NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry (Ex 11.1) Exercise 11.1

The Central Board of Secondary Education (CBSE) serves both public and private schools in New Delhi. Having been regulated by the Government of India, CBSE is governed and supervised by it. Founded in 1921, this board educated students from Rajputana, Central India, and Gwalior under the name of the Uttar Pradesh Board of High School and Intermediate Education. Ajmer, Merwara, Central India, and Gwalior were all included in the Board of High School and Intermediate Education, which was established by the Indian government in 1929. Later, only the cities of Ajmer, Bhopal, and Vindhya Pradesh were included in the program. A new name was given to the Board in 1952: the Central Board of Secondary Education.

The typical promotion criteria for students in Class 12 is a 33% overall average, as well as a 33% average in both theoretical and practical examinations (if applicable). It may be possible for students with failing grades in one subject to write the compartment for that subject. If a student fails any compartment or two or more subjects, the entire course must be retaken the following year.

NCERT is governed by the CBSE Board. In 1961, the Government of India established the National Council of Educational Research and Training (NCERT) to provide assistance and guidance to the State and Central governments. To improve the quality of school education, the organisation develops policies and programs. Research on school education is the primary responsibility of NCERT, which includes conducting, promoting, and coordinating research. In addition, the NCERT collaborates with international organizations, visits foreign delegations, and provides training to educators in developing countries. A variety of educational materials, such as model textbooks, supplementary materials, newsletters, journals, educational kits, and multimedia digital products, are produced by the National Council for Educational Research and Training (NCERT).

As a fundamental component of human intellect and logic, Mathematics plays a vital role in understanding the world. In addition to promoting logical reasoning and mental clarity, Mathematics contributes to the development of mental discipline. Furthermore, understanding the topics when studying for the Class 12 board exams requires an understanding of Mathematics. The NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 are frequently used to simplify Mathematical concepts in Class 12 Maths. Mathematical concepts and terminologies are required for careers in Science, Engineering, Commerce, Astronomy, and Economics.

In Chapter 11, Three Dimensional Geometry, students will apply the knowledge they gained in previous chapters regarding Vectors. In addition to the direction Cosines and Ratios of a Line Connecting Two Points, the Equation of Lines and Planes in space under different conditions are also included. Students in Class 12 Mathematics are advised to practise topics such as the Equation of a Line in Space, the angle between Two Lines, and the shortest distance between two lines. To prepare for the Class 12 Mathematics Board Examinations, students should improve their conceptual understanding.

To practise and verify their answers, students should take assistance from the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. By reviewing the solutions, students will be able to identify where they made mistakes and improve in those areas. In Exercise 11.1 Class 12, students are required to answer 15+ numerical questions to assist them in clarifying their understanding of the topics. By practising additional questions, students can gain a deeper understanding of the concepts.

Extramarks provides students with a variety of study materials that can be used to help them understand the concepts presented in this chapter. Throughout the study materials, including the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, the key concepts outlined in the CBSE syllabus are comprehensively reviewed. Many qualified teachers prepare the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 to assist students in preparing for the standardised CBSE board examinations. Students may also use the NCERT Solutions for Class 12 Mathematics to prepare for college entrance exams such as the JEE, VITEEE, and other examinations. For students to fully understand the questions in the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, it is recommended that they practise the exercise at least twice.

As a resource for preparing for the Class 12 board exams, NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 are developed to help students prepare as efficiently as possible for their exams. The NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 contains detailed explanations and examples that are relevant and easy to understand. Each topic ends with a set of practise questions that will assist the students in improving their performance in Chapter 11. The NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 can be used as a quick and effective study aid during exam preparation. It is the objective of Exercise 11.1 Class 12 Maths NCERT Solutions to provide students with problem-solving solutions to assess their conceptual understanding.

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry (Ex 11.1) Exercise 11.1

To prepare for their Class 12 board examinations, students should download the PDF version of the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 from the Extramarks’ website. As students may not always have access to the internet, the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 can be downloaded as a PDF file and used at any time and anywhere.

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Access NCERT Solutions for Class 12 Chapter 11 – Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1

Students need to have access to a sufficient number of questions before the examination. To prepare well for the exams, students often require the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. It is recommended that the students download and review the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. Through this, the students can comprehend and learn the study material, which results in a better performance in the examination. Additionally, this allows students to devote more time to other aspects of their learning while remaining calm. As a result of understanding the principles of the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, it is possible for students to progress to the next level of their studies. Although the concepts presented in this chapter may seem confusing and difficult at times, the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 is an excellent study resource. These learning resources can be used by students to improve their understanding of Mathematics.

Students can learn the concepts clearly and efficiently with Extramarks’ NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 created by skilled specialists. In the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, textbook problems are explained in detail. Students can go through the revision notes, sample papers with solutions, past years’ papers, and worksheets that are easily available on the Extramarks’ website for students in Class 12. Furthermore, Extramarks offers solutions for all classes, including NCERT Solutions Class 11, NCERT Solutions Class 10, and NCERT Solutions Class 9.

Besides the NCERT Solutions Class 12, students in secondary schools can also access the NCERT Solutions Class 8, NCERT Solutions Class 7, NCERT Solutions Class 6, and many others through Extramarks.

In preparation for the Class 12 Mathematics board exams, students should refer to the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 for guidance. The NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 are created following the revised CBSE syllabus and guidelines. To prepare for board exams and competitive examinations, NCERT Solutions are frequently recommended to students.Since they have been developed by highly qualified professionals, these solutions are relevant, accurate, and comprehensive. Aside from the answer key, several unanswered questions are included for the student to practise and revise.

In addition to long and short answers, NCERT also includes questions with multiple choice answers (MCQs). By providing topic- and chapter-specific additional problems, the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 enables students to test their conceptual knowledge and apply it to practical situations.

Important of Class 12 Maths Exercise 11.1

Going through the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 while studying for Chapter 11 of Class 12 Mathematics can help the students with their exam preparation. In light of the popularity of NCERT textbooks in Class 12, the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 are in high demand. For preparation for the Class 12 board exams, students should review the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. Students can learn more about the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 through Extramarks.

Understanding the process used to achieve the solution is more important than simply verifying the answer. To gain a better understanding of the procedure and steps involved, students can review NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. By answering board examination questions in this manner, students can perform well on the exam.

In Ex 11.1 Class 12 of the NCERT Solutions Class 12, all topics mentioned in the CBSE Class 12 Mathematics Syllabus are covered. To facilitate a better understanding of the concepts, all major theorems and formulas are explained in detail in the book.

With the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, students can prepare for college entrance exams such as JEE Mains, BITSAT, VITEEE, and other aptitude-based exams.

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Q.1 If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

Ans

Let the direction cosines of the line be  l, m and n. Then,  l= cos90°=0,  m=cos 135°=−12,n=cos 45°=12Therefore, the direction cosines of the line are 0,−12 and  12.

Q.2 Find the direction cosines of a line which makes equal angles with the coordinate axes.

Ans

Let the direction cosines of the line, which makes equal angle α with each of the coordinate axes be  l, m and n. Then,  l= cosα,  m=cos α,n=cos αSince,  l2+m2+n2=1⇒cos2α+cos2α+cos2α=1⇒3cos2α=1⇒    cos2α=13⇒     cosα=±13Therefore, the direction cosines of the line, which is equally inclined to the coordinate axes, are ±13,±13,±13.

Q.3 If a line has the direction ratios –18, 12, –4 then what are its direction cosines?

Ans

If a line has the direction ratios –18, 12, –4, then directioncosines are:l=−18(−18)2+(12)2+(−4)2  =−18324+144+16  =−1822  =−911m=12(−18)2+(12)2+(−4)2  =12324+144+16  =1222  =611n=−4(−18)2+(12)2+(−4)2  =−4324+144+16  =−422  =−211Thus, the direction cosines are −911,611,−211.

Q.4 Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

Ans

The given points are: A(2, 3, 4), B(–1, –2, 1) and C(5, 8, 7).
Direction ratios of line joining A (2, 3, 4) and B (–1, –2, 1) are:
–1–2, –2 – 3, 1 – 4 i.e., –3, –5, –3.
Direction ratios of line joining B (–1, –2, 1) and C (5, 8, 7) are:
5–(–1), 8 –(–2), 7 –1 i.e., 6, 10, 6.
It can be seen that the direction ratios of BC are –2 times of that of AB.
Therefore, AB is parallel to BC. Since, B is common to both AB and BC, so, points A, B and C are collinear.

Q.5 Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (–1, 1, 2) and (–5, –5, –2).

Ans

Let the vertices of triangle ABC be A 3,5,−4, B−1,1,2 and C−5,−5,−2. The direction ratios of side AB​ are:−1−3,  1−5,  2−−4  i.e.,−4,−4,6.So, the direction cosines of AB are:  −4−42+−42+62,−4−42+−42+62,6−42+−42+62i.e.,  −4217,−4217,6217⇒−217,−217,317The direction ratios of side BC are:−5−−1,  −5−1, −2−2  i.e.,−4,− 6,− 4.So, the direction cosines of AB are:  − 4−42+−42+62,− 6−42+−42+62,− 4−42+−42+62i.e.,  − 4217,− 6217,− 4217⇒−217,−317,−217The direction ratios of side AC are:−5−3,  −5−5, − 2−−4  i.e.,−8,  −10,  2.So, the direction cosines of AB are:  −8−82+−102+22,−10−82+−102+22,2−82+−102+22i.e.,  − 8242,−10242,2242⇒ − 442,−542,142

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FAQs (Frequently Asked Questions)

1. Are NCERT Solutions Class 12 effective at clarifying concepts for students in Class 12?

When students have completed their preparation, they can use the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 to practice questions and score well on the Mathematics examination. It is possible for students to identify their mistakes and improve by utilizing the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1, which clarifies their concepts. Moreover, Extramarks provides access to study materials and other resources that can assist students in preparing for the examination.

2. When it comes to the NCERT Class 12 Mathematics examination, how can a student perform well?

Students must be able to comprehend the concepts and terminologies to succeed in the Class 12 board examinations. Students should solve Class 12 Mathematics past years’ papers to be successful. The learning process becomes more engaging and motivating when students have a clear understanding of the concepts. It is recommended that students practice the extra questions after mastering the concepts for Class 12 Mathematics. Each student should solve these papers daily so that they can determine where they need more practice and where they need to improve. Students need to devise an effective study plan and adhere to it strictly to achieve good grades.

3. Does Extramarks provide NCERT solutions for students in primary schools?

There are times when students in primary school have difficulty answering questions from NCERT textbooks. As they may not always have access to teachers and parents, they can refer to the NCERT solutions on the Extramarks website. Extramarks offers NCERT Solutions Class 5 and NCERT Solutions Class 4 on its website and mobile application. Additionally, students in Classes 1, 2, and 3 can prepare for their exams with NCERT Solutions Class 3, NCERT Solutions Class 2, and NCERT Solutions Class 1.

 

4. Is it possible to find NCERT Solutions for Class 12 Mathematics Chapter 11 online?

On the Extramarks’ website, the students can find the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1. Answering textbook questions provides a foundation for understanding the topics covered in the chapter. It is also important to note that the questions constitute a valuable source of learning materials for students. Students can access the NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 on the Extramarks’ website, and they have been prepared by subject experts to assist them with completing their curriculum on time. Consequently, students can focus more effectively on learning concepts, which results in a stronger academic foundation.

 

5. How does Extramarks help students prepare for Class 12 board examinations?

Students can access the curriculum-mapped learning modules for CBSE and other major boards on the Extramarks’ website. As well as preparing for competitive exams such as JEE and NEET, this resource can also be used by students to prepare for board exams. A wide variety of NCERT Solutions are available from Extramarks, including NCERT Solutions Class 12. To prepare for the exam, students can access various study materials through the online platform. A collection of past year’s papers, sample papers, assignments, worksheets, and doubt-solving groups can be used as study materials. Extramarks’ Learning App allows students to bridge the gap between their strong and weak areas. They are thus capable of solving the Class 12 exam with confidence.