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12th Maths Sectionwise paper - 04 (Vectors, 3D, Probability, LP)

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MATHEMATICS Class II Pre-Uni Plus 2016-17 Topics Board Section-wise Test - 04 Vector Algebra, Probability, Three dimensional Geometry, Linear programming Max. Marks 100 Duration Date 3 hrs 15 min 03-01-2017 Instructions 1. Do not overwrite. 2. Draw a line using pen at the end of each answer. 3. Steps adopted in solving problems should be clearly shown. 4. In case of violation of the above conditions, the answer scripts will not be considered for valuation. PART - A I. Answer all the ten questions [10 1 = 10] 1. Find the unit vector in the direction of a b given that a 2i 2 j 5k and b 4i 2 j 3k . 2. Find the projection of a 5i 2 j 3k on b 4i 2 j 5k . 3. If the vectors i 2 j k and i 3j 2k are orthogonal. Find . 4. Find the vector equation of the line passing through the origin and the point (1, 3, 2). 5. Find the distance of the point (1, 2, 3) from the plane r 2i 3j 4k 5 . 6. Find the equation of the plane through the point (2, 3, 4) and parallel to the plane 5x 6y + 5z = 6. 7. If P A 0.7, P B 0.6 and P(B | A) 0.5 find P(A B). 8. A family has two children. What is the probability that both the children are girls given that at least one of them is a girl? 9. Define feasible region in a linear programming problem. 10. Define optimal solution of a linear programming problem. PART - B II. Answer any TEN of the following questions [10 2 = 20] 11. If a and b are unit vectors inclined at an angle . Show that a b 2cos . 2 12. If a 2, 3, 1 and b 3, 2,5 find a b 2a b . 13. If a, b, c are unit vectors such that a b c 0 . Find the value of a b b c c a . 14. For any two vectors a and b show that a b a b . 15. Find the area of parallelogram whose 3i j 5k and 2i 4 j 3k . adjacent sides are represented by the k i . 16. Evaluate i j, j k, 17. If cos , cos , cos are the direction cosines of a line, show that sin 2 sin 2 sin 2 2 . 18. Find the direction cosines of the line passing through the points ( 3, 1, 2) and (2, 3, 5). 2P1617M(B)SWT4 1 vectors

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