The Three Dimensional Geometry Class 12 NCERT Solutions page compiles NCERT Class 12 Mathematics Chapter 11 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
- CBSE Weightage: 2 to 3 marks from Chapter 11, typically a 1-mark VSA on direction cosines and a 2-mark question built on direction ratios through two points.
- Exercise size: 5 solved problems, the shortest exercise in the Three Dimensional Geometry Class 12 NCERT Solutions, fully clearable in one short study sitting.
Student Pulse - Three Dimensional Geometry Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Every solved problem in the Three Dimensional Geometry Class 12 NCERT Solutions opens with the controlling relation, either l = cosα, m = cosβ, n = cosγ or the proportionality l : m :
n = a : b : c , substitutes the numbers, then confirms with l2 + m2 + n2 = 1 . The ± sign on direction cosines is written out every time so the second sense of the line is never dropped.
The Collegedunia editorial team has checked each answer against the official NCERT key and the current 2026-27 textbook. The normalisation template in Q4 and the collinearity-by-proportional-ratios test in Q3 are the two highest-yield patterns in this exercise.
How Collegedunia's NCERT Solutions Help You Clear Exercise 11.1
The Three Dimensional Geometry Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 11.1 looks light, but it is where sign errors and the direction-ratio versus direction-cosine confusion are born. Every Collegedunia solution states the formula in words, substitutes, then runs the unit-length check, so the reasoning is examination-ready rather than just the final triple.
- Formula stated first: each answer opens with l=cosα or the normalising factor 1/√a2+b2+c2 , so markers see the method.
- Sign discipline: cos 135∘ = -1/√2 is flagged in Q1 to close the second-quadrant trap early.
- Unit-length verification via l2+m2+n2=1 at the end of every direction-cosine problem.
- Collinearity argument in Q3 shows why a shared point upgrades parallel direction ratios to a single line.
Three Dimensional Geometry Ex 11 1 Video Walkthrough
Source: Magnet Brains on YouTube
Three Dimensional Geometry Class 12 NCERT Solutions Exercise 11.1: Question-Wise Answer Map
The five problems split across three reusable skills. The table pairs each question with the relation it needs and the final answer, so the Three Dimensional Geometry Class 12 NCERT Solutions can be used as a self-check grid after a first attempt.
| Q No. | Problem | Core relation | Answer |
|---|---|---|---|
| 1 | Line makes 90∘, 135∘, 45∘ with axes | l=cosα | (0, -1√2, 1√2) |
| 2 | Line equally inclined to all three axes | 3l2=1 | ±(1√3,1√3,1√3) |
| 3 | Show (2,3,4),(-1,-2,1),(5,8,7) collinear | AB⃗∥BC⃗ | Collinear, BC⃗=-2 AB⃗ |
| 4 | DCs from DRs (-18,12,-4) | Normalise by √a2+b2+c2 | (-911,611,-211) |
| 5 | DCs of sides of triangle (3,5,-4),(-1,1,2),(-5,-5,-2) | Two-point DC formula | Three DC triples (isosceles, AB=BC ) |
Q1 and Q2 use the direct cosine relation, Q4 uses normalisation, and Q3 and Q5 use the two-point template. Q5 quietly reveals an isosceles triangle since AB=BC=2√17 , a fact CBSE often asks about in a follow-up part.
Direction Cosines and Direction Ratios Explained for Class 12 Maths Chapter 11
Fix the two definitions before opening the Three Dimensional Geometry Class 12 NCERT Solutions. A line in space makes angles α, β, γ with the positive x, y, z axes.
The cosines of these angles are the direction cosines (l, m, n) . Any triple (a, b, c) proportional to (l, m, n) is a set of direction ratios for the same line.
Core identity: l2 + m2 + n2 = 1 . This holds only for direction cosines, never for direction ratios. To convert DRs (a,b,c) to DCs, use l = ± a / √a2+b2+c2 , and similarly for m and n .
For a segment joining P(x1, y1, z1) and Q(x2, y2, z2) , the direction ratios are simply (x2-x1, y2-y1, z2-z1) . This is why direction ratios are read off a problem statement with no square roots until the final normalisation step.
Step-by-Step Method Used in NCERT Solutions for Class 12 Maths Exercise 11.1
Every direction-cosine problem in the Three Dimensional Geometry Class 12 NCERT Solutions runs the same three-line routine. Internalising it clears any Exercise 11.1 sum within a minute.
- Read the data: identify whether the question gives angles, direction ratios, or two points.
- Apply one relation: l=cosα for angles, l=± a/√a2+b2+c2 for direction ratios, or the two-point formula for points.
- Verify: substitute into l2+m2+n2=1 . If it does not return 1, an arithmetic slip has occurred.
The verification step is what separates a board-ready answer from a lucky one. CBSE markers reward the explicit unit-length check, and it doubles as a free error detector during the exam.
Concept Tags Across the Five Problems of Class 12 Maths Chapter 11 Exercise 11.1
The Three Dimensional Geometry Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Recognising the tag at sight saves about forty seconds per problem in the board exam.
- Direction-angle to DC conversion (Q1, Q2): apply l=cosα , with care on sign when an angle exceeds 90∘ .
- Direction-ratio normalisation (Q4): divide (a,b,c) by √a2+b2+c2 ; both signs are valid because a line has two senses.
- Two-point DCs and collinearity (Q3, Q5): build DRs from coordinate differences; three points are collinear iff DRs of two of the segments are proportional.
Common Mistakes Students Make in Class 12 Maths Chapter 11 Exercise 11.1
The Three Dimensional Geometry Class 12 NCERT Solutions are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Dropping the ± sign when normalising direction ratios. Both senses of the line are valid unless one direction is fixed by the problem.
- Writing cos 135∘ = +1/√2 . The correct value is -1/√2 since 135∘ is in the second quadrant.
- Declaring three points collinear from one pair of proportional ratios without naming the shared point.
- Confusing order of subtraction in the two-point formula, which flips the sign of every component.
Class 12 Maths Chapter 11 Three Dimensional Geometry: All Exercises
Exercise 11.1 is the foundation; the table below links the rest of the Three Dimensional Geometry Class 12 NCERT Solutions so you can move on once the direction-cosine drill is done.
| Exercise | Topic | Questions |
|---|---|---|
| Exercise 11.2 | Line equations, angle between lines, shortest distance | 15 |
| Miscellaneous Exercise | Mixed problems across all chapter topics | 5 |
Related Resources for Class 12 Maths Chapter 11 Three Dimensional Geometry
The Three Dimensional Geometry Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
- Formula Sheet for Class 12 Maths Chapter 11 Three Dimensional Geometry
- NCERT Notes for Class 12 Maths Chapter 11 Three Dimensional Geometry
- NCERT Book PDF for Class 12 Maths Chapter 11 Three Dimensional Geometry
- NCERT Exemplar Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry
Three Dimensional Geometry Class 12 NCERT Solutions: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Three Dimensional Geometry Chapter
The Three Dimensional Geometry chapter splits into 2 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 11.1 | Direction cosines, direction ratios of a line |
| Exercise 11.2 | Vector and Cartesian equations of a line in 3D |
| Miscellaneous Exercise | Mixed three-dimensional geometry problems |
PDF Download Formats and Languages for the Three Dimensional Geometry Chapter
The Three Dimensional Geometry Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The three dimensional geometry class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the three dimensional geometry class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The three dimensional geometry class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The three dimensional geometry class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The three dimensional geometry class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this three dimensional geometry class 12 pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the three dimensional geometry class 12 formulas pdf), and the full three dimensional geometry class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Three Dimensional Geometry Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Three Dimensional Geometry chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable three dimensional geometry class 12 important questions you will see on board day.
- three dimensional geometry class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The three dimensional geometry class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching three dimensional geometry class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Three Dimensional Geometry chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full three dimensional geometry class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Three Dimensional Geometry Notes Pair with NCERT Solutions and the Formula Sheet
The Three Dimensional Geometry Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Three Dimensional Geometry Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| three dimensional geometry class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| three dimensional geometry class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The three dimensional geometry class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The three dimensional geometry class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The three dimensional geometry class 12 formulas reference sheet is the same A4 file students sometimes refer to as three dimensional geometry class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board three dimensional geometry class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 three dimensional geometry solutions, class 12 three dimensional geometry ncert solutions, ncert class 12 three dimensional geometry solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Three Dimensional Geometry Class 12 |
|---|---|
| RD Sharma Class 12 Three Dimensional Geometry | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Three Dimensional Geometry | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo three dimensional geometry class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa three dimensional geometry class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board three dimensional geometry class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Three Dimensional Geometry | Advanced problems for JEE Main/JEE Advanced preparation. |
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
All NCERT Solutions for Three Dimensional Geometry Ex 11.1 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 11 Three Dimensional Geometry Ex 11.1 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
If a line makes angles 90∘, 135∘, 45∘ with the x, y and z-axes respectively, find its direction cosines.
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If a line has the direction ratios -18, 12, -4, then what are its direction cosines?
Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
How to Use the Three Dimensional Geometry Notes Page Most Effectively
The recommended study plan for the Three Dimensional Geometry Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the three dimensional geometry class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The three dimensional geometry class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Three Dimensional Geometry Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 11 Exercise 11.1?
Ans. Five questions. Q1 and Q2 cover direction-angle to direction-cosine conversion, Q3 tests collinearity of three points through proportional direction ratios, Q4 normalises given direction ratios into direction cosines, and Q5 finds the direction cosines of the three sides of a triangle in space.
Ques. What is the difference between direction cosines and direction ratios in Class 12 Exercise 11.1?
Ans. Direction cosines (l, m, n) are the cosines of the angles a line makes with the positive x, y, z axes and satisfy l2+m2+n2=1 . Direction ratios (a, b, c) are any three numbers proportional to the direction cosines. Direction cosines are unique up to sign; direction ratios are unique only up to a non-zero scalar.
Ques. Why do direction cosines carry a ± sign?
Ans. A line in space has two opposite senses. Reversing the sense flips all three direction cosines together, so both sign choices describe the same line and CBSE accepts either form unless one direction is fixed by the problem.
Ques. How do I prove three points collinear in Q3 of Exercise 11.1?
Ans. Compute the direction ratios of AB⃗ and BC⃗ . If a1/a2=b1/b2=c1/c2 , the segments are parallel, and because they share the point B , the three points lie on one line. Here BC⃗=-2 AB⃗ , so the points are collinear.
Ques. Is Class 12 Maths Chapter 11 Exercise 11.1 important for JEE Main 2026?
Ans. Yes. JEE Main usually carries one question per shift on direction cosines, direction ratios, or the identity l2+m2+n2=1 . Exercise 11.1 is the most efficient drill for that pattern and also feeds the line and shortest-distance problems later in the paper.
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