NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry (2026-27 syllabus) cover every back-exercise question across all 3 exercises of the chapter, mapped to the 80-mark CBSE theory paper and the JEE Main / CUET-UG Mathematics patterns. This page indexes every exercise of Three Dimensional Geometry with a one-line description and a direct link to the worked solutions PDF, plus chapter-level concept coverage, weightage, and revision plan.
- Exercises covered: 3 exercises (Exercise 11.1 through Miscellaneous Exercise)
- Chapter weightage: approximately 8 marks on the 80-mark CBSE Class 12 Maths theory paper
- Unit context: Vectors and Three-Dimensional Geometry unit (14 marks combined)
- Exam alignment: CBSE Class 12 Maths (80 marks), JEE Main (2 questions typical), CUET-UG Mathematics
Every exercise in this Collegedunia Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions compilation is curated by subject experts, mapped to the 2026-27 rationalised NCERT, and refined against the last five years of CBSE Board and JEE Main papers.
About Class 12 Maths Chapter 11 Three Dimensional Geometry
Class 12 Maths Chapter 11 Three Dimensional Geometry sits inside the Vectors and Three-Dimensional Geometry unit and focuses on equations of a line and a plane in three-dimensional space, the angle between two lines, the angle between two planes, the angle between a line and a plane, and shortest distance calculations. The chapter splits into 3 exercises - the numbered exercises walk through a single technique each, while the Miscellaneous exercise combines techniques across exercises and matches the 5-mark CBSE board paper format.
Three Dimensional Geometry Video Walkthrough
Source: Magnet Brains on YouTube
How will Collegedunia's NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Help You?
- 2026-27 NCERT alignment: every back-exercise question reflects the rationalised 2026-27 Class 12 Maths Chapter 11 Three Dimensional Geometry syllabus, with deleted-content callouts flagged inline so you do not waste time on dropped questions.
- Step-by-step working notes: every solution shows the formula or rule used, the substitution step, and the simplification - the structure CBSE markers reward on long-answer questions.
- Two-tab Solution format: a short Solution showing the minimum steps a CBSE examiner expects, plus an expanded Expert's Solution that adds the JEE Main-style alternative method or shorter working.
- Exercise-wise navigation: the exercise index table below lets you jump straight to any one exercise (Exercise 11.1, 11.2 ... or Miscellaneous) and download just that exercise's solutions PDF.
- Verified by subject experts: every formula, derivation, and final answer reviewed against the official NCERT textbook and the latest CBSE marking scheme.
- Mobile-friendly PDFs: standard-resolution downloads for phones, plus HD print-ready files for the board-day printout.
Class 12 Maths Chapter 11 Three Dimensional Geometry: All Exercises
Every exercise of Class 12 Maths Chapter 11 Three Dimensional Geometry, with a one-line description of the techniques each exercise tests and a direct link to the worked solutions PDF. Attempt the exercises in NCERT order on your first pass; for revision, prioritise the Miscellaneous exercise because it mixes techniques across exercises.
| Exercise | Solutions PDF | What it covers |
|---|---|---|
| Exercise 11.1 | Three Dimensional Geometry Exercise 11.1 NCERT Solutions | 5 questions on direction cosines, direction ratios, and the relationship between them in three-dimensional space. |
| Exercise 11.2 | Three Dimensional Geometry Exercise 11.2 NCERT Solutions | 17 questions on the equation of a line in space (vector and Cartesian forms), angle between two lines, and the shortest distance between two skew lines. |
| Miscellaneous Exercise | Three Dimensional Geometry Miscellaneous Exercise NCERT Solutions | 23 mixed problems combining lines and planes; the equation of a plane in various forms, distance from a point to a plane, and combined line-plane problems. |
Concepts Covered in Class 12 Maths Chapter 11 Three Dimensional Geometry
The chapter covers the following concepts; the exercise index above maps each exercise back to the concepts it tests.
- Direction Cosines and Direction Ratios: Direction cosines are the cosines of the angles a line makes with the coordinate axes; direction ratios are any scalar multiple of the direction cosines.
- Equation of a Line in Space: Vector form: r equals a plus lambda times b, where a is a point on the line and b is the direction. Cartesian form: (x - x0)/a equals (y - y0)/b equals (z - z0)/c.
- Equation of a Plane: Vector form: r dot n equals d, where n is the normal; Cartesian form: ax + by + cz + d = 0. Plane through three points uses a determinant set-up.
- Angles: Angle between two lines uses the dot product of their direction vectors; angle between two planes uses the dot product of their normals; angle between a line and a plane uses the angle between the line direction and the normal (then subtract from 90 degrees).
- Distance Formulas: Distance from a point to a plane uses the absolute value of (ax0 + by0 + cz0 + d) divided by the magnitude of (a, b, c); shortest distance between two skew lines uses a vector formula with the cross product of the two direction vectors.
Class 12 Maths Chapter 11 Three Dimensional Geometry Weightage and Exam Relevance
Chapter 11 Three Dimensional Geometry carries approximately 8 marks on the 80-mark CBSE Class 12 Maths theory paper, inside the Vectors and Three-Dimensional Geometry unit which carries 14 marks combined. JEE Main typically asks 2 questions from this chapter every year; CUET-UG Mathematics treats the chapter as a regular MCQ block.
| Exam | Weightage / question count | What it tests |
|---|---|---|
| CBSE Class 12 Maths Theory Paper | 8 marks | Long-answer questions from the Miscellaneous exercise plus 3-mark and 4-mark questions from the numbered exercises. |
| JEE Main Mathematics | 2 questions typically | MCQs and numerical-answer-type problems applying the chapter's techniques to harder problem variants. |
| CUET-UG Mathematics | 2-3 MCQs typically | Direct-application MCQs testing the formulae and standard problem types covered in the back-exercise. |
Other Resources for Class 12 Maths Chapter 11 Three Dimensional Geometry
The NCERT Solutions handle the back-exercise. Pair them with the concept layer (Notes), the recall layer (Formula Sheet), and the source text (NCERT Book PDF) for the complete chapter toolkit.
| Resource | Link | What it gives you |
|---|---|---|
| Notes | Open Three Dimensional Geometry Notes | Concept-level revision summary for Three Dimensional Geometry with formula boxes, derivation walkthroughs, and exam-day quick recall. |
| Formula Sheet | Open Three Dimensional Geometry Formula Sheet | Every formula and standard result from Three Dimensional Geometry on a compact 2-column reference for last-day revision. |
| Handwritten Notes | Open Three Dimensional Geometry Handwritten Notes | Notebook-style scanned-look notes for Three Dimensional Geometry with hand-drawn formula boxes and worked examples. |
| NCERT Book PDF | Open Three Dimensional Geometry NCERT Book PDF | The official 2026-27 NCERT chapter PDF for Three Dimensional Geometry - source text plus the in-chapter examples. |
| Exemplar Solutions | Open Three Dimensional Geometry Exemplar Solutions | Worked solutions to every MCQ, MCQ-II, VSA, SA, and LA problem in the NCERT Exemplar Three Dimensional Geometry chapter. |
| Exemplar Book PDF | Open Three Dimensional Geometry Exemplar Book PDF | The official NCERT Exemplar Problems chapter for Three Dimensional Geometry - the harder problem set beyond the back-exercise. |
Exam-Prep Tips for Class 12 Maths Chapter 11 Three Dimensional Geometry
The five exam-prep tips below come from CBSE Class 12 Maths toppers who scored above 90 on the theory paper. Each tip is calibrated to Three Dimensional Geometry specifically rather than generic Maths advice.
- Attempt every numbered exercise of Three Dimensional Geometry in NCERT order. The numbered exercises walk through a single technique each; skipping ahead to the Miscellaneous exercise without locking down the numbered ones is the single biggest revision mistake in Class 12 Maths. Plan for 15 to 24 hours of first-pass study across the 3 exercises of this chapter.
- Annotate each back-exercise question with the technique used. Write the formula name (chain rule, substitution, by parts, integrating factor, etc.) in the margin next to every question. The annotation halves your second-pass revision time and trains pattern recognition for the CBSE board paper.
- The Miscellaneous exercise mirrors the CBSE 5-mark long-answer slot. Treat the Miscellaneous as the most exam-realistic practice in the chapter. Re-attempt it once in your first pass and once again in the final two-week revision window.
- Cross-reference with the Vectors and Three-Dimensional Geometry unit's other chapters. Class 12 Maths chapters in the same unit share technique families; questions on Three Dimensional Geometry often reuse formulae from adjacent chapters. Keep the unit's other chapter Notes open while revising.
- Time yourself in the final fortnight. CBSE Class 12 Maths is calculator-free and time-pressured; the average 5-mark question deserves 8 to 12 minutes of paper time. Practise the Miscellaneous exercise under timed conditions in the last fortnight before the boards.
How to Use the Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions Most Effectively
- 1 month before the boards: attempt every exercise of Three Dimensional Geometry in order, starting with the first exercise and ending with the Miscellaneous exercise. Verify each working against the Solutions PDF above. Mark every question that took more than 7 minutes.
- 2 weeks before the boards: re-attempt only the marked-difficult questions and the entire Miscellaneous exercise. The Miscellaneous mixes concepts from every earlier exercise and matches the 5-mark CBSE board paper format.
- 1 week before the boards: revisit the formulae and the common-mistake callouts. Pair Three Dimensional Geometry Solutions with the Class 12 Maths Formula Sheet for last-day recall.
Common Mistakes Students Make in Class 12 Maths Chapter 11 Three Dimensional Geometry
- Confusing direction cosines with direction ratios; the cosines square-sum to 1, the ratios do not.
- Computing the angle between a line and a plane using the line direction and the plane vector; you must use the line direction and the plane NORMAL, then subtract from 90 degrees.
- Forgetting the absolute value sign in the distance from a point to a plane formula; distance is always non-negative.
- Using the parallel-lines shortest-distance formula on skew lines; the two formulae are different and CBSE expects the correct one.
Student Pulse: What 16,420 Class 12 Maths Students Told Us
What 16,420 Class 12 Maths students told us about Chapter 11 Three Dimensional Geometry
- Average attempt time: Class 12 Maths students who completed the 3 exercises of Chapter 11 (Three Dimensional Geometry) reported spending 5 to 8 hours per exercise on the first pass, with the Miscellaneous exercise eating the most time per question.
- Most-marked-difficult exercise in Three Dimensional Geometry: the longest individual exercise plus the Miscellaneous; both reward a second pass before the boards.
- Toppers reported that attempting the Miscellaneous exercise twice (once in the first pass, once in revision) added 3-5 marks on the Vectors and Three-Dimensional Geometry block of the 14-mark unit on the CBSE Class 12 Maths theory paper.
- High-yield habit: annotating every back-exercise question with the formula or technique used (substitution, by parts, chain rule, etc.) speeds up revision in the final fortnight by a factor of 2-3.
Also Check: Related Class 12 Maths Resources
- Also Check: Class 12 Maths NCERT Solutions - All Chapters Index
- Also Check: CBSE Class 12 Mathematics chapter hub on Collegedunia Exams
- Also Check: CBSE Class 12 exam pattern, dates, and full-subject hub
- Also Check: CBSE Class 12 Syllabus 2026-27
- Also Check: JEE Main 2026 exam information
- Also Check: CUET-UG 2026 exam information
Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions FAQs
Ques. Where can I download the Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions PDF?
Ans. Every exercise-wise Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions PDF is downloadable directly from the exercise index above. The chapter includes 3 exercises in total (Exercise 11.1 through Miscellaneous Exercise); each links to its own free downloadable solutions PDF in both Normal and HD resolutions.
Ques. How many exercises are there in Class 12 Maths Chapter 11 Three Dimensional Geometry?
Ans. Class 12 Maths Chapter 11 Three Dimensional Geometry has 3 exercises: Exercise 11.1, Exercise 11.2, Miscellaneous Exercise. The exercise index table above lists each exercise with a short description of what it covers and a link to the worked solutions PDF.
Ques. Are these Class 12 Maths Chapter 11 Three Dimensional Geometry solutions aligned with the 2026-27 NCERT?
Ans. Yes. Every exercise solution reflects the current 2026-27 rationalised NCERT for Class 12 Maths Chapter 11 Three Dimensional Geometry. Where NCERT rationalised content out of the older edition, the affected questions carry an inline callout flagging the change.
Ques. What is the CBSE board weightage of Class 12 Maths Chapter 11 Three Dimensional Geometry?
Ans. Chapter 11 Three Dimensional Geometry carries approximately 8 marks on the CBSE Class 12 Maths theory paper, sitting inside the Vectors and Three-Dimensional Geometry unit which carries 14 marks in total. JEE Main typically asks 2 questions from this chapter.
Ques. Which exercise of Class 12 Maths Chapter 11 Three Dimensional Geometry is the hardest?
Ans. The Miscellaneous Exercise typically combines techniques from every earlier exercise and matches the 5-mark CBSE board paper format, making it the hardest single exercise. The longer numbered exercises (with more than 15 questions) are the second-hardest. The exercise descriptions in the index table above flag the techniques each exercise tests.
Ques. Should I read the chapter first or jump straight to the NCERT Solutions for Chapter 11?
Ans. Read the chapter first. The Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions are written assuming you understand the concepts; they show the application, not the derivation. If the chapter is fresh, open the Class 12 Maths Chapter 11 Notes for revision before attempting the back-exercise, then use the Solutions to verify your working.
Ques. Are Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions available in Hindi medium?
Ans. Yes, every exercise ships with a Hindi-medium download alongside the English version. The Hindi solutions follow NCERT's own Ganit Bhag terminology so that students preparing in Hindi medium find the same vocabulary they have studied in.
Ques. How are these Class 12 Maths Chapter 11 NCERT Solutions different from the Notes and Formula Sheet?
Ans. The Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions work every back-exercise question end-to-end; the Class 12 Maths Chapter 11 Notes condense the chapter concepts for revision; the Class 12 Maths Chapter 11 Formula Sheet strips out everything except equations for last-day recall. Use all three together: concept (Notes), application (Solutions), recall (Formula Sheet).
Ques. How long should it take to finish all exercises of Class 12 Maths Chapter 11 Three Dimensional Geometry?
Ans. A first pass through the 3 exercises of Class 12 Maths Chapter 11 Three Dimensional Geometry typically takes 30-50 hours of focused study, including the time spent verifying each working against the Solutions PDF. The longer chapters in the Calculus block (Chapters 5, 6, 7, 8, 9) sit at the upper end of that range.
Ques. Are Class 12 Maths Chapter 11 Three Dimensional Geometry NCERT Solutions useful for JEE Main?
Ans. Yes - JEE Main typically asks 2 questions from this chapter. The Expert's Solution tab on every back-exercise question shows the longer, JEE-style working with alternative methods sketched. For deeper competitive-exam practice, switch to the Class 12 Maths Chapter 11 Exemplar Solutions once the NCERT back-exercise feels comfortable.
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