NCERT Textbook Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry (3D Geometry) is an essential Maths chapter for understanding the position and orientation of objects in a three-dimensional space. NCERT Exercise Solutions for Class 12 Math Book Chapter 11 provide a structured approach to solve problems related to planes, lines and their intersections in 3D space. Mathematical concepts like direction cosines, direction ratios and the Cartesian equation of a line are introduced here. NCERT Mathematics chapter’s solved examples are highly beneficial for grasping these concepts.

NCERT Textbook Solutions Class 12 Maths Chapter 11 Three Dimensional Geometry

Practicing the NCERT exercises and CBSE previous year questions ensures clarity of fundamental principles and enhances problem-solving skills. NCERT Textbook question bank and CBSE Board practice papers further help students prepare effectively for exams. NCERT Class 12th Mathematics chapter 11 emphasizes understanding through formulas and solving NCERT worksheet questions that strengthen conceptual clarity. Class 12 Maths Chapter 11 NCERT Solutions of Three dimensional geometry (3D) in Hindi and English Medium are given here for session 2024-25. As per the new syllabus and latest NCERT textbook, there are only 3 exercises (including miscellaneous) are there in chapter 11 of 12th Maths.

Importance of Solved Examples and Key Formulas in 3D Geometry Mathematics

NCERT Textbook solved examples in Chapter 11 of Class 12 Mathematics are designed to provide a step-by-step mathematical approach to solving math problems on lines and planes. These NCERT Book examples focus on important math concepts such as finding the angle between two lines or between a line and a plane. NCERT Class 12 Textbook Math Chapter 11 notes and revision materials emphasize the significance of practicing these examples to develop mathematical accuracy.

Important formulas like the shortest distance between two skew lines or the equation of a plane passing through three points are frequently covered in sample papers. To perform well in CBSE or state board exams, students can use practice worksheets, CBSE mock tests and NCERT assignments that focus on specific topics in 3D Geometry. Class 12th NCERT Mathematics Solutions ensure students are well-prepared with key concepts and essential problem-solving strategies.

Preparing for Class 12 Mathematics CBSE Exams with Chapter 11 Resources

To ace Chapter 11 of NCERT Class 12 Mathematics, students must utilize NCERT comprehensive resources for Maths, such as CBSE practice papers, NCERT Book assignments and Board MCQ. The video lectures and NCERT PDF solutions provide detailed explanations for complex problems, ensuring students understand the mathematical approach behind each solution. A thorough review of previous year questions and question banks enables students to recognize patterns in exam papers and focus on frequently asked questions.

NCERT Book Summary notes and CBSE revision math worksheets are ideal for last-minute preparation, reinforcing key formulas and concepts in Three Dimensional Geometry. Solving mock tests and sample papers allows students to assess their understanding and improve their speed and accuracy. With consistent practice and a focus on NCERT materials, students can confidently approach even the trickiest 3D Geometry problems in their exams.

Class: 12Mathematics
Chapter: 11 Three Dimensional Geometry
Number of Exercises:2 + Miscellaneous
Content:Textbook and Extra Questions
Mode of Content:PDF, Text and Online Videos
Session:CBSE 2024-25
Medium:Hindi and English Medium
Class 12 Maths Chapter 11 Solutions

Important Points in Class 12 Maths Chapter 11 Three Dimensional Geometry for Exams

Focus on key concepts like direction cosines, direction ratios and the Cartesian equations of lines and planes. Memorize important formulas such as the shortest distance between skew lines and angles between planes. Practice NCERT solved examples, question banks and previous year papers for a strong exam preparation strategy.

Day Topics to Cover Practice Activities
Day 1 Direction Cosines and Direction Ratios Read NCERT examples, solve Exercise 11.1
Day 2 Equation of a Line in 3D Solve problems from Exercise 11.2
Day 3 Equation of a Plane Practice NCERT solved examples
Day 4 Angle Between Two Planes and Lines Review Exercise 11.3
Day 5 Shortest Distance Between Skew Lines Solve NCERT question bank and mock tests
Day 6 Revision and Key Formula Review Solve previous year questions and sample papers

NCERT Solutions for Class 12 Maths Chapter 11

Class XII Mathematics chapter 11 Exercise 11.1, 11.2 and Miscellaneous exercises in Hindi and English Medium PDF format for new academic session. Study material, solutions, Sample question papers, test papers based on different levels, latest CBSE syllabus for the current academic year. Join the Discussion Forum and ask your doubts.

12th Maths Chapter 11 Solutions

NCERT Solutions for Class 12 Maths Chapter 11 are given below to download in PDF format free in Hindi and English Medium. Join the discussion forum to ask your doubts related to NIOS board and CBSE board. Download latest NCERT Books 2024-25 following the current CBSE Syllabus.

Class 12 Maths Chapter 11 Three Dimensional Geometry

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Important Questions on 12th Maths Chapter 11

What is the relation between the direction cosines of a line?

If l, m, n are the direction cosines of a line, then l² + m² + n² = 1

What is meant by Direction ratios of a line?

Direction ratios of a line are the numbers which are proportional to the direction cosines of the line.

What are Skew lines in 3-D geometry?

Skew lines are lines in the space which are neither parallel nor intersecting. They lie in the different planes.

What is the shortest distance between two skew lines?

The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines.

Find the Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n.

Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n is lx + my + nz = p.

Previous Years Important Questions

    1. Find the Cartesian and Vector equations of the line which passes through the point (-2, 4, -5) and parallel to the line given by (x+3)/3 = (y-4)/5 = (8-z)/-6. [CBSE Sample Paper 2017]
    2. If the vectors p = ai + j + k, q = i + bj + k and r = i + j + ck are co-planar, then for a, b, c ≠ 1, show that 1/(1-a) + 1/(1-b) + 1/(1-c) = 1. [CBSE Sample Paper 2017]
    3. A plane meets the coordinate axes in A, B and C such that the centroid of triangle ABC is the point (α, β, γ). Show that the equation of the plane is x/α + y/β + z/γ = 3. [CBSE Sample Paper 2017]
    4. Define skew lines. Using only vector approach, find the shortest distance between the following two skew lines: r = (8 + 3m)i – (9 + 16m)j + (10 + 7m)k and r = 15i + 29j + 5k + n(3i + 8j – 5k). [CBSE Sample Paper 2017]
    5. Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z. [Delhi 2017]
Questions from Board Papers

1. Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines (x – 8)/3 = (y + 19)/-16 = (z – 10)/7 and (x – 15)/3 = (y – 29)/8 = (z – 5)/-5. [Delhi 2017]
2. Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is 2i – 3j + 6k. [Delhi 2016]
3. Show that the vectors a, b, and c are co-planar if a + b + c and c + a are co-planar. [Delhi 2016]

4. Find the coordinate of the point P where the line through A(3, – 4, –5) and B (2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0). Also, find the ratio in which P divides the line segment AB. [Delhi 2016]
5. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Then find the distance of plane thus obtained from the point A(1, 3, 6). [Delhi 2015C]

What are the key topics covered in NCERT Class 12 Maths Chapter 11 Three Dimensional Geometry?

The key math topics in Chapter 11 include understanding the equations of a line and plane in three-dimensional space, direction cosines, direction ratios and angles between lines and planes. NCERT Class 12 Mathematics Chapter 11 solutions provide a detailed explanation of these concepts. Important formulas like the shortest distance between two skew lines and the Cartesian equation of a line are extensively discussed. Solved examples in the NCERT solutions help students master the application of these concepts. Practicing exercises, worksheets and question banks further enhances clarity and prepares students effectively for exams.

How can NCERT Solutions for Chapter 11 help in exam preparation?

NCERT Detailed Solutions for Class 12 Mathematics Chapter 11 are a vital resource for exam preparation as they provide a systematic approach to solving problems on Three Dimensional Geometry. NCERT Class 12th Mathematics solutions explain critical concepts, including the angle between a line and a plane and ensure students understand each step of the process. Mock tests, revision notes and boards previous year questions available in NCERT resources help students focus on important questions and key formulas. Solving sample papers, practice worksheets and assignments reinforces understanding and builds confidence for solving complex 3D Geometry problems during exams.

What are the best ways to practice Three Dimensional Geometry for Class 12 exams?

To practice 3D Geometry effectively, start with NCERT Class 12 Maths Chapter 11 solved examples to understand the step-by-step solutions to complex problems. Next, work on exercise questions and focus on key formulas like the equation of a plane passing through three points. Practice using question banks, MCQs and sample papers to strengthen conceptual clarity. Solving mock tests and reviewing previous year questions can help identify common patterns in exams. Revision notes and worksheets from NCERT resources are excellent tools for last-minute preparation, ensuring a solid grasp of the chapter’s important concepts.

Last Edited: January 25, 2025
Content Reviewed: January 25, 2025
Content Reviewer

Shikhar Tiwari

Having graduated from Electronics and Communication Engineering from AKTU – Noida, India, in 2021, working for Tiwari Academy as a content writer and reviewer. My main focus is to provide an easy to understand methods in all subjects specially mathematics and making study material with step by step explanation.