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

## 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.

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

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 $0,\ -\frac{1}{\sqrt2},and\ \ \frac{1}{\sqrt2}$ .

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 $\pm \frac{1}{\sqrt3},\ \pm \frac{1}{\sqrt3},and\ \pm \frac{1}{\sqrt3}$

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 $\frac{-9}{11},\ \frac{6}{11},\ \frac{-2}{11}$ .

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.

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.

## 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-

## Subject Wise NCERT Exemplar Solutions

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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### Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

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Yuvan 01 September,2024

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.

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.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

Hello student,

If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:

• No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
• You have to appear for the 2025 12th board exams.
• Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
• Aim to register before late October to avoid extra fees.
• Schools might not offer classes for private students, so focus on self-study or coaching.

Remember , these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

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 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

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 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

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