Page 1 CBSE TEST PAPER-07 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. x 2 â€“ x 1, y 2 â€“ y, and z 2-z 1 are the direction ratio of the line segment PQ. Ans2. Comparing the given equation with the standard equation form ? ? ? ? 1 1 1 ( 3 5 6 ) (2 4 2 ) x x y y z z a b c r i j k i j k ? - - - = = = - + + + + + ? ? Ans3. x 1=-3, y 1 = 1, z 1 = 5 a 1 = -3, b 1=1, c 1= 5 x 2 = -1, y 2=2, z 2 = 5 a 2 = -1, b 2 = 2, c 2 = 5 2 1 2 1 2 1 1 1 1 2 2 2 2 1 0 3 1 5 0 1 2 5 x x y y z z a b c z b c - - - = - = - Therefore lines are coplanar. Ans4. ? ? ? 1 1 , 2 a i j b i j k = + = - + ? ? ? ? ? ? ? ? ? ? ? 2 2 2 1 1 2 1 2 1 2 2 1 1 2 2 , 3 5 2 2 1 1 3 5 2 3 7 59 3 0 7 ( ).( ) 10 59 59 a i j k b i j k a a i k i j k b b i j k b b b b a a d b b = + - = - + - = - × = - - = - - × = - + × - = = = × ? ? ? ? ? Page 2 CBSE TEST PAPER-07 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. x 2 â€“ x 1, y 2 â€“ y, and z 2-z 1 are the direction ratio of the line segment PQ. Ans2. Comparing the given equation with the standard equation form ? ? ? ? 1 1 1 ( 3 5 6 ) (2 4 2 ) x x y y z z a b c r i j k i j k ? - - - = = = - + + + + + ? ? Ans3. x 1=-3, y 1 = 1, z 1 = 5 a 1 = -3, b 1=1, c 1= 5 x 2 = -1, y 2=2, z 2 = 5 a 2 = -1, b 2 = 2, c 2 = 5 2 1 2 1 2 1 1 1 1 2 2 2 2 1 0 3 1 5 0 1 2 5 x x y y z z a b c z b c - - - = - = - Therefore lines are coplanar. Ans4. ? ? ? 1 1 , 2 a i j b i j k = + = - + ? ? ? ? ? ? ? ? ? ? ? 2 2 2 1 1 2 1 2 1 2 2 1 1 2 2 , 3 5 2 2 1 1 3 5 2 3 7 59 3 0 7 ( ).( ) 10 59 59 a i j k b i j k a a i k i j k b b i j k b b b b a a d b b = + - = - + - = - × = - - = - - × = - + × - = = = × ? ? ? ? ? Ans5. ? ? 1 3 2 6 b i j k = + + ? ? ? 2 2 2 b i j k = + + ? The angle ? between them is given by ? ? ? ? ? ? ? ? 1 2 1 2 . cos (3 2 6 ).( 2 2 ) 3 2 6 2 2 3 4 12 49 9 19 19 7 3 21 b b b b i j k i j k i j k i j k ? = + + + + + + + + + + = = × ? ? ? ? Ans6. 5 ( 2) 0 7 5 1 x y z - - - - = = - 1 1 1 2 2 2 0 0 0 1 2 3 7, 5, 1 1, 2, 3 x y z a b c a b c - - - = = = = - = = = = For ? a 1a 2+b 1b 2+c 1c 2=0 L.H. S 1 2 7 1 ( 5 2) 1 3 7 10 3 0 hence l l = × + - × + × = - + = ? Ans7. Let ? ? ? ? ? ? 2 5 3 2 3 5 5 3 3 a i j k b i j k c i j k = + - = - - + = + - ? ? ? Vector equation is ? ? ? ? ? ( ).[( ) ( )] 0 [ (2 5 3 )].[( 4 8 8 ) (3 2 )] 0 r a b a c a r i j k i j k i j - - × - = - + - - - + × - = ? ? ? Page 3 CBSE TEST PAPER-07 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Three Dimensional Geometry [ANSWERS] Ans1. x 2 â€“ x 1, y 2 â€“ y, and z 2-z 1 are the direction ratio of the line segment PQ. Ans2. Comparing the given equation with the standard equation form ? ? ? ? 1 1 1 ( 3 5 6 ) (2 4 2 ) x x y y z z a b c r i j k i j k ? - - - = = = - + + + + + ? ? Ans3. x 1=-3, y 1 = 1, z 1 = 5 a 1 = -3, b 1=1, c 1= 5 x 2 = -1, y 2=2, z 2 = 5 a 2 = -1, b 2 = 2, c 2 = 5 2 1 2 1 2 1 1 1 1 2 2 2 2 1 0 3 1 5 0 1 2 5 x x y y z z a b c z b c - - - = - = - Therefore lines are coplanar. Ans4. ? ? ? 1 1 , 2 a i j b i j k = + = - + ? ? ? ? ? ? ? ? ? ? ? 2 2 2 1 1 2 1 2 1 2 2 1 1 2 2 , 3 5 2 2 1 1 3 5 2 3 7 59 3 0 7 ( ).( ) 10 59 59 a i j k b i j k a a i k i j k b b i j k b b b b a a d b b = + - = - + - = - × = - - = - - × = - + × - = = = × ? ? ? ? ? Ans5. ? ? 1 3 2 6 b i j k = + + ? ? ? 2 2 2 b i j k = + + ? The angle ? between them is given by ? ? ? ? ? ? ? ? 1 2 1 2 . cos (3 2 6 ).( 2 2 ) 3 2 6 2 2 3 4 12 49 9 19 19 7 3 21 b b b b i j k i j k i j k i j k ? = + + + + + + + + + + = = × ? ? ? ? Ans6. 5 ( 2) 0 7 5 1 x y z - - - - = = - 1 1 1 2 2 2 0 0 0 1 2 3 7, 5, 1 1, 2, 3 x y z a b c a b c - - - = = = = - = = = = For ? a 1a 2+b 1b 2+c 1c 2=0 L.H. S 1 2 7 1 ( 5 2) 1 3 7 10 3 0 hence l l = × + - × + × = - + = ? Ans7. Let ? ? ? ? ? ? 2 5 3 2 3 5 5 3 3 a i j k b i j k c i j k = + - = - - + = + - ? ? ? Vector equation is ? ? ? ? ? ( ).[( ) ( )] 0 [ (2 5 3 )].[( 4 8 8 ) (3 2 )] 0 r a b a c a r i j k i j k i j - - × - = - + - - - + × - = ? ? ? Ans8. Let ? ? ? ? ? ? .( ) 2 ( )( ) 2 2 r xi yi zk r i j k xi yi zk i j k x y z = + + + - = + + + - = + - = ? ? ? ? ? ? Which is the required equation of plane. Ans9. Equation of any plane through the intersection of given planes can be taken as (3 2 4) ( 2) 0.....( ) x y z x y z i ? - + - + + + - = The point (2,2,1) lies in this plane 2 / 3 ? = - put in eq â€¦.(i) ( ) 2 3 2 4 ( 2) 0 3 7 5 4 8 0 x y z x y z x y z - + - - + + - = - + - = Ans10. The given plane is ? ? ? ? ? ? .(3 4 12 ) 13 0 ( ).(3 4 12 ) 13 0 3 4 12 13 0....( ) r i j k xi y j zk i j k x y z i + - + = + + + - + = + - + = ? ? ? This plane is equidistant from the points (1,1,P) and (-3,0,1) 2 2 2 2 2 2 3(1) 4(1) 12 13 3( 3) 4(0) 12(1) 13 3 4 ( 12) 3 4 ( 12) 20 12 8 20 12 8 7 1 3 p p p p or + - + - + - + = + + - + + - - = - - = ± = -Read More

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