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

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

Edited By Komal Miglani | Updated on May 07, 2025 04:10 PM IST | #CBSE Class 12th

Three-dimensional geometry allows us to construct and model everything around us. From understanding motion, building houses and even computer graphics requires an understanding of the 3D space around us and how to construct them mathematically.

A line in 3D space is the most fundamental object. In Class 11 Maths Chapter 11 Exercise 11.1 solutions of NCERT, we uncover how to construct lines in 3D space, the angles they make and the criteria for constructing them. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2025-26. Class 12th Maths exercise 11.1 Solutions are designed as per the students' demand covering comprehensive, step by step solutions of every problem.

This Story also Contains
  1. Class 12 Maths Chapter 11 Exercise 11.1 Solutions: Download PDF
  2. NCERT Solutions Class 12 Maths Chapter 11: Exercise 11.1
  3. Topics covered in Chapter 11 Three Dimensional Geometry: Exercise 11.1
  4. NCERT Solutions Subject Wise
  5. Subject Wise NCERT Exemplar Solutions
LiveCBSE Result 2025 (LIVE): Class 10, 12 marksheet likely today at cbse.gov.in, Digilocker; websites to checkMay 12, 2025 | 10:32 PM IST

No, the Central Board of Secondary Education is yet to declare CBSE Class 10 results 2025. Once declared, the CBSE 10th result link 2025 will be available on the official website, results.cbse.nic.in.

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Class 12 Maths Chapter 11 Exercise 11.1 Solutions: Download PDF

Download the Exercise 11.1 class 12 maths as a PDF for easy offline access. This will help you study and prepare with ease, anytime-anywhere.

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NCERT Solutions Class 12 Maths Chapter 11: Exercise 11.1

Question 1: If a line makes angles 90∘,135∘,45∘ 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=cos⁡90∘=0

m=cos⁡135∘=−12

n=cos⁡45∘=12

Therefore the direction cosines of the lines are 0, −12,and  12 .

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 α with each coordinate axes.

Therefore, we can write;

l=cos⁡α, m=cos⁡α,and n=cos⁡α

And as we know the relation; l2+m2+n2=1

⇒cos2⁡α+cos2⁡α+cos2⁡α=1

⇒cos2⁡α=13

or cos⁡α=±13

Thus the direction cosines of the line are ±13, ±13,and ±13

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:

−18(−18)2+(12)2+(−4)2=−1822=−911

Line having direction ratio 12 has direction cosine:

12(−18)2+(12)2+(−4)2=1222=611

Line having direction ratio -4 has direction cosine:

12(−4)2+(12)2+(−4)2=−422=−211

Thus, the direction cosines are −911, 611, −211 .

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 (x1,y1,z1) and (x2,y2,z2) is given by x2−x1,y2−y1, and z2−z1.

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.

∴ 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 △ABC (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).

Finding each side direction ratios;

⇒ 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;

−4(−4)2+(−4)2+(6)2, −4(−4)2+(−4)2+(6)2, 6(−4)2+(−4)2+(6)2 or −4217,−4217,6217 or −217,−217,317

SImilarly for side BC;

⇒ 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;

−4(−4)2+(−6)2+(−4)2, −6(−4)2+(−6)2+(−4)2, −4(−4)2+(−6)2+(−4)2 or −4217,−6217,−4217 or −217,−317,−217

⇒ 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;

−8(−8)2+(10)2+(2)2, −5(−8)2+(10)2+(2)2, 2(−8)2+(10)2+(2)2 or −8242,−10242,2242 or −442,−542,142

Topics covered in Chapter 11 Three Dimensional Geometry: Exercise 11.1

Ex 11.1 Class 12 covers straight lines in 3D space. This includes direction cosines and direction ratios of a straight line. This tells us about the angles made by the line with the coordinate axes. We also understand how to imagine a line only using its direction cosines.

l=cos⁡α,m=cos⁡β,n=cos⁡γl2+m2+n2=1cos2⁡α+cos2⁡β+cos2⁡γ=1

Direction Cosines of the Line Passing Through Two Points

Let P(x1,y1,z1) and Q(x2,y2,z2) be two points on the line L .
Let 1, m, and n be the direction cosines of the line PQ, and let it make angles α,β, and y with the x-axis, y-axis, and z-axis respectively.
The direction cosines of the line segment joining the points P and Q are given by

(x2−x1PQ,y2−y1PQ,z2−z1PQ)

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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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Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.

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Manasvini Gupta 30 January,2025
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.

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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,

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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
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