Page 1 CBSE TEST PAPER-02 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ? ? 2 3 a i j k = + + ? ? ? 1 4 9 14 3 2 1 9 4 1 14 a b i j k b = + + = = + + = + + = ? Ans3. D.R of 1,1,1 r are 1 1 1 3 1 1 1 . , , 3 3 3 r D C of are = + + = Ans4. ( ) ( ) 2 . a b a b a b - = - - 2 . . . . 2 . a a a b b a b b a a b b = - - + = - + 4 2 4 9 5 5 a b = - × + = - = Ans5. ? 3 2 b i k = + ? ? ? 2 8 6 4 a i j k = + + ? Page 2 CBSE TEST PAPER-02 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ? ? 2 3 a i j k = + + ? ? ? 1 4 9 14 3 2 1 9 4 1 14 a b i j k b = + + = = + + = + + = ? Ans3. D.R of 1,1,1 r are 1 1 1 3 1 1 1 . , , 3 3 3 r D C of are = + + = Ans4. ( ) ( ) 2 . a b a b a b - = - - 2 . . . . 2 . a a a b b a b b a a b b = - - + = - + 4 2 4 9 5 5 a b = - × + = - = Ans5. ? 3 2 b i k = + ? ? ? 2 8 6 4 a i j k = + + ? ? ? 2 3 0 2 8 6 4 i j k b a × = ? ( ) ? ( ) ? ( ) 0 12 12 16 18 0 i j k = - - - + - ? ? ? ( ) ( ) ( ) 2 2 2 12 4 18 2 12 4 18 i j k b a - + + × = - + + ? 484 22 = = Ans6. (i) ( ) ( ) 2 1 3 2 2 1 a b a b OR + + - = + 5 3 a = (ii) ( ) ( ) 2 3 2 2 1 a b a b OR + - - = - 2 2 3 2 1 4 a b a b b a + - + = = - Ans7. 2 3 5 OA a b c = - + + 2 3 7 3 2 OB a b c OC a c OA OB OA a b c = + + = - = - = - - ( ) 6 2 4 2 3 2 2 BC OC OB a b c a b c BC AB = - = - - = - - = Thus || AB BC but one point B is common to both vectors hence A, B, C are collinear. Page 3 CBSE TEST PAPER-02 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ? ? 2 3 a i j k = + + ? ? ? 1 4 9 14 3 2 1 9 4 1 14 a b i j k b = + + = = + + = + + = ? Ans3. D.R of 1,1,1 r are 1 1 1 3 1 1 1 . , , 3 3 3 r D C of are = + + = Ans4. ( ) ( ) 2 . a b a b a b - = - - 2 . . . . 2 . a a a b b a b b a a b b = - - + = - + 4 2 4 9 5 5 a b = - × + = - = Ans5. ? 3 2 b i k = + ? ? ? 2 8 6 4 a i j k = + + ? ? ? 2 3 0 2 8 6 4 i j k b a × = ? ( ) ? ( ) ? ( ) 0 12 12 16 18 0 i j k = - - - + - ? ? ? ( ) ( ) ( ) 2 2 2 12 4 18 2 12 4 18 i j k b a - + + × = - + + ? 484 22 = = Ans6. (i) ( ) ( ) 2 1 3 2 2 1 a b a b OR + + - = + 5 3 a = (ii) ( ) ( ) 2 3 2 2 1 a b a b OR + - - = - 2 2 3 2 1 4 a b a b b a + - + = = - Ans7. 2 3 5 OA a b c = - + + 2 3 7 3 2 OB a b c OC a c OA OB OA a b c = + + = - = - = - - ( ) 6 2 4 2 3 2 2 BC OC OB a b c a b c BC AB = - = - - = - - = Thus || AB BC but one point B is common to both vectors hence A, B, C are collinear. Ans8. A vector which is ? to both ( ) ( ) a b are a b + - is giving by ( ) ( ) ? ? 2 3 4 0 1 2 i j k a b a b + × - = - - ? ? ? ? ? 2 4 2 2 4 2 i j k Let c i j k = - + - = - + - ? ? 4 16 4 24 2 6 c = + + = = Req. unit vector is ? ? 1 2 1 6 6 6 c i j k c = - + - ? Ans9. ? ? 2 4 5 a i j k = + - ? ? ? ( ) ? ? 3 2 6 2 b i j k a b i j k ? ? = + + + = + + - ? ? Unit vector along ( ) ? ? ( ) ( ) ( ) ? ? ( ) 2 2 2 2 2 6 2 2 6 + (-2) 2 6 2 2 40 a b a b a b i j k i j k ? ? ? ? + + = + + + - = + + + + - = + + ? ? ( ) ? ? ( ) ( ) ? ? ( ) 2 . 1 2 6 2 . 1 2 40 ATQ c a b i j k i j k ? ? + = ? ? + + - + + = ? ? ? ? + + ? ? ? ? Page 4 CBSE TEST PAPER-02 CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) Topic: - Vectors [ANSWERS] Ans1. Vector Ans2. ? ? 2 3 a i j k = + + ? ? ? 1 4 9 14 3 2 1 9 4 1 14 a b i j k b = + + = = + + = + + = ? Ans3. D.R of 1,1,1 r are 1 1 1 3 1 1 1 . , , 3 3 3 r D C of are = + + = Ans4. ( ) ( ) 2 . a b a b a b - = - - 2 . . . . 2 . a a a b b a b b a a b b = - - + = - + 4 2 4 9 5 5 a b = - × + = - = Ans5. ? 3 2 b i k = + ? ? ? 2 8 6 4 a i j k = + + ? ? ? 2 3 0 2 8 6 4 i j k b a × = ? ( ) ? ( ) ? ( ) 0 12 12 16 18 0 i j k = - - - + - ? ? ? ( ) ( ) ( ) 2 2 2 12 4 18 2 12 4 18 i j k b a - + + × = - + + ? 484 22 = = Ans6. (i) ( ) ( ) 2 1 3 2 2 1 a b a b OR + + - = + 5 3 a = (ii) ( ) ( ) 2 3 2 2 1 a b a b OR + - - = - 2 2 3 2 1 4 a b a b b a + - + = = - Ans7. 2 3 5 OA a b c = - + + 2 3 7 3 2 OB a b c OC a c OA OB OA a b c = + + = - = - = - - ( ) 6 2 4 2 3 2 2 BC OC OB a b c a b c BC AB = - = - - = - - = Thus || AB BC but one point B is common to both vectors hence A, B, C are collinear. Ans8. A vector which is ? to both ( ) ( ) a b are a b + - is giving by ( ) ( ) ? ? 2 3 4 0 1 2 i j k a b a b + × - = - - ? ? ? ? ? 2 4 2 2 4 2 i j k Let c i j k = - + - = - + - ? ? 4 16 4 24 2 6 c = + + = = Req. unit vector is ? ? 1 2 1 6 6 6 c i j k c = - + - ? Ans9. ? ? 2 4 5 a i j k = + - ? ? ? ( ) ? ? 3 2 6 2 b i j k a b i j k ? ? = + + + = + + - ? ? Unit vector along ( ) ? ? ( ) ( ) ( ) ? ? ( ) 2 2 2 2 2 6 2 2 6 + (-2) 2 6 2 2 40 a b a b a b i j k i j k ? ? ? ? + + = + + + - = + + + + - = + + ? ? ( ) ? ? ( ) ( ) ? ? ( ) 2 . 1 2 6 2 . 1 2 40 ATQ c a b i j k i j k ? ? + = ? ? + + - + + = ? ? ? ? + + ? ? ? ? ( ) ( ) ( ) 2 2 2 6 2 1 2 40 2 4 2 40 ? ? ? ? + + - = + + + + = + + ( ) 2 2 . 36 12 2 40 1 sq both site ? ? ? ? + + = + + = Ans10. A (1, 1, 2) B(2, 3, 5) C (1, 5, 5) ? ? 2 OA i j k = + + ? ? ? ? ? 2 3 5 5 5 OB i j k OC i j k = + + = + + ? ? ? ? ? ? 2 2 4 3 OA OB OA i j k AC OC OA j k = - = + + = - = + ? Ë† Ë† Ë† 1 2 2 0 4 3 Ë† Ë† Ë† 2 3 4 i j k AB AC i j k × = = - - + ( ) ( ) ( ) 2 2 2 1 2 1 2 3 4 2 1 29 . 2 Arae of ABC AB AC sq unit ? = × = - + - + =Read More

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