Page 1 CBSE TEST PAPER-03 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ( ) ( ) 2 2 1 2 5 a = + = ( ) ( ) 2 2 2 1 5 b = + = Ans3. D.R of 1,2,3 r are 1 4 9 4 r = + + = D.C of 1 2 3 , , 14 14 14 r are Ans4. ( ) ( ) 3 5 . 2 7 a b a b - + 2 2 2 2 6 11 . 35 . . . . a a b b a a a b b b a b b a = - - ? = = = ? ? ? Ans5. ? ? 2 1 3 3 5 2 i j k a b × = - ? ( ) ? ( ) ? ( ) ? ? 2 15 4 9 10 3 17 13 7 i j k i j k = - - - - - + - = - + + ? ? Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7) ? ? ? ? ? ? 1 2 8 5 0 2 11 3 7 OA i j k OB i j k OC i j k = - - = - - = + + ? ? ? ? ? 4 2 6 AB OB OA i j k = - = + + ? Page 2 CBSE TEST PAPER-03 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ( ) ( ) 2 2 1 2 5 a = + = ( ) ( ) 2 2 2 1 5 b = + = Ans3. D.R of 1,2,3 r are 1 4 9 4 r = + + = D.C of 1 2 3 , , 14 14 14 r are Ans4. ( ) ( ) 3 5 . 2 7 a b a b - + 2 2 2 2 6 11 . 35 . . . . a a b b a a a b b b a b b a = - - ? = = = ? ? ? Ans5. ? ? 2 1 3 3 5 2 i j k a b × = - ? ( ) ? ( ) ? ( ) ? ? 2 15 4 9 10 3 17 13 7 i j k i j k = - - - - - + - = - + + ? ? Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7) ? ? ? ? ? ? 1 2 8 5 0 2 11 3 7 OA i j k OB i j k OC i j k = - - = - - = + + ? ? ? ? ? 4 2 6 AB OB OA i j k = - = + + ? ? ? ( ) 3 2 3 BC OC OB i j k = - = + + ? ? ? ( ) 3 4 2 6 2 3 2 i j k BC AB = + + = ? Thus BC AB and one point B is common there fore A, B, C are collinear. Ans7. ? ? 4 2 a i j k = + + ? ? ? ? ? 3 2 7 2 4 b i j k c i j k = - + = - + ? ? ? ? . 0, . 0 . 15 Let d xi y j zk ATQ d a d b and c d = + + = = = ? 4 2 0 (1) 3 2 7 0 (2) 2 4 15 (3) x y z x y z x y z + + = - - - - - - - - - + = - - - - - - - - + = - - - - - - - On solving equation (i) and (ii) 28 4 6 7 2 12 32 , , 14 x y z K x k y k z k = = = + - - - = = - = - Put x, y, z in equation (iii) ( ) ( ) ( ) 2 32 4 14 15 64 56 15 9 15 k k k k k k k - - + - = + - = = 15 9 5 3 k k = = 5 160 32 3 3 x = × = Page 3 CBSE TEST PAPER-03 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ( ) ( ) 2 2 1 2 5 a = + = ( ) ( ) 2 2 2 1 5 b = + = Ans3. D.R of 1,2,3 r are 1 4 9 4 r = + + = D.C of 1 2 3 , , 14 14 14 r are Ans4. ( ) ( ) 3 5 . 2 7 a b a b - + 2 2 2 2 6 11 . 35 . . . . a a b b a a a b b b a b b a = - - ? = = = ? ? ? Ans5. ? ? 2 1 3 3 5 2 i j k a b × = - ? ( ) ? ( ) ? ( ) ? ? 2 15 4 9 10 3 17 13 7 i j k i j k = - - - - - + - = - + + ? ? Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7) ? ? ? ? ? ? 1 2 8 5 0 2 11 3 7 OA i j k OB i j k OC i j k = - - = - - = + + ? ? ? ? ? 4 2 6 AB OB OA i j k = - = + + ? ? ? ( ) 3 2 3 BC OC OB i j k = - = + + ? ? ? ( ) 3 4 2 6 2 3 2 i j k BC AB = + + = ? Thus BC AB and one point B is common there fore A, B, C are collinear. Ans7. ? ? 4 2 a i j k = + + ? ? ? ? ? 3 2 7 2 4 b i j k c i j k = - + = - + ? ? ? ? . 0, . 0 . 15 Let d xi y j zk ATQ d a d b and c d = + + = = = ? 4 2 0 (1) 3 2 7 0 (2) 2 4 15 (3) x y z x y z x y z + + = - - - - - - - - - + = - - - - - - - - + = - - - - - - - On solving equation (i) and (ii) 28 4 6 7 2 12 32 , , 14 x y z K x k y k z k = = = + - - - = = - = - Put x, y, z in equation (iii) ( ) ( ) ( ) 2 32 4 14 15 64 56 15 9 15 k k k k k k k - - + - = + - = = 15 9 5 3 k k = = 5 160 32 3 3 x = × = ? ? 5 5 3 3 5 70 14 3 3 160 5 70 3 3 3 y z d i j k = - = - = - × = - = - - ? Ans8. ( ) ( ) ( ) ( ) . 0, . 0 . 0, a b c b c a c a b Given + = + = + = ( ) ( ) ( ) ( ) ( ) 2 . . . . . . a b c a b c a b c a a a b c b b b a c c c a b + + = + + + + = + + + + + + + + 2 2 2 9 16 25 a b c = + + = + + 50 50 5 2 a b c = + + = = Ans9. ? ? 9 4 4 a b i j k + = + - ? ? ( ) ( ) Ë† 2 . 17 a b i k a b a b - = - + + - = - 113 5 a b a b + = - = ( ) ( ) . cos 17 113. 5 a b a b a b a b ? + - = + - - = 1 17 cos 565 17 cos 565 ? ? - - = - ? ? = ? ? ? ? Ans10. ? ? 2 1 3 1 3 2 i j k a b × = - ? ? ? ( ) ( ) ( ) 2 2 2 11 7 11 1 7 i j k a b = - - + × = - + - + ? 171 3 19 3 19 3 sin 19 14 14. 14 a b a b ? = = × = = =Read More