Page 1 CBSE TEST PAPER-05 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Scalar Ans2. ? ? 1 3 5 a i j k = + + ? ? ? ? ? ( ) 2 4 6 2 2 3 2 b i j k b i j k b a = + + = + + = ? ? Ans3. . 1,1, 2 D R of r are - 1 1 4 6 1 1 2 . , , 6 6 6 r D C of r are = + + = - Ans4. . cos a b a b ? = ( )( ) 1 1 cos 1 2 2 cos cos 3 3 ? p p ? ? = = = ? = Ans5. ? a a a = ? ( ) 2 5 i j - = ? Vector having magnitude equal to 7 and in the direction of a is ? ? ? ( ) 2 7 7 5 7 2 5 i j a i j ? ? - = ? ? ? ? ? ? = - ? ? Page 2 CBSE TEST PAPER-05 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Scalar Ans2. ? ? 1 3 5 a i j k = + + ? ? ? ? ? ( ) 2 4 6 2 2 3 2 b i j k b i j k b a = + + = + + = ? ? Ans3. . 1,1, 2 D R of r are - 1 1 4 6 1 1 2 . , , 6 6 6 r D C of r are = + + = - Ans4. . cos a b a b ? = ( )( ) 1 1 cos 1 2 2 cos cos 3 3 ? p p ? ? = = = ? = Ans5. ? a a a = ? ( ) 2 5 i j - = ? Vector having magnitude equal to 7 and in the direction of a is ? ? ? ( ) 2 7 7 5 7 2 5 i j a i j ? ? - = ? ? ? ? ? ? = - ? ? Ans6. ? ? a i j k = + + ? ? ? ? ? 1 2 3 b j k Let c c i c j c k = - = + + ? ? ? 1 2 3 1 1 1 a c b i j k b c c c × = = ? ( ) ? ( ) ? ( ) ? ? 3 2 3 1 2 1 3 2 . 0 i c c j c c k c c j k c c - - - + - = - - = ? 3 2 3 1 ( ) 1 ( ) c c i c c ii = - - - - - - - - - + = - - - - - 2 1 1 ( ) . 3 c c iii also a c - = - - - - - - = 1 2 3 3 ( ) on solving equaiton (i) (ii) (iii) and (iv) c c c iv + + = - - - - ? 1 2 3 2 5 2 , , 3 3 3 2 5 2 j . 3 3 3 c c c c i k = = = = + + ? ? Ans7. (i) Given vectors are parallel ? ? 2 4 1 0 4 8 2 i j k if ? ? - = - = ? (ii) For ? ? ? ( ) ? ? ( ) 4 8 . 2 4 0 8 32 0 40 i j k i j k ? ? ? - + - + = + + = = - ? ? Ans8. 1 a = 1 1 b c = = Page 3 CBSE TEST PAPER-05 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Scalar Ans2. ? ? 1 3 5 a i j k = + + ? ? ? ? ? ( ) 2 4 6 2 2 3 2 b i j k b i j k b a = + + = + + = ? ? Ans3. . 1,1, 2 D R of r are - 1 1 4 6 1 1 2 . , , 6 6 6 r D C of r are = + + = - Ans4. . cos a b a b ? = ( )( ) 1 1 cos 1 2 2 cos cos 3 3 ? p p ? ? = = = ? = Ans5. ? a a a = ? ( ) 2 5 i j - = ? Vector having magnitude equal to 7 and in the direction of a is ? ? ? ( ) 2 7 7 5 7 2 5 i j a i j ? ? - = ? ? ? ? ? ? = - ? ? Ans6. ? ? a i j k = + + ? ? ? ? ? 1 2 3 b j k Let c c i c j c k = - = + + ? ? ? 1 2 3 1 1 1 a c b i j k b c c c × = = ? ( ) ? ( ) ? ( ) ? ? 3 2 3 1 2 1 3 2 . 0 i c c j c c k c c j k c c - - - + - = - - = ? 3 2 3 1 ( ) 1 ( ) c c i c c ii = - - - - - - - - - + = - - - - - 2 1 1 ( ) . 3 c c iii also a c - = - - - - - - = 1 2 3 3 ( ) on solving equaiton (i) (ii) (iii) and (iv) c c c iv + + = - - - - ? 1 2 3 2 5 2 , , 3 3 3 2 5 2 j . 3 3 3 c c c c i k = = = = + + ? ? Ans7. (i) Given vectors are parallel ? ? 2 4 1 0 4 8 2 i j k if ? ? - = - = ? (ii) For ? ? ? ( ) ? ? ( ) 4 8 . 2 4 0 8 32 0 40 i j k i j k ? ? ? - + - + = + + = = - ? ? Ans8. 1 a = 1 1 b c = = Hence they are unit vectors . 0 . 0 a b b c = = . 0 , c a a b b c and c a = ? ? ? So they are ? to each other. Ans9. (i) ? ? 3 1 2 2 2 4 i j k a b × = - ? ? ? 8 8 8 8 3 i j k a b = - - × = ? (ii) a b a b × = + ? ? 8 8 8 8 3 i j k - - = ? (iii) . 12 a b = 14 24 a b = = . cos a b a b ? = 12 3 14. 24 7 sin a b Also a b ? = = + = 8 3 14 24 2 7 = = Ans10. . 0 a b a b ? = ? ? ? ( ) 6 (1) y x a b Given - = - - - - - - - - = 2 2 2 2 2 2 2 3 1 2 1 5 x y y x + + = + + - = ( )( ) ( ) 5 6 5 y x y x y x - + = - = 5 6 31 41 , 12 12 y x x y - = - = =Read More

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