The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.2 solve every question of the second exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
Exercise 11.2 is built entirely on the distance formula in space. The exercise has 5 questions that use it to test triangles, collinearity and point sets:
- Finding the distance between two points in three dimensions.
- Proving points are collinear or form a given type of triangle.
- Finding the equation of a set of points from a distance condition.
Topics Covered in Exercise 11.2
This exercise gives one working tool: the distance between two points in space. Once students square the differences of the three coordinates and add them, most parts become direct. The later questions apply the same formula to shapes and to sets of points.
- Distance formula: the length between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).
- Collinearity test: three points are collinear when the largest distance equals the sum of the other two.
- Triangle types: checking equal sides for isosceles and the Pythagoras relation for right-angled.
- Parallelogram test: opposite sides equal in length.
- Locus of points: forming an equation from a distance condition, such as equal distances from two fixed points.
Important: in the locus questions, set the two distances equal, square both sides, and simplify. The squaring step removes the square roots and leaves a clean equation.
Question-wise Breakdown of Exercise 11.2
The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.2 solve all 5 questions in order. The table shows what each question asks. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | Find the distance between four given pairs of points | Distance formula |
| Q2 | Show that three given points are collinear | Collinearity test |
| Q3 | Verify isosceles, right-angled and parallelogram cases | Triangle and quadrilateral tests |
| Q4 | Find the equation of points equidistant from two given points | Locus from a distance condition |
| Q5 | Find the equation of points whose distances from A and B sum to 10 | Locus with a distance sum |
The first three questions are direct applications, while Q4 and Q5 introduce the idea of a locus, which returns in coordinate geometry later.
Key Formulas and Definitions for Exercise 11.2
Exercise 11.2 needs only the distance formula and a few tests that follow from it. Students should keep these results ready before starting the questions.
| Result | Formula / condition |
|---|---|
| Distance PQ | √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] |
| Distance from origin | √(x² + y² + z²) |
| Collinear points A, B, C | AB + BC = AC (largest distance equals the sum of the other two) |
| Isosceles triangle | any two sides equal in length |
| Right-angled triangle | sum of squares of two sides equals the square of the third |
| Parallelogram | both pairs of opposite sides equal in length |
Tip: to prove a right angle, compare the squares of the sides, not the sides themselves. That avoids the square roots and keeps the working short.
Common Mistakes in Exercise 11.2
Students lose easy marks in Exercise 11.2 on sign and squaring slips. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.2.
- Subtracting coordinates in the wrong order and then forgetting that the square fixes the sign anyway, but mismatching x with z.
- Leaving the answer under the root when a perfect square simplifies, for example writing √36 instead of 6.
- Testing collinearity by eye instead of checking that AB + BC = AC with actual distances.
- Forgetting to square both sides fully in the locus questions, which leaves stray square-root terms in the final equation.
Practice Questions for Exercise 11.2
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for Exercise 11.2.
Practice Card: Solved Practice Questions for Class 11 Maths Three Dimensional Geometry Exercise 11.2 — attempt each question, then check the worked solution.
Other Exercises of Chapter 11 Introduction to Three Dimensional Geometry
Chapter 11 has two exercises plus a Miscellaneous Exercise. Use the table to move between exercises once Exercise 11.2 is done.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 11.1 | Class 11 Maths Three Dimensional Geometry Exercise 11.1 Solutions |
| Miscellaneous Exercise | Class 11 Maths Three Dimensional Geometry Miscellaneous Exercise Solutions |
More Class 11 Maths Three Dimensional Geometry Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 11 are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Three Dimensional Geometry Notes |
| Handwritten Notes | Class 11 Maths Three Dimensional Geometry Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Three Dimensional Geometry NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | You are here |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 11,280 Class 11 students conducted before the 2026 exams, 61% of students said the locus questions (Q4 and Q5) of Exercise 11.2 took the most time, while the plain distance calculations in Q1 were rated the easiest.
FAQs on NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.2
Introduction to Three Dimensional Geometry NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 11 Exercise 11.2?
Ans. Exercise 11.2 has 5 questions. They cover finding the distance between pairs of points, proving collinearity, verifying isosceles, right-angled and parallelogram cases, and forming the equation of a set of points from a distance condition.
Ques. What is the distance formula in three dimensions?
Ans. The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. It extends the two dimensional distance formula by adding the square of the difference of the z-coordinates. Every question in Exercise 11.2 uses this result.
Ques. How do you prove three points are collinear in Exercise 11.2?
Ans. Find the three distances between the points. If the largest distance equals the sum of the other two, the points lie on a straight line and are collinear. Question 2 of Exercise 11.2 uses this test on the points (−2, 3, 5), (1, 2, 3) and (7, 0, −1).
Ques. How do you check a right-angled triangle in three dimensions?
Ans. Find the squares of the three side lengths using the distance formula. If the sum of the two smaller squares equals the largest square, the triangle has a right angle. Working with squares avoids the square roots and keeps Question 3 short.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 11 Exercise 11.2 free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.2 are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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