EXAM PRACTICE CBSE PREVIOUS YEARS’ QUESTIONS Chapter Name: Vectors Subject - Mathematics Class: XII Section A (1 mark each) Question 1. Write a unit vector in the direction of a = 2i - 6j + 3k. Question 2. Write the value of p so that a = 3i + 2j + 9k and b = i + pj + 3k are parallel vectors. Question 3. Find a unit vector perpendicular to each of the vectors a + b and a - b, where a =3i + 2j + 2k and b = i + 2j - 2k. Question 4. Write a vector in the direction of vector -2i + j + 2k that has magnitude 9 units. Question 5. For what value of ‘a’, the vectors 2i - 3j + 4k and ai + 6j - 8k are collinear? Question 6. Write the projection of the vector i - j on the vector i + j. Question 7. Write the value of (i x j) . k +(i . j). Question 8. Find the sum of vectors a = i - 2j + k, b = -2i + 4j + 5k and c = i - 6j -7k. Question 9. Write the value of (i x j).k +(j x k).i . Question 10. Find the area of the parallelogram determined by the vectors 2i and 3j. Question 11. What is the cosine of the angle which the vector 2i + j + k makes with y-axis? Question 12. Find the scalar components of the vector AB with initial point A (2,1) and terminal point B (-5,7). Question 13. Find the projection of a on b, if a. b = 8 and b = 2i + 6j + 3k. Question 14. If a = xi + 2j - zk and b = 3i - yj + k are two equal vectors, then write the value of x + y + z. Question 15. p p If a unit vector p makes angles with i, with j and an acute angle q with k, then find the 3 4 value of q. Section B Question 16. (4 marks each) If a x b = c x d and a x c = b x d, show that a - d is parallel to b - c, where a = d and b = c. Question 17. Find the position vector of a point R which divides the line externally in the ratio 1:2, joining the two points P and Q with position vectors as (2a + b) and (a - 3b) respectively. Also, show that P is the mid point of the line segment RQ. Question 18. The point A (4,5,10), B(2,3,4) and C (1, 2, -1) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D. Question 19. If a = i + j + k, b = 4i - 2j+3k and c = i - 2j - k, find a vector of magnitude 6 units which is parallel to the vector 2a - b = 3c. Question 20. If a = i + 4j + 2k, b = 3i - 2j + 7k and c = 2i - j + 4k, find the vector d which is perpendicular to both a and b. Also, c . d = 18. Question 21. Using vectors, find the area of the triangle with vertices A (1,1,2), B (2,3,5) and C (1,5,5). Question 22. The line passing through the points A (3,4,1) and B (5,1,6) crosses the XY-plane at point C. Find the co-ordinates of this point. Question 23. Find ‘l’, if the projection of a = li + j + 4k on b = 2i + 6j + 3k is 4 units. Question 24. If a, b, c are three vectors such that a = 5, b = 12 and c = 13, and a + b + c = 0, find the value of a.b + b.c + c.a . Question 25. If a = 3i + 4j + 5k and b = 2i + j - 4k, then express b in the form b = b1 + b2, where b1 is parallel to a and b2 is perpendicular to a. Question 26. If a and b are two vectors such that a + b = a , then prove that vector 2a + b is perpendicular to vector b.
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