past year question.cdr

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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