Class 12 > Solution- Three Dimensional Geometry Test-1

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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

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.

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