Class 12 > Solution- Three Dimensional Geometry Test-1
Solution- Three Dimensional Geometry Test-1 - Class 12
| Download, print and study this document offline |
Page 1 CBSE TEST PAPER-06 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. 1,0,0, 0,1,0 0,0,1 Ans2. Let a and b be the p.v of the points A (-1,0,2) and B ( 3, 4 6) ? ? ? ( ) ( 2 ) (4 4 4 ) r a b a i j i j k ? ? = + - = - + + + + ? ? Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 0 1 cos 2 60 a a b b c c a b c a b c ? ? ? ? + + ? ? = = ? ? + + + + ? ? = Ans4. ? ? ? ? 5 2 4 , 3 2 8 a i j k b i j k = + - = + - ? ? Vector equation of line is ? ? ? ? 5 2 4 (3 2 8 ) r a b i j k i j k ? ? = + = + - + + - ? ? Cartesian equation is ? ? ? ? ? ? ? ? ? ? 5 2 4 (3 2 8 ) (5 3 ) (2 2 ) ( 4 8 ) 5 3 , 2 2 , 4 8 5 2 4 3 2 8 xi y j zk i j k i j k xi y j zk i j k x y z x y z ? ? ? ? ? ? ? + + = + - + + - + + = + + + + - - = + = + = - - - - + = = - ? ? ? ? ? Ans5. Let ? is the angle between the given lines ? ? ? ? 1 2 2 1 2 1 2 3 5 4 cos b i j k and b i j k b b b b ? = - - = - - · = ? ? Page 2 CBSE TEST PAPER-06 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. 1,0,0, 0,1,0 0,0,1 Ans2. Let a and b be the p.v of the points A (-1,0,2) and B ( 3, 4 6) ? ? ? ( ) ( 2 ) (4 4 4 ) r a b a i j i j k ? ? = + - = - + + + + ? ? Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 0 1 cos 2 60 a a b b c c a b c a b c ? ? ? ? + + ? ? = = ? ? + + + + ? ? = Ans4. ? ? ? ? 5 2 4 , 3 2 8 a i j k b i j k = + - = + - ? ? Vector equation of line is ? ? ? ? 5 2 4 (3 2 8 ) r a b i j k i j k ? ? = + = + - + + - ? ? Cartesian equation is ? ? ? ? ? ? ? ? ? ? 5 2 4 (3 2 8 ) (5 3 ) (2 2 ) ( 4 8 ) 5 3 , 2 2 , 4 8 5 2 4 3 2 8 xi y j zk i j k i j k xi y j zk i j k x y z x y z ? ? ? ? ? ? ? + + = + - + + - + + = + + + + - - = + = + = - - - - + = = - ? ? ? ? ? Ans5. Let ? is the angle between the given lines ? ? ? ? 1 2 2 1 2 1 2 3 5 4 cos b i j k and b i j k b b b b ? = - - = - - · = ? ? ? ? ? ? ? ? ? ? ( 2 ) (3 5 4 ) 2 3 5 4 3 5 8 16 6 50 50 16 6 5 2 16 3 2 3 5 2 3 816 i j k i j k i j k i j k - - · - - - - - - + + = = × × × × = ? ? ? ? 3 2 1 3 5 8 3 cos 15 8 3 cos 15 ? ? - × × = ? ? = ? ? ? ? ? ? Ans6. ? ? ? ? 1 1 2 , a i j k b i j k = + + = - + ? ? ? ? ? ? ( ) ? ? ? ? ? ? ? ? ? 2 1 2 1 1 2 1 2 2 1 1 2 2 , 2 2 .( ) 3 2 1 1 1 2 1 2 3 3 ( 3 2 ).( 3 3 3 3 3 6 9 3 9 9 3 2 2 a i j k b i j k a a b b d b b a a i j k i j k b b i k i j k i k d i k = - - = + + - × = × - = - - × = - = - + - - - + = - + - - = = = + ? ? ? ? ? ? ? ? Page 3 CBSE TEST PAPER-06 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. 1,0,0, 0,1,0 0,0,1 Ans2. Let a and b be the p.v of the points A (-1,0,2) and B ( 3, 4 6) ? ? ? ( ) ( 2 ) (4 4 4 ) r a b a i j i j k ? ? = + - = - + + + + ? ? Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 0 1 cos 2 60 a a b b c c a b c a b c ? ? ? ? + + ? ? = = ? ? + + + + ? ? = Ans4. ? ? ? ? 5 2 4 , 3 2 8 a i j k b i j k = + - = + - ? ? Vector equation of line is ? ? ? ? 5 2 4 (3 2 8 ) r a b i j k i j k ? ? = + = + - + + - ? ? Cartesian equation is ? ? ? ? ? ? ? ? ? ? 5 2 4 (3 2 8 ) (5 3 ) (2 2 ) ( 4 8 ) 5 3 , 2 2 , 4 8 5 2 4 3 2 8 xi y j zk i j k i j k xi y j zk i j k x y z x y z ? ? ? ? ? ? ? + + = + - + + - + + = + + + + - - = + = + = - - - - + = = - ? ? ? ? ? Ans5. Let ? is the angle between the given lines ? ? ? ? 1 2 2 1 2 1 2 3 5 4 cos b i j k and b i j k b b b b ? = - - = - - · = ? ? ? ? ? ? ? ? ? ? ( 2 ) (3 5 4 ) 2 3 5 4 3 5 8 16 6 50 50 16 6 5 2 16 3 2 3 5 2 3 816 i j k i j k i j k i j k - - · - - - - - - + + = = × × × × = ? ? ? ? 3 2 1 3 5 8 3 cos 15 8 3 cos 15 ? ? - × × = ? ? = ? ? ? ? ? ? Ans6. ? ? ? ? 1 1 2 , a i j k b i j k = + + = - + ? ? ? ? ? ? ( ) ? ? ? ? ? ? ? ? ? 2 1 2 1 1 2 1 2 2 1 1 2 2 , 2 2 .( ) 3 2 1 1 1 2 1 2 3 3 ( 3 2 ).( 3 3 3 3 3 6 9 3 9 9 3 2 2 a i j k b i j k a a b b d b b a a i j k i j k b b i k i j k i k d i k = - - = + + - × = × - = - - × = - = - + - - - + = - + - - = = = + ? ? ? ? ? ? ? ? Ans7. ? ? .(6 3 2 ) 1 r i j k - - = - ? ? ? ? ? .( 6 3 2 ) 1....(1) 6 3 2 36 9 4 7 r i j k i j k - + + = - + + = + + = ? ? ? Dividing equation 1 by 7 ? ? ? ? ? 6 3 2 1 . 7 7 7 7 6 3 2 [ . 7 7 7 r i j k n i j k r n d - ? ? + + = ? ? ? ? - = + + = ? ? ? Hence direction cosines of ? n is 6 3 2 , , 7 5 7 - Ans8. Comparing the giving eq of the planes with the equations A 1 x +B 1y +C 1Z + D = 0 , A 2 x + B 2y + C 2 Z + D 2 = 0 A 1 = 3, B 1 = -6, C 1 = 2 A 2 = 2, B 2 = 2, C 2 = -2 1 2 1 2 1 2 2 2 2 2 2 1 12 1 2 2 2 1 cos 10 7 2 3 5 5 3 21 7 3 5 3 21 A A B B C C A B C A B C COS ? ? - + + = + + + + - = × = = ? ? = ? ? ? ? ? ? Ans9. ? ? ? ? 1 2 1 2 , 2 3 4 5, 6 n i j k n i j k d d = + + = + + = - = ? ? Using the relation ? ? ? ? ? ? ? ? 1 2 1 2 .( ) .[(1 2 ) (1 3 ) (1 4 ) ] 6 5 ......(1) ( )[(1 2 ) (1 3 ) (1 4 ) ] 6 5 (1 2 ) (1 3 ) (1 4 ) 6 5 r n n d d r i j k taking r xi y j zk xi y j zk i j k x y z ? ? ? ? ? ? ? ? ? ? ? ? ? ? + = + + + + + + = - = + + + + + + + + + = - + + + + + = - ? ? ? ? Page 4 CBSE TEST PAPER-06 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. 1,0,0, 0,1,0 0,0,1 Ans2. Let a and b be the p.v of the points A (-1,0,2) and B ( 3, 4 6) ? ? ? ( ) ( 2 ) (4 4 4 ) r a b a i j i j k ? ? = + - = - + + + + ? ? Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 0 1 cos 2 60 a a b b c c a b c a b c ? ? ? ? + + ? ? = = ? ? + + + + ? ? = Ans4. ? ? ? ? 5 2 4 , 3 2 8 a i j k b i j k = + - = + - ? ? Vector equation of line is ? ? ? ? 5 2 4 (3 2 8 ) r a b i j k i j k ? ? = + = + - + + - ? ? Cartesian equation is ? ? ? ? ? ? ? ? ? ? 5 2 4 (3 2 8 ) (5 3 ) (2 2 ) ( 4 8 ) 5 3 , 2 2 , 4 8 5 2 4 3 2 8 xi y j zk i j k i j k xi y j zk i j k x y z x y z ? ? ? ? ? ? ? + + = + - + + - + + = + + + + - - = + = + = - - - - + = = - ? ? ? ? ? Ans5. Let ? is the angle between the given lines ? ? ? ? 1 2 2 1 2 1 2 3 5 4 cos b i j k and b i j k b b b b ? = - - = - - · = ? ? ? ? ? ? ? ? ? ? ( 2 ) (3 5 4 ) 2 3 5 4 3 5 8 16 6 50 50 16 6 5 2 16 3 2 3 5 2 3 816 i j k i j k i j k i j k - - · - - - - - - + + = = × × × × = ? ? ? ? 3 2 1 3 5 8 3 cos 15 8 3 cos 15 ? ? - × × = ? ? = ? ? ? ? ? ? Ans6. ? ? ? ? 1 1 2 , a i j k b i j k = + + = - + ? ? ? ? ? ? ( ) ? ? ? ? ? ? ? ? ? 2 1 2 1 1 2 1 2 2 1 1 2 2 , 2 2 .( ) 3 2 1 1 1 2 1 2 3 3 ( 3 2 ).( 3 3 3 3 3 6 9 3 9 9 3 2 2 a i j k b i j k a a b b d b b a a i j k i j k b b i k i j k i k d i k = - - = + + - × = × - = - - × = - = - + - - - + = - + - - = = = + ? ? ? ? ? ? ? ? Ans7. ? ? .(6 3 2 ) 1 r i j k - - = - ? ? ? ? ? .( 6 3 2 ) 1....(1) 6 3 2 36 9 4 7 r i j k i j k - + + = - + + = + + = ? ? ? Dividing equation 1 by 7 ? ? ? ? ? 6 3 2 1 . 7 7 7 7 6 3 2 [ . 7 7 7 r i j k n i j k r n d - ? ? + + = ? ? ? ? - = + + = ? ? ? Hence direction cosines of ? n is 6 3 2 , , 7 5 7 - Ans8. Comparing the giving eq of the planes with the equations A 1 x +B 1y +C 1Z + D = 0 , A 2 x + B 2y + C 2 Z + D 2 = 0 A 1 = 3, B 1 = -6, C 1 = 2 A 2 = 2, B 2 = 2, C 2 = -2 1 2 1 2 1 2 2 2 2 2 2 1 12 1 2 2 2 1 cos 10 7 2 3 5 5 3 21 7 3 5 3 21 A A B B C C A B C A B C COS ? ? - + + = + + + + - = × = = ? ? = ? ? ? ? ? ? Ans9. ? ? ? ? 1 2 1 2 , 2 3 4 5, 6 n i j k n i j k d d = + + = + + = - = ? ? Using the relation ? ? ? ? ? ? ? ? 1 2 1 2 .( ) .[(1 2 ) (1 3 ) (1 4 ) ] 6 5 ......(1) ( )[(1 2 ) (1 3 ) (1 4 ) ] 6 5 (1 2 ) (1 3 ) (1 4 ) 6 5 r n n d d r i j k taking r xi y j zk xi y j zk i j k x y z ? ? ? ? ? ? ? ? ? ? ? ? ? ? + = + + + + + + = - = + + + + + + + + + = - + + + + + = - ? ? ? ? ? ? ? ? ? ? ( 6) (2 3 4 4 5) 0....(2) int (1,1,1) 3 14 (1) 3 9 6 15 . 1 1 1 6 7 14 7 14 10 23 13 69 . 7 14 7 14 .(20 23 26 ) 69 x y z x y y z plane passes through the po put in eq r i j k r i j k r i j k ? ? ? + + - + + + + + = = ? ? ? ? ? ? ? ? + + + + + = - ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + = ? ? ? ? + + = ? ? ? Ans10. Given points are A(3,-4,-5) B(2,-3,1) Direction ration of AB are 3-2, -4+3, -5-1 1,-1,-6 Eq. of line AB 3 4 5 ( ) 1 1 6 3, 4, 6 5 ( 3, 4, 6 5) 2 7 2( 3) ( 4) ( 6 5) 7 2 (1, 2,7) x y Z say x y Z let lies in the plane x y Z ? ? ? ? ? ? ? ? ? ? ? - + + = = = - - = + = - - = - - + - - - - + + = + + - - + - - = = - - are the required pointRead More
FAQs on Solution- Three Dimensional Geometry Test-1 - Class 12
| 1. What is three-dimensional geometry? |  |
Ans. Three-dimensional geometry is a branch of mathematics that deals with the study of objects in three-dimensional space. It involves understanding the properties and relationships of points, lines, curves, surfaces, and solids in three dimensions.
| 2. How is three-dimensional geometry different from two-dimensional geometry? | |
Ans. Three-dimensional geometry involves studying objects in three dimensions, which means considering length, width, and height. On the other hand, two-dimensional geometry only deals with objects in a plane, considering only length and width. Three-dimensional geometry adds an extra dimension, allowing for a more comprehensive understanding of spatial relationships.
| 3. What is the importance of three-dimensional geometry in real life? | |
Ans. Three-dimensional geometry is highly relevant in various real-life applications. It is used in architecture and engineering to design and analyze structures, in computer graphics and animation to create realistic 3D models, in medical imaging to visualize and understand body structures, and in navigation systems to determine positions and distances accurately.
| 4. How can I visualize three-dimensional objects? | |
Ans. Visualizing three-dimensional objects can be challenging, but there are several techniques to help. One common approach is using perspective drawings, where objects appear smaller as they move farther away. Another method is using physical models or computer-aided design (CAD) software to create virtual representations of objects. Additionally, you can use your imagination and spatial reasoning skills to mentally visualize and manipulate three-dimensional objects.
| 5. What are some common three-dimensional shapes? | |
Ans. Some common three-dimensional shapes include cubes, spheres, cylinders, cones, pyramids, and prisms. These shapes have distinct properties and formulas to calculate their volume, surface area, and other characteristics. Understanding these shapes is essential in many fields, such as architecture, engineering, and manufacturing.
Document Description: Solution- Three Dimensional Geometry Test-1 for Class 12 2023 is part of Class 12 preparation. The notes and questions for Solution- Three Dimensional Geometry Test-1 have been prepared according to the Class 12 exam syllabus. Information about Solution- Three Dimensional Geometry Test-1 covers topics like and Solution- Three Dimensional Geometry Test-1 Example, for Class 12 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Solution- Three Dimensional Geometry Test-1.
Introduction of Solution- Three Dimensional Geometry Test-1 in English is available as part of
our Class 12 preparation & Solution- Three Dimensional Geometry Test-1 in Hindi for Class 12
courses. Download more important topics, notes, lectures and mock test series for Class 12
Exam by signing up for free. Class 12: Solution- Three Dimensional Geometry Test-1 - Class 12
Description
Full syllabus notes, lecture & questions for Solution- Three Dimensional Geometry Test-1 - Class 12 - Class 12 | Plus excerises question with solution to help you revise complete syllabus | Best notes, free PDF download
Information about Solution- Three Dimensional Geometry Test-1
In this doc you can find the meaning of Solution- Three Dimensional Geometry Test-1 defined & explained in the simplest way possible.
Besides explaining types of Solution- Three Dimensional Geometry Test-1 theory,
EduRev gives you an ample number of questions to practice Solution- Three Dimensional Geometry Test-1 tests, examples and also practice Class 12 tests.
Download as PDF
|
Explore Courses for Class 12 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Download free EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches
Extra Questions
,Solution- Three Dimensional Geometry Test-1 - Class 12
,Previous Year Questions with Solutions
,Viva Questions
,Objective type Questions
,Solution- Three Dimensional Geometry Test-1 - Class 12
,Free
,Summary
,Important questions
,video lectures
,MCQs
,past year papers
,Sample Paper
,ppt
,Semester Notes
,mock tests for examination
,Solution- Three Dimensional Geometry Test-1 - Class 12
,shortcuts and tricks
,study material
,Exam
,practice quizzes
;
Additional Information about Solution- Three Dimensional Geometry Test-1 for Class 12 Preparation
Solution- Three Dimensional Geometry Test-1 Free PDF Download
The Solution- Three Dimensional Geometry Test-1 is an invaluable resource that delves deep into the core of the Class 12 exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Solution- Three Dimensional Geometry Test-1 now and kickstart your journey towards success in the Class 12 exam.
Importance of Solution- Three Dimensional Geometry Test-1
The importance of Solution- Three Dimensional Geometry Test-1 cannot be overstated, especially for Class 12 aspirants.
This document holds the key to success in the Class 12 exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Solution- Three Dimensional Geometry Test-1 Notes
Solution- Three Dimensional Geometry Test-1 Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Solution- Three Dimensional Geometry Test-1.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Solution- Three Dimensional Geometry Test-1 Notes on EduRev are your ultimate resource for success.
Solution- Three Dimensional Geometry Test-1 Class 12 Questions
The "Solution- Three Dimensional Geometry Test-1 Class 12 Questions" guide is a valuable resource for all aspiring students preparing for the
Class 12 exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Solution- Three Dimensional Geometry Test-1 on the App
Students of Class 12 can study Solution- Three Dimensional Geometry Test-1 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Solution- Three Dimensional Geometry Test-1,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Solution- Three Dimensional Geometry Test-1 is prepared as per the latest Class 12 syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup
