Question 1.
The position vector of the point (1, 0, 2) is
(a) \(\vec{i}\) +\(\vec{j}\) + 2\(\vec{k}\)
(b) \(\vec{i}\) + 2\(\vec{j}\)
(c) \(\vec{2}\) + 3\(\vec{k}\)
(d) \(\vec{i}\) + 2\(\vec{K}\)

Answer

Answer: (d) \(\vec{i}\) + 2\(\vec{K}\)

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Question 2.
The modulus of 7\(\vec{i}\) – 2\(\vec{J}\) + \(\vec{K}\)
(a) \(\sqrt{10}\)
(b) \(\sqrt{55}\)
(c) 3\(\sqrt{6}\)
(d) 6

Answer

Answer: (c) 3\(\sqrt{6}\)

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Question 3.
If O be the origin and \(\vec{OP}\) = 2\(\hat{i}\) + 3\(\hat{j}\) – 4\(\hat{k}\) and \(\vec{OQ}\) = 5\(\hat{i}\) + 4\(\hat{j}\) -3\(\hat{k}\), then \(\vec{PQ}\) is equal to
(a) 7\(\hat{i}\) + 7\(\hat{j}\) – 7\(\hat{k}\)
(b) -3\(\hat{i}\) + \(\hat{j}\) – \(\hat{k}\)
(c) -7\(\hat{i}\) – 7\(\hat{j}\) + 7\(\hat{k}\)
(d) 3\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)

Answer

Answer: (d) 3\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)

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Question 4.
The scalar product of 5\(\hat{i}\) + \(\hat{j}\) – 3\(\hat{k}\) and 3\(\hat{i}\) – 4\(\hat{j}\) + 7\(\hat{k}\) is
(a) 10
(b) -10
(c) 15
(d) -15

Answer

Answer: (b) -10

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Question 5.
If \(\vec{a}\).\(\vec{b}\) = 0, then
(a) a ⊥ b
(b) \(\vec{a}\) || \(\vec{b}\)
(c) \(\vec{a}\) + \(\vec{b}\) = 0
(d) \(\vec{a}\) – \(\vec{b}\) = 0

Answer

Answer: (a) a ⊥ b

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Question 6.
\(\vec{i}\) – \(\vec{j}\) =
(a) 0
(b) 1
(c) \(\vec{k}\)
(d) –\(\vec{k}\)

Answer

Answer: (a) 0

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Question 7.
\(\vec{k}\) × \(\vec{j}\) =
(a) 0
(b) 1
(c) \(\vec{i}\)
(d) –\(\vec{i}\)

Answer

Answer: (d) –\(\vec{i}\)

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Question 8.
\(\vec{a}\). \(\vec{a}\) =
(a) 0
(b) 1
(c) |\(\vec{a}\)|²
(d) |\(\vec{a}\)|

Answer

Answer: (c) |\(\vec{a}\)|²

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Question 9.
The projection of the vector 2\(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\) on the vector \(\hat{i}\) – 2\(\hat{j}\) + \(\hat{k}\) is
(a) \(\frac{4}{√6}\)
(b) \(\frac{5}{√6}\)
(c) \(\frac{4}{√3}\)
(d) \(\frac{7}{√6}\)

Answer

Answer: (b) \(\frac{5}{√6}\)

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Question 10.
If \(\vec{a}\) = \(\vec{i}\) – \(\vec{j}\) + 2\(\vec{k}\) and b = 3\(\vec{i}\) + 2\(\vec{j}\) – \(\vec{k}\) then the value of (\(\vec{a}\) + 3\(\vec{b}\))(2\(\vec{a}\) – \(\vec{b}\))=.
(a) 15
(b) -15
(c) 18
(d) -18

Answer

Answer: (b) -15

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Question 11.
If |\(\vec{a}\)|= \(\sqrt{26}\), |b| = 7 and |\(\vec{a}\) × \(\vec{b}\)| = 35, then \(\vec{a}\).\(\vec{b}\) =
(a) 8
(b) 7
(c) 9
(d) 12

Answer

Answer: (b) 7

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Question 12.
If \(\vec{a}\) = 2\(\vec{i}\) – 3\(\vec{j}\) + 4\(\vec{k}\) and \(\vec{b}\) = \(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\) then \(\vec{a}\) + \(\vec{b}\) =
(a) \(\vec{i}\) + \(\vec{j}\) + 3\(\vec{k}\)
(b) 3\(\vec{i}\) – \(\vec{j}\) + 5\(\vec{k}\)
(c) \(\vec{i}\) – \(\vec{j}\) – 3\(\vec{k}\)
(d) 2\(\vec{i}\) + \(\vec{j}\) + \(\vec{k}\)

Answer

Answer: (b) 3\(\vec{i}\) – \(\vec{j}\) + 5\(\vec{k}\)

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Question 13.
If \(\vec{a}\) = \(\vec{i}\) + 2\(\vec{j}\) + 3\(\vec{k}\) and \(\vec{b}\) = 3\(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\), then cos θ =
(a) \(\frac{6}{7}\)
(b) \(\frac{5}{7}\)
(c) \(\frac{4}{7}\)
(d) \(\frac{1}{2}\)

Answer

Answer: (b) \(\frac{5}{7}\)

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Question 14.
If |\(\vec{a}\) + \(\vec{b}\)| = |\(\vec{a}\) – \(\vec{b}\)|, then
(a) \(\vec{a}\) || \(\vec{a}\)
(b) \(\vec{a}\) ⊥ \(\vec{b}\)
(c) |\(\vec{a}\)| = |\(\vec{b}\)|
(d) None of these

Answer

Answer: (b) \(\vec{a}\) ⊥ \(\vec{b}\)

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Question 15.
The projection of the vector 2\(\hat{i}\) + 3\(\hat{j}\) – 6\(\hat{k}\) on the line joining the points (3, 4, 2) and (5, 6,3) is
(a) \(\frac{2}{3}\)
(b) \(\frac{4}{3}\)
(c) –\(\frac{4}{3}\)
(d) \(\frac{5}{3}\)

Answer

Answer: (b) \(\frac{4}{3}\)

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Question 16.
If |\(\vec{a}\) × \(\vec{b}\)| – |\(\vec{a}\).\(\vec{b}\)|, then the angle between \(\vec{a}\) and \(\vec{b}\), is
(a) 0
(b) \(\frac{π}{2}\)
(c) \(\frac{π}{4}\)
(d) π

Answer

Answer: (c) \(\frac{π}{4}\)

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Question 17.
The angle between two vector \(\vec{a}\) and \(\vec{b}\) with magnitude √3 and 4, respectively and \(\vec{a}\).\(\vec{b}\) = 2√3 is
(a) \(\frac{π}{6}\)
(b) \(\frac{π}{3}\)
(c) \(\frac{π}{2}\)
(d) \(\frac{5π}{2}\)

Answer

Answer: (b) \(\frac{π}{3}\)

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Question 18.
Unit vector perpendicular to each of the vector 3\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\) and 2\(\hat{i}\) – 2\(\hat{j}\) + 4\(\hat{k}\) is
(a) \(\frac{\hat{i}+\hat{j}+\hat{k}}{√3}\)
(b) \(\frac{\hat{i}-\hat{j}+\hat{k}}{√3}\)
(c) \(\frac{\hat{i}-\hat{j}-\hat{k}}{√3}\)
(d) \(\frac{\hat{i}+\hat{j}-\hat{k}}{√3}\)

Answer

Answer: (c) \(\frac{\hat{i}-\hat{j}-\hat{k}}{√3}\)

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Question 19.
If \(\vec{a}\) = 2\(\vec{i}\) – 5\(\vec{j}\) + k and \(\vec{b}\) = 4\(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\) then \(\vec{a}\).\(\vec{b}\) =
(a) 0
(b) -1
(c) 1
(d) 2

Answer

Answer: (b) -1

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Question 20.
If 2\(\vec{i}\) + \(\vec{j}\) + \(\vec{k}\), 6\(\vec{i}\) – \(\vec{j}\) + 2\(\vec{k}\) and 14\(\vec{i}\) – 5\(\vec{j}\) + 4\(\vec{k}\) be the position vector of the points A, B and C respectively, then
(a) The A, B and C are collinear
(b) A, B and C are not colinear
(c) \(\vec{AB}\) ⊥ \(\vec{BC}\)
(d) None of these

Answer

Answer: (a) The A, B and C are collinear

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Question 21.
According to the associative lass of addition of addition of s ector
(\(\vec{a}\) + …….) + \(\vec{c}\) = …… + (\(\vec{b}\) + \(\vec{c}\))
(a) \(\vec{b}\), \(\vec{a}\)
(b) \(\vec{a}\), \(\vec{b}\)
(c) \(\vec{a}\), 0
(d) \(\vec{b}\), 0

Answer

Answer: (a) \(\vec{b}\), \(\vec{a}\)

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Question 22.
Which one of the following can be written for (\(\vec{a}\) – \(\vec{b}\)) × (\(\vec{a}\) + \(\vec{b}\))
(a) \(\vec{a}\) × \(\vec{b}\)
(b) 2\(\vec{a}\) × \(\vec{b}\)
(c) \(\vec{a}\)² – \(\vec{b}\)
(d) 2\(\vec{b}\) × \(\vec{b}\)

Answer

Answer: (b) 2\(\vec{a}\) × \(\vec{b}\)

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Question 23.
The points with position vectors (2. 6), (1, 2) and (a, 10) are collinear if the of a is
(a) -8
(b) 4
(c) 3
(d) 12

Answer

Answer: (c) 3

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Question 24.
|\(\vec{a}\) + \(\vec{b}\)| = |\(\vec{a}\) – \(\vec{b}\)| then the angle between \(\vec{a}\) and \(\vec{b}\)
(a) \(\frac{π}{2}\)
(b) 0
(c) \(\frac{π}{4}\)
(d) \(\frac{π}{6}\)

Answer

Answer: (a) \(\frac{π}{2}\)

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Question 25.
|\(\vec{a}\) × \(\vec{b}\)| = |\(\vec{a}\).\(\vec{b}\)| then the angle between \(\vec{a}\) and \(\vec{b}\)
(a) 0
(b) \(\frac{π}{2}\)
(c) \(\frac{π}{4}\)
(d) π

Answer

Answer: (a) 0

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Question 26.
If ABCDEF is a regular hexagon then \(\vec{AB}\) + \(\vec{EB}\) + \(\vec{FC}\) equals
(a) zero
(b) 2\(\vec{AB}\)
(c) 4\(\vec{AB}\)
(d) 3\(\vec{AB}\)

Answer

Answer: (d) 3\(\vec{AB}\)

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Question 27.
Which one of the following is the modulus of x\(\hat{i}\) + y\(\hat{j}\) + z\(\hat{k}\)?
(a) \(\sqrt{x^2+y^2+z^2}\)
(b) \(\frac{1}{\sqrt{x^2+y^2+z^2}}\)
(c) x² + y² + z²
(d) none of these

Answer

Answer: (a) \(\sqrt{x^2+y^2+z^2}\)

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Question 28.
If C is the mid point of AB and P is any point outside AB then,
(a) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\)
(b) \(\vec{PA}\) + \(\vec{PB}\) = \(\vec{PC}\)
(c) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\) = 0
(d) None of these

Answer

Answer: (a) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\)

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Question 29.
If \(\vec{OA}\) = 2\(\vec{i}\) – \(\vec{j}\) + \(\vec{k}\), \(\vec{OB}\) = \(\vec{i}\) – 3\(\vec{j}\) – 5\(\vec{k}\) then |\(\vec{OA}\) × \(\vec{OB}\)| =
(a) 8\(\vec{i}\) + 11\(\vec{j}\) – 5\(\vec{k}\)
(b) \(\sqrt{210}\)
(c) sin θ
(d) \(\sqrt{40}\)

Answer

Answer: (b) \(\sqrt{210}\)

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Question 30.
If |a| = |b| = |\(\vec{a}\) + \(\vec{b}\)| = 1 then |\(\vec{a}\) – \(\vec{b}\)| is equal to
(a) 1
(b) √3
(c) 0
(d) None of these

Answer

Answer: (b) √3

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Question 31.
If \(\vec{a}\) and \(\vec{b}\) are any two vector then (\(\vec{a}\) × \(\vec{b}\))² is equal to
(a) (\(\vec{a}\))²(\(\vec{b}\))² – (\(\vec{a}\).\(\vec{b}\))²
(b) (\(\vec{a}\))² (\(\vec{b}\))² + (\(\vec{a}\).\(\vec{b}\))²
(c) (\(\vec{a}\).\(\vec{b}\))²
(d) (\(\vec{a}\))²(\(\vec{b}\))²

Answer

Answer: (a) (\(\vec{a}\))²(\(\vec{b}\))² – (\(\vec{a}\).\(\vec{b}\))²

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Question 32.
If \(\hat{a}\) and \(\hat{b}\) be two unit vectors and 0 is the angle between them, then |\(\hat{a}\) – \(\hat{b}\)| is equal to
(a) sin \(\frac{θ}{2}\)
(b) 2 sin \(\frac{θ}{2}\)
(c) cos \(\frac{θ}{2}\)
(d) 2 cos \(\frac{θ}{2}\)

Answer

Answer: (b) 2 sin \(\frac{θ}{2}\)

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Question 33.
The angle between the vector 2\(\hat{i}\) + 3\(\hat{j}\) + \(\hat{k}\) and 2\(\hat{i}\) – \(\hat{j}\) – \(\hat{k}\) is
(a) \(\frac{π}{2}\)
(b) \(\frac{π}{4}\)
(c) \(\frac{π}{3}\)
(d) 0

Answer

Answer: (a) \(\frac{π}{2}\)

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Question 34.
If \(\vec{a}\) = \(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\) = \(\hat{i}\) + 2\(\hat{j}\) – \(\hat{k}\), \(\vec{c}\) = 3\(\hat{i}\) – p\(\hat{j}\) – 5\(\hat{k}\) are coplanar then P =
(a) 6
(b) -6
(c) 2
(d) -2

Answer

Answer: (a) 6

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Question 35.
The distance of the point (- 3, 4, 5) from the origin
(a) 50
(b) 5√2
(c) 6
(d) None of these

Answer

Answer: (b) 5√2

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Question 36.
If \(\vec{AB}\) = 2\(\hat{i}\) + \(\hat{j}\) – 3\(\hat{k}\) and the co-ordinates of A are (1, 2, -1) then coordinate of B are
(a) (2, 2, -3)
(b) (3, 2, -4)
(c) (4, 2, -1)
(d) (3, 3, -4)

Answer

Answer: (d) (3, 3, -4)

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Question 37.
If \(\vec{b}\) is a unit vector in xy-plane making an angle of \(\frac{π}{4}\) with x-axis. then \(\vec{b}\) is equal to
(a) \(\hat{i}\) + \(\hat{j}\)
(b) \(\vec{i}\) – \(\vec{j}\)
(c) \(\frac{\vec{i}+\vec{j}}{√2}\)
(d) \(\frac{\vec{i}-\vec{j}}{√2}\)

Answer

Answer: (c) \(\frac{\vec{i}+\vec{j}}{√2}\)

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Question 38.
\(\vec{a}\) = 2\(\hat{i}\) + \(\hat{j}\) – 8\(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) + 3\(\hat{j}\) – 4\(\hat{k}\) then the magnitude of \(\vec{a}\) + \(\vec{b}\) is equal to
(a) 13
(b) \(\frac{13}{4}\)
(c) \(\frac{3}{13}\)
(d) \(\frac{4}{13}\)

Answer

Answer: (a) 13

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Question 39.
The vector in the direction of the vector \(\hat{i}\) – 2\(\hat{j}\) + 2\(\hat{k}\) that has magnitude 9 is
(a) \(\hat{i}\) – 2\(\hat{j}\) + 2\(\hat{k}\)
(b) \(\frac{\hat{i}-2\hat{j}+2\hat{k}}{3}\)
(c) 3(\(\hat{i}\) – 2\(\hat{j}\) + 2\(\hat{k}\))
(d) 9(\(\hat{i}\) – 2\(\hat{j}\) + 2\(\hat{k}\))

Answer

Answer: (c) 3(\(\hat{i}\) – 2\(\hat{j}\) + 2\(\hat{k}\))

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Question 40.
The position vector of the point which divides the join of points 2\(\vec{a}\) – 3\(\vec{b}\) and \(\vec{a}\) + \(\vec{b}\) in the ratio 3 : 1 is
(a) \(\frac{3\vec{a}-2\vec{b}}{2}\)
(b) \(\frac{7\vec{a}-8\vec{b}}{2}\)
(c) \(\frac{3\vec{a}}{2}\)
(d) \(\frac{5\vec{a}}{4}\)

Answer

Answer: (d) \(\frac{5\vec{a}}{4}\)

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Question 41.
The vector having, initial and terminal points as (2, 5, 0) and (- 3, 7, 4) respectively is
(a) –\(\hat{i}\) + 12\(\hat{j}\) + 4\(\hat{k}\)
(b) 5\(\hat{i}\) + 2\(\hat{j}\) – 4\(\hat{k}\)
(c) -5\(\hat{i}\) + 2\(\hat{j}\) + 4\(\hat{k}\)
(d) \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)

Answer

Answer: (c) -5\(\hat{i}\) + 2\(\hat{j}\) + 4\(\hat{k}\)

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Question 42.
Find the value of λ such that the vectors \(\vec{a}\) = 2\(\hat{i}\) + λ\(\hat{j}\) + \(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) + 2\(\hat{j}\) + 3\(\hat{k}\) are orthogonal
(a) 0
(b) 1
(c) \(\frac{3}{2}\)
(d) –\(\frac{5}{2}\)

Answer

Answer: (d) –\(\frac{5}{2}\)

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Question 43.
The value of λ for which the vectors 3\(\hat{i}\) – 6\(\hat{j}\) + \(\hat{k}\) and 2\(\hat{i}\) – 4\(\hat{j}\) + λ\(\hat{k}\) are parallel is
(a) \(\frac{2}{3}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{5}{2}\)
(d) –\(\frac{2}{5}\)

Answer

Answer: (a) \(\frac{2}{3}\)

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Question 44.
The vectors from origin to the points A and B are \(\vec{a}\) = 2\(\hat{i}\) – 3\(\hat{j}\) +2\(\hat{k}\) and \(\vec{b}\) = 2\(\hat{i}\) + 3\(\hat{j}\) + \(\hat{k}\) respectively, then the area of triangle OAB is
(a) 340
(b) \(\sqrt{25}\)
(c) \(\sqrt{229}\)
(d) \(\frac{1}{2}\) \(\sqrt{229}\)

Answer

Answer: (d) \(\frac{1}{2}\) \(\sqrt{229}\)

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Question 45.
For any vector \(\vec{a}\) the value of (\(\vec{a}\) × \(\vec{i}\))² + (\(\vec{a}\) × \(\hat{j}\))² + (\(\vec{a}\) × \(\hat{k}\))² is equal to
(a) \(\vec{a}\)²
(b) 3\(\vec{a}\)²
(c) 4\(\vec{a}\)²
(d) 2\(\vec{a}\)²

Answer

Answer: (d) 2\(\vec{a}\)²

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Question 46.
If |\(\vec{a}\)| = 10, |\(\vec{b}\)| = 2 and \(\vec{a}\).\(\vec{b}\) = 12, then the value of |\(\vec{a}\) × \(\vec{b}\)| is
(a) 5
(b) 10
(c) 14
(d) 16

Answer

Answer: (d) 16

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Question 47.
The vectors λ\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\), \(\hat{i}\) + λ\(\hat{j}\) – \(\hat{k}\) and 2\(\hat{i}\) – \(\hat{j}\) + λ\(\hat{k}\) are coplanar if
(a) λ = -2
(b) λ = 0
(c) λ = 1
(d) λ = -1

Answer

Answer: (a) λ = -2

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Question 48.
If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are unit vectors such that \(\vec{a}\) + \(\vec{b}\) + \(\vec{c}\) = \(\vec{0}\), then the value of \(\vec{a}\).\(\vec{b}\) + \(\vec{b}\).\(\vec{c}\) + \(\vec{c}\).\(\vec{a}\)
(a) 1
(b) 3
(c) –\(\frac{3}{2}\)
(d) None of these

Answer

Answer: (c) –\(\frac{3}{2}\)

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Question 49.
Projection vector of \(\vec{a}\) on \(\vec{b}\) is
(a) (\(\frac{\vec{a}.\vec{b}}{|\vec{b}|^2}\))\(\vec{b}\)
(b) \(\frac{\vec{a}.\vec{b}}{|\vec{b}|}\)
(c) \(\frac{\vec{a}.\vec{b}}{|\vec{a}|}\)
(d) (\(\frac{\vec{a}.\vec{b}}{|\vec{a}|^2}\))\(\hat{b}\)

Answer

Answer: (b) \(\frac{\vec{a}.\vec{b}}{|\vec{b}|}\)

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Question 50.
If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are three vectors such that \(\vec{a}\) + \(\vec{b}\) + \(\vec{c}\) = 5 and |\(\vec{a}\)| = 2, |\(\vec{b}\)| = 3, |\(\vec{c}\)| = 5, then the value of \(\vec{a}\).\(\vec{b}\) +\(\vec{b}\).\(\vec{c}\) + \(\vec{c}\).\(\vec{a}\) is
(a) 0
(b) 1
(c) -19
(d) 38

Answer

Answer: (c) -19

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Question 51.
If |\(\vec{a}\)| 4 and – 3 ≤ λ ≤ 2, then the range of |λ\(\vec{a}\)| is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]

Answer

Answer: (b) [-12, 8]

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Question 52.
The number of vectors of unit length perpendicular to the vectors \(\vec{a}\) = 2\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\) and \(\vec{b}\) = \(\hat{j}\) + \(\hat{k}\) is
(a) one
(b) two
(c) three
(d) infinite

Answer

Answer: (b) two

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Question 53.
If (\(\frac{1}{2}\), \(\frac{1}{3}\), n) are the direction cosines of a line, then the value of n is
(a) \(\frac{\sqrt{23}}{6}\)
(b) \(\frac{23}{6}\)
(c) \(\frac{2}{3}\)
(d) –\(\frac{3}{2}\)

Answer

Answer: (a) \(\frac{\sqrt{23}}{6}\)

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Question 54.
Find the magnitude of vector 3\(\hat{i}\) + 2\(\hat{j}\) + 12\(\hat{k}\)
(a) \(\sqrt{157}\)
(b) 4\(\sqrt{11}\)
(c) \(\sqrt{213}\)
(d) 9√3

Answer

Answer: (a) \(\sqrt{157}\)

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Question 55.
Three points (2, -1, 3), (3, – 5, 1) and (-1, 11, 9) are
(a) Non-collinear
(b) Non-coplanar
(c) Collinear
(d) None of these

Answer

Answer: (c) Collinear

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Question 56.
The vectors 3\(\hat{i}\) + 5\(\hat{j}\) + 2\(\hat{k}\), 2\(\hat{i}\) – 3\(\hat{j}\) – 5\(\hat{k}\) and 5\(\hat{i}\) + 2\(\hat{j}\) – 3\(\hat{k}\) form the sides of
(a) Isosceles triangle
(b) Right triangle
(c) Scalene triangle
(d) Equilateral triangle

Answer

Answer: (a) Isosceles triangle

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Question 57.
The points with position vectors 60\(\hat{i}\) + 3\(\hat{j}\), 40\(\hat{i}\) – 8\(\hat{j}\) and a\(\hat{i}\) – 52\(\hat{j}\) are collinear if
(a) a = -40
(b) a = 40
(c) a = 20
(d) None of these

Answer

Answer: (a) a = -40

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Question 58.
The ratio in which 2x + 3y + 5z = 1 divides the line joining the points (1, 0, -3) and (1, -5, 7) is
(a) 5 : 3
(b) 3 : 2
(c) 2 : 1
(d) 1 : 3

Answer

Answer: (a) 5 : 3

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Question 59.
If O is origin and C is the mid point of A (2, -1) and B (-4, 3) then the value of \(\bar{OC}\) is
(a) \(\hat{i}\) + \(\hat{j}\)
(b) \(\hat{i}\) – \(\hat{j}\)
(c) –\(\hat{i}\) + \(\hat{j}\)
(d) –\(\hat{i}\) – \(\hat{j}\)

Answer

Answer: (c) –\(\hat{i}\) + \(\hat{j}\)

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Question 60.
If ABCDEF is regular hexagon, then \(\vec{AD}\) + \(\vec{EB}\) + \(\vec{FC}\) is equal
(a) 0
(b) 2\(\vec{AB}\)
(c) 3\(\vec{AB}\)
(d) 4\(\vec{AB}\)

Answer

Answer: (d) 4\(\vec{AB}\)

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Question 61.
If \(\vec{a}\) = \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\) = 2\(\hat{i}\) – 4\(\hat{k}\), \(\vec{c}\) = \(\hat{i}\) + λ\(\hat{j}\) + 3\(\hat{j}\) are coplanar, then the value of λ is
(a) \(\frac{5}{2}\)
(b) \(\frac{3}{5}\)
(c) \(\frac{7}{3}\)
(d) –\(\frac{5}{3}\)

Answer

Answer: (d) –\(\frac{5}{3}\)

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Question 62.
The vectors \(\vec{a}\) = x\(\hat{i}\) – 2\(\hat{j}\) + 5\(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) + y\(\hat{j}\) – z\(\hat{k}\) are collinear, if
(a) x = 1, y = -2, z = -5
(b) x = \(\frac{3}{2}\), y = -4, z = -10
(c) x = \(\frac{3}{2}\), y = 4, z = 10
(d) All of these

Answer

Answer: (d) All of these

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Question 63.
The vectors (x, x + 1, x + 2), (x + 3, x + 4, x + 5) and (x + 6, x + 7, x + 8) are coplanar for
(a) all values of x
(b) x < 0
(c) x ≤ 0
(d) None of these

Answer

Answer: (a) all values of x

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Question 64.
The vectors \(\vec{AB}\) = 3\(\hat{i}\) +4\(\hat{k}\) and \(\vec{AC}\) = 5\(\hat{i}\) – 2\(\hat{j}\) + 4\(\hat{k}\) are the sides of ΔABC. The length of the median through A is
(a) \(\sqrt{18}\)
(b) \(\sqrt{72}\)
(c) \(\sqrt{33}\)
(d) \(\sqrt{288}\)

Answer

Answer: (c) \(\sqrt{33}\)

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Question 65.
The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vector is
(a) √3
(b) 1 – √3
(c) 1 + √3
(d) -√3

Answer

Answer: (a) √3

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