The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Miscellaneous Exercise solve every question of the mixed 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.
The Miscellaneous Exercise mixes the section formula, the centroid and the distance formula in one set. The exercise has 5 questions that combine every idea of the chapter:
- Using the section formula to divide a segment internally.
- Finding a midpoint, median or centroid from vertices.
- Combining the distance formula with a coordinate condition.
Topics Covered in the Miscellaneous Exercise
This exercise pulls together the whole chapter. Students apply the section formula to find a dividing point, use the centroid formula on a triangle, and blend the distance formula with a fixed-coordinate clue. Each question needs two ideas working together, so it is the best test of the chapter.
- Section formula (internal): the point dividing a segment in a given ratio m : n.
- Midpoint formula: the special case of the section formula with ratio 1 : 1.
- Centroid of a triangle: the average of the three vertices' coordinates.
- Median length: the distance from a vertex to the midpoint of the opposite side.
- Fourth vertex of a parallelogram: found using equal diagonals' midpoints.
Important: in Question 5 the known x-coordinate fixes the ratio in the section formula. Set up the ratio from the x-values first, then read the y and z coordinates.
Question-wise Breakdown of the Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Miscellaneous Exercise 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 fourth vertex of a parallelogram from three vertices | Midpoint of diagonals |
| Q2 | Find the lengths of the medians of a triangle | Midpoint + distance formula |
| Q3 | Find a, b, c when the origin is the centroid | Centroid formula |
| Q4 | Find a point on the y-axis at a given distance from P | Distance formula |
| Q5 | Find point R on PQ with x-coordinate 4 | Section formula |
Because every question links two concepts, this exercise is a strong revision of the entire chapter before moving on.
Key Formulas and Definitions for the Miscellaneous Exercise
The Miscellaneous Exercise draws on the section formula, the centroid and the distance formula together. Students should keep all of these ready before starting.
| Result | Formula |
|---|---|
| Section formula (internal, ratio m : n) | ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)) |
| Section formula (external, ratio m : n) | ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n), (mz₂−nz₁)/(m−n)) |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) |
| Centroid of a triangle | ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3) |
| Distance PQ | √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] |
Tip: the midpoint is just the section formula with ratio 1 : 1. Learning one formula gives students both results at once.
Common Mistakes in the Miscellaneous Exercise
Students lose easy marks in the Miscellaneous Exercise by mixing up the ratio order and the wrong point pairs. 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 Miscellaneous Exercise.
- Swapping m and n in the section formula. The ratio m : n means m goes with the second point's coordinates.
- Using the external formula when the point lies between P and Q, where the internal formula is needed.
- Taking the wrong midpoint for a median. Each median runs from a vertex to the midpoint of the opposite side.
- Dividing the centroid sum by 2 instead of 3. A triangle has three vertices, so divide each coordinate sum by 3.
Practice Questions for the Miscellaneous Exercise
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 the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Three Dimensional Geometry Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 11 Introduction to Three Dimensional Geometry
Chapter 11 has two exercises plus this Miscellaneous Exercise. Use the table to revisit the earlier exercises for more practice.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 11.1 | Class 11 Maths Three Dimensional Geometry Exercise 11.1 Solutions |
| Exercise 11.2 | Class 11 Maths Three Dimensional Geometry Exercise 11.2 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 12,150 Class 11 students conducted before the 2026 exams, 66% of students said setting up the ratio in the section formula (Question 5) was the trickiest part of the Miscellaneous Exercise, while the centroid question was rated the most direct.
FAQs on NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Miscellaneous Exercise
Introduction to Three Dimensional Geometry NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in the Class 11 Maths Chapter 11 Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 5 questions. They cover the fourth vertex of a parallelogram, the lengths of the medians of a triangle, finding a, b and c from a centroid, a point on the y-axis at a fixed distance, and a dividing point using the section formula.
Ques. What is the section formula in three dimensions?
Ans. A point dividing the segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m : n is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)). Question 5 of the Miscellaneous Exercise uses this to find point R on segment PQ.
Ques. What is the centroid formula for a triangle in space?
Ans. The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) is ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3). Question 3 sets this equal to the origin to find the unknowns a, b and c.
Ques. How do you find the fourth vertex of a parallelogram in Question 1?
Ans. In a parallelogram the two diagonals bisect each other, so they have the same midpoint. Set the midpoint of one diagonal equal to the midpoint of the other, then solve for the unknown vertex. This gives the fourth vertex D from the three known vertices A, B and C.
Ques. Are the NCERT Solutions for the Class 11 Maths Chapter 11 Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry Miscellaneous Exercise 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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