistakes students make in Introduction to Three Dimensional Geometry?+

Common errors include: confusing octant sign conventions, forgetting to include the z-term in distance formula, mixing up which coordinate is zero on which plane, calculation errors in section formula, and not showing sufficient working in exam answers.

How does CBSETUTOR.ai help with Class 11 Mathematics Chapter 11 doubts?+

CBSETUTOR.ai provides 24×7 AI tutoring where you can upload photos of any 3D Geometry problem and receive instant step-by-step solutions with explanations. It covers all CBSE Class 11 Mathematics topics at ₹999/month for Classes 6-12, with a 3-day free trial available to start learning interactively.

Related resources

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