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

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

Edited By Ramraj Saini | Updated on Dec 04, 2023 09:06 AM IST | #CBSE Class 12th
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NCERT Solutions For Class 12 Maths Chapter 11 Exercise 11.1

NCERT Solutions for Exercise 11.1 Class 12 Maths Chapter 11 Three Dimensional Geometry are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for exercise 11.1 Class 12 Maths chapter 11 focus on the topic direction ratio and direction cosines of a line.aths moves aro Exercise 11.1 Class 12 Mund the topic enumerated in syllabus. After having a thorough look at the concepts and example questions, students can solve exercise 11.1 for good clarification of the concepts.

This Story also Contains
  1. NCERT Solutions For Class 12 Maths Chapter 11 Exercise 11.1
  2. Access NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
  3. Three Dimensional Geometry Class 12th Chapter 11 -Exercise: 11.1
  4. More About NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
  5. Benefits of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
  6. Key Features Of NCERT Solutions for Exercise 11.1 Class 12 Maths Chapter 11
  7. NCERT Solutions Subject Wise
  8. Subject Wise NCERT Exemplar Solutions

12th class Maths exercise 11.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

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Three Dimensional Geometry Class 12th Chapter 11 -Exercise: 11.1

Question:1 If a line makes angles 90^{\circ}, 135^{\circ},45^{\circ} with the x, y and z-axes respectively, find its direction cosines.

Answer:

Let the direction cosines of the line be l,m, and n.

So, we have

l = \cos90^{\circ}=0

m = \cos135^{\circ}=-\frac{1}{\sqrt2}

n= \cos45^{\circ}=\frac{1}{\sqrt2}

Therefore the direction cosines of the lines are 0,\ -\frac{1}{\sqrt2},and\ \ \frac{1}{\sqrt2} .

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

Answer:

If the line is making equal angle with the coordinate axes. Then,

Let the common angle made is \alpha with each coordinate axes.

Therefore, we can write;

l = \cos \alpha,\ m= \cos \alpha,and\ n= \cos \alpha

And as we know the relation; l^2+m^2+n^2 = 1

\Rightarrow \cos^2 \alpha +\cos^2 \alpha+\cos^2 \alpha = 1

\Rightarrow \cos^2 \alpha = \frac{1}{3}

or \cos \alpha =\pm \frac{1}{\sqrt3}

Thus the direction cosines of the line are \pm \frac{1}{\sqrt3},\ \pm \frac{1}{\sqrt3},and\ \pm \frac{1}{\sqrt3}

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

Answer:

GIven a line has direction ratios of -18, 12, – 4 then its direction cosines are;

Line having direction ratio -18 has direction cosine:

\frac{-18}{\sqrt{(-18)^2+(12)^2+(-4)^2}} = \frac{-18}{22} = \frac{-9}{11}

Line having direction ratio 12 has direction cosine:

\frac{12}{\sqrt{(-18)^2+(12)^2+(-4)^2}} = \frac{12}{22} =\frac{6}{11}

Line having direction ratio -4 has direction cosine:

\frac{12}{\sqrt{(-4)^2+(12)^2+(-4)^2}} = \frac{-4}{22} = \frac{-2}{11}

Thus, the direction cosines are \frac{-9}{11},\ \frac{6}{11},\ \frac{-2}{11} .

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

Answer:

We have the points, A (2, 3, 4),B (– 1, – 2, 1),C (5, 8, 7);

And as we can find the direction ratios of the line joining the points (x_{1},y_{1},z_{1}) \ and\ (x_{2},y_{2},z_{2}) is given by x_{2}-x_{1}, y_{2}-y_{1}, \ and\ z_{2}-z_{1}.

The direction ratios of AB are (-1-2), (-2-3),\ and\ (1-4) i.e., -3,\ -5,\ and\ -3

The direction ratios of BC are (5-(-1)), (8-(-2)),\ and\ (7-1) i.e., 6,\ 10,\ and\ 6 .

We can see that the direction ratios of AB and BC are proportional to each other and is -2 times.

\therefore AB is parallel to BC. and as point B is common to both AB and BC,

Hence the points A, B and C are collinear.

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

Answer:

Given vertices of the triangle \triangle ABC (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).

Finding each side direction ratios;

\Rightarrow Direction ratios of side AB are (-1-3), (1-5),\ and\ (2-(-4)) i.e.,

-4,-4,\ and\ 6.

Therefore its direction cosines values are;

\frac{-4}{\sqrt{(-4)^2+(-4)^2+(6)^2}},\ \frac{-4}{\sqrt{(-4)^2+(-4)^2+(6)^2}},\ \frac{6}{\sqrt{(-4)^2+(-4)^2+(6)^2}} or\ \frac{-4}{2\sqrt{17}},\frac{-4}{2\sqrt{17}},\frac{6}{2\sqrt{17}}\ or\ \frac{-2}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}

SImilarly for side BC;

\Rightarrow Direction ratios of side BC are (-5-(-1)), (-5-1),\ and\ (-2-2) i.e.,

-4,-6,\ and\ -4.

Therefore its direction cosines values are;

\frac{-4}{\sqrt{(-4)^2+(-6)^2+(-4)^2}},\ \frac{-6}{\sqrt{(-4)^2+(-6)^2+(-4)^2}},\ \frac{-4}{\sqrt{(-4)^2+(-6)^2+(-4)^2}} or\ \frac{-4}{2\sqrt{17}},\frac{-6}{2\sqrt{17}},\frac{-4}{2\sqrt{17}}\ or\ \frac{-2}{\sqrt{17}},\frac{-3}{\sqrt{17}},\frac{-2}{\sqrt{17}}

\Rightarrow Direction ratios of side CA are (-5-3), (-5-5),\ and\ (-2-(-4)) i.e.,

-8,-10,\ and\ 2.

Therefore its direction cosines values are;

\frac{-8}{\sqrt{(-8)^2+(10)^2+(2)^2}},\ \frac{-5}{\sqrt{(-8)^2+(10)^2+(2)^2}},\ \frac{2}{\sqrt{(-8)^2+(10)^2+(2)^2}} or\ \frac{-8}{2\sqrt{42}},\frac{-10}{2\sqrt{42}},\frac{2}{2\sqrt{42}}\ or\ \frac{-4}{\sqrt{42}},\frac{-5}{\sqrt{42}},\frac{1}{\sqrt{42}}

More About NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1

The NCERT book exercise 11.1 Class 12 Maths have 5 questions. The problems in the NCERT solutions for Class 12 Maths chapter 11 exercise 11.1 are to find the direction cosines and one among the five questions in the Class 12th Maths chapter 11 exercise 11.1 is to find whether the given points are collinear or not. No multiple-choice questions are given in the Class 12 Maths chapter 11 exercise 11.1.

Also Read| Three Dimensional Geometry Class 12th Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1

  • The exercise 11.1 Class 12 Maths clears the concepts of direction cosines and ratios.
  • All the questions of NCERT syllabus for Class 12 Maths chapter 11 exercise 11.1 are important and will be useful in CBSE board exam preparation.
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Key Features Of NCERT Solutions for Exercise 11.1 Class 12 Maths Chapter 11

  • Comprehensive Coverage: The solutions encompass all the topics covered in ex 11.1 class 12, ensuring a thorough understanding of the concepts.
  • Step-by-Step Solutions: In this class 12 maths ex 11.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
  • Accuracy and Clarity: Solutions for class 12 ex 11.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
  • Conceptual Clarity: In this 12th class maths exercise 11.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
  • Inclusive Approach: Solutions for ex 11.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
  • Relevance to Curriculum: The solutions for class 12 maths ex 11.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT Solutions Subject Wise

Subject Wise NCERT Exemplar Solutions

Frequently Asked Questions (FAQs)

1. What is the condition for three points A, B and C to be collinear in terms of direction ratios?

The direction ratio of AB and BC will be proportional and since B is a comment point for both AB and BC, the points A, B and C will be collinear. 

2. How many solved examples are given under the discussion of direction cosines and ratios?

5 examples are given

3. What is the main prior knowledge required to solve exercise 11.1 Class 12 Maths?

The idea of vectors and direction cosines and ratios.

4. What number of questions and solutions are detailed in the NCERT solutions for Class 12 Maths chapter 11 exercise 11.1?

Five questions are explained in the exercise 11.1 Class 12 Maths

5. What types of questions are covered in the Class 12th Maths chapter 11 exercise 11.1?

The questions are to find the direction cosines and to check whether the given points are collinear using the concept of direction ratios

6. Give the direction cosine of x axis?

(1, 0, 0)

7. Mention the direction cosine of y axis?

(0, 1, 0)

8. Give the direction cosine of z axis?

(0, 0, 1)

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quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

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Jefferson 25 September,2024
i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

  1. Re-evaluate Your Study Strategies:

    • Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
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  2. Consider Professional Help:

    • Tutoring: A tutor can provide personalized guidance and support.
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Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.







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Kanishka kaushikii 17 September,2024
i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
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How to Apply:
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Hope you find this useful!

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Yuvan 30 August,2024
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024
Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

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