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Vector Algebra Class 12 Maths MCQs Pdf
Question 1.
Answer:
(c) \(\frac{3 \hat{i}-2 \hat{j}+6 \hat{k}}{7}\)
Question 2.
The area of parallelogram whose adjacent sides are \(\hat{i}-2 \hat{j}+3 \hat{k}\) and \(2 \hat{i}+\hat{j}-4 \hat{k}\) is
(a) 10√6
(b) 5√6
(c) 10√3
(d) 5√3
Answer:
(b) 5√6
Question 3.
If AB × AC = \(2 \hat{i}-4 \hat{j}+4 \hat{k}\), then the are of ΔABC is
(a) 3 sq. units
(b) 4 sq. units
(c) 16 sq. units
(d) 9 sq. units
Answer:
(a) 3 sq. units
Question 4.
Answer:
(a) \(\frac{5 \sqrt{3}}{3}(\hat{i}+\hat{j}+\hat{k})\)
Question 5.
|a × b|2 + |a.b|2 = 144 and |a| = 4, then |b| is equal to
(a) 12
(b) 3
(c) 8
(d) 4
Answer:
(b) 3
Question 6.
If |a × b| = 4 and |a.b| = 2, then |a|2 |b|2 is equal to
(a) 2
(b) 6
(c) 8
(d) 20
Answer:
(d) 20
Question 7.
Answer:
(c) \(\hat{i}\)
Question 8.
The two vectors a = \(2 \hat{i}+\hat{j}+3 \hat{k}\) and b = 4 \hat{i}-\lambda \hat{j}+6 \hat{k} ae parallel, if λ is equal to
(a) 2
(b) -3
(c) 3
(d) 2
Answer:
(d) 2
Question 9.
If |a|= 5, |b|= 13 and |a × b|= 25, find a.b
(a) ±10
(b) ±40
(c) ±60
(d) ±25
Answer:
(c) ±60
Question 10.
Find the value of λ so that the vectors \(2 i-4 \hat{j}+\hat{k}\) and \(4 i-8 \hat{j}+\lambda \hat{k}\) are parallel.
(a) -1
(b) 3
(c) -4
(d) 2
Answer:
(d) 2
Question 11.
If O is origin and C is the mid point of A(2, -1) and B(-4, 3), then the value of OC is
(a) \(\hat{i}+\hat{j}\)
(b) \(\hat{i}-\hat{j}\)
(c) \(-\hat{i}+\hat{j}\)
(d) \(-\hat{i}-\hat{j}\)
Answer:
(c) \(-\hat{i}+\hat{j}\)
Question 12.
The vectors AB = \(3 \hat{i}+4 \hat{k}\) and AC = \(A C=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the side of a ΔABC. The length of the median through A is
(a) √18
(b) √72
(c) √33
(d) √288
Answer:
(c) √33
Question 13.
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:
(a) √3
Question 14.
Answer:
(d) \(\frac{1}{\sqrt{6}}(2 \hat{i}-\hat{j}+\hat{k})\)
Question 15.
Answer:
(c) \(\pi \geq \theta>\frac{2 \pi}{3}\)
Question 16.
The value of λ for which the vectors \(3 \hat{i}-6 \hat{j}+\hat{k}\) and \(2 \hat{i}-4 \hat{j}+\lambda \hat{k}\) are parallel is
(a) \(\frac{2}{3}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{5}{2}\)
(d) \(\frac{2}{5}\)
Answer:
(a) \(\frac{2}{3}\)
Question 17.
The vectors from origin to the points A and B are a = \(2 \hat{i}-3 \hat{j}+2 \hat{k}\) and b = \(2 \hat{i}+3 \hat{j}+\hat{k}\), respectively then the area of triangle OAB is
(a) 340
(b) √25
(c) √229
(d) \(\frac{1}{2}\) √229
Answer:
(d) \(\frac{1}{2}\) √229
Question 18.
The vectors \(\lambda \hat{i}+\hat{j}+2 \hat{k}, \hat{i}+\lambda \hat{j}-\hat{k}\) and \(2 \hat{i}-\hat{j}+\lambda \hat{k}\) are coplanar if
(a) λ = -2
(b) λ = 0
(c) λ = 1
(d) λ = -1
Answer:
(a) λ = -2
Question 19.
If a, b, c are unit vectors such that a + b + c = 0, then the value of a.b + b.c + c.a is
(a) 1
(b) 3
(c) \(-\frac{3}{2}\)
(d) None of these
Answer:
(c) \(-\frac{3}{2}\)
Question 20.
If |a| = 4 and -3 ≤ λ ≤ 2, then the range of |λa| is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]
Answer:
(c) [0, 12]
Question 21.
The number of vectors of unit length perpendicular to the vectors a = \(2 \hat{i}+\hat{j}+2 \hat{k}\) and b = \(\hat{j}+\hat{k}\) is
(a) one
(b) two
(c) three
(d) infinite
Answer:
(b) two
Question 22.
Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a.b + b.c + c.a is
(a) 47
(b) 25
(c) 50
(d) -25
Answer:
(d) -25
Question 23.
If |a| = |b| = 1 and |a + b| = √3, then the value of (3a – 4b).(2a + 5b) is
(a) -21
(b) \(-\frac{21}{2}\)
(c) 21
(d) \(\frac{21}{2}\)
Answer:
(b) \(-\frac{21}{2}\)
Question 24.
Answer:
(c) \(\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})\)
Question 25.
If |a – b| = |a| = |b| = 1, then the angle between a and b is
(a) \(\frac{\pi}{3}\)
(b) \(\frac{3 \pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) 0
Answer:
(a) \(\frac{\pi}{3}\)
Question 26.
Answer:
(d) |a|2
Question 27.
a, b, c are three vectors, such that a + b + c = 0, |a|= 1, |b|= 2, |c|= 3, then a.b + b.c + c is equal to
(a) 0
(b) -7
(c) 7
(d) 1
Answer:
(b) -7
Question 28.
If |a + b| = |a – b|, then angle between a and b is (a ≠ 0, b ≠ 0)
(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{6}\)
(c) \(\frac{\pi}{4}\)
(d) \(\frac{\pi}{2}\)
Answer:
(d) \(\frac{\pi}{2}\)
Question 29.
If a and b are two unit vectors inclined to x-axis at angles 30° and 120° respectively, then |a + b| equals
(a) \(\sqrt{\frac{2}{3}}\)
(b) √2
(c) √3
(d) 2
Answer:
(d) 2
Question 30.
If the angle between \(\hat{i}+\hat{k}\) and \(\hat{i}+\hat{j}+a \hat{k}\) is \(\frac{\pi}{3}\), then the value of a is
(a) 0 or 2
(b) -4 or 0
(c) 0 or -3
(d) 2 or -2
Answer:
(b) -4 or 0
Question 31.
The length of longer diagronai of the parallelogram constructed on 5a + 2b and a – 3b. If it is given that
|a| = 2√2, |b| = 3 and angle between a and b is \(\frac{\pi}{4}\), is
(a) 15
(b) √113
(c) √593
(d) √369
Answer:
(c) √593
Question 32.
If \(\left(\frac{1}{2}, \frac{1}{3}, n\right)\) 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:
(a) \(\frac{\sqrt{23}}{6}\)
Question 33.
Find the magnitude of vector \(3 \hat{i}+2 \hat{j}+12 \hat{k}\).
(a) √157
(b) 4√11
(c) √213
(d) 9√3
Answer:
(a) √157
Direction (34 – 36): Study the given parallelogram and answer the following questions.
Question 34.
Which of the following represents equal vectors?
(a) a, c
(b) b, d
(c) b, c
(d) m, d
Answer:
(b) b, d
Question 35.
Which of the following represents collinear but not equal vectors?
(a) a, c
(b) b, d
(c) b, m
(d) Both (a) and (b)
Answer:
(a) a, c
Question 36.
Which of the following represents coinitial vector?
(a) c, d
(b) m, b
(c) b, d
(d) Both (a) and (b)
Answer:
(d) Both (a) and (b)
Question 37.
The unit vector in the direction of the sum of vectors
Answer:
(a) \(\frac{1}{5 \sqrt{2}}(3 \hat{i}+4 \hat{j}+5 \hat{k})\)
Question 38.
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) Equilaterala triangle
Answer:
(d) Equilaterala triangle
Question 39.
Answer:
(d) α = ±1, β = 1
Question 40.
The vectors \(a=x \hat{i}-2 \hat{j}+5 \hat{k}\) and \(b=\hat{i}+y \hat{j}-z \hat{k}\) are collinear, if
(a) x =1, y = -2, z = -5
(b) x= 1.2, y = -4, z = -10
(c) x = -1/2, y = 4, z = 10
(d) All of these
Answer:
(d) All of these
Question 41.
The vector \(\hat{i}+x \hat{j}+3 \hat{k}\) is rotated through an angle θ and doubled in magnitude, then it becomes \(4 \hat{i}+(4 x-2) \hat{i}+2 \hat{k}\). The value of x is
(a) \(\left\{-\frac{2}{3}, 2\right\}\)
(b) \(\left\{\frac{1}{3}, 2\right\}\)
(c) \(\left\{\frac{2}{3}, 0\right\}\)
(d) {2, 7}
Answer:
(a) \(\left\{-\frac{2}{3}, 2\right\}\)
Question 42.
If a + b + c = 0, then a × b =
(a) c × a
(b) b × c
(c) 0
(d) Both (a) and (b)
Answer:
(d) Both (a) and (b)
Question 43.
If a is perpendicular to b and c, |a| = 2, |b| = 3, |c| = 4 and the angle between b and c is \(\frac{2 \pi}{3}\), |abc| is equal to
(a) 4√3
(b) 6√3
(c) 12√3
(d) 18√3
Answer:
(c) 12√3
Question 44.
Answer:
(b) a
Question 45.
Answer:
(a) neither x nor y
Question 46.
If a, b, c are three non-coplanar vectors, then (a + b + c).[(a + b) × (a + c)] is
(a) 0
(b) 2[abc]
(c) -[abc]
(d) [abc]
Answer:
(c) -[abc]
Question 47.
If u, v and w are three non-coplanar vectors, then (u + v – w).[(u – v) × (v – w)] equals
(a) 0
(b) u.v × w
(c) u.w × v
(d) 3u.v × w
Answer:
(b) u.v × w
Question 48.
If unit vector c makes an angle \(\frac{\pi}{3}\) with \(\hat{i} \times \hat{j}\), then minimum and maximum values of \((\hat{i} \times \hat{j}) \cdot c\) respectively are
(a) 0, \(\frac{\sqrt{3}}{2}\)
(b) \(-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\)
(c) -1, \(\frac{\sqrt{3}}{2}\)
(d) None of these
Answer:
(b) \(-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\)
Question 49.
The volume of the tetrahedron whose conterminous edges are \(\hat{j}+\hat{k}, \hat{i}+\hat{k}, i+\hat{j}\) is
(a) \(\frac{1}{6}\) cu. unit
(b) \(\frac{1}{3}\) cu. unit
(c) \(\frac{1}{2}\) cu. unit
(d) \(\frac{2}{3}\) cu. unit
Answer:
(b) \(\frac{1}{3}\) cu. unit
Question 50.
If the vectors \(2 \hat{i}-3 \hat{j}, i+\hat{j}-\hat{k}\) and \(3 \hat{i}-\hat{k}\) form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is
(a) 8
(b) 10
(c) 4
(d) 14
Answer:
(c) 4
Question 51.
The volume of the parallelopiped whose edges are represented by \(-12 \hat{i}+\alpha \hat{k}, 3 j-\hat{k}\) and \(2 \hat{i}+j-15 \hat{k}\) is 546 cu. units. Then α =
(a) 3
(b) 2
(c) -3
(d) -2
Answer:
(c) -3
Question 52.
Answer:
(d) None of these
Question 53.
Answer:
(a) -2
Question 54.
Answer:
(a) all values of x
Question 55.
If the vectors \(\hat{i}-2 \hat{j}+3 \hat{k},-2 \hat{i}+3 \hat{j}-4 \hat{k}, \lambda \hat{i}-\hat{j}+2 \hat{k}\) are coplanar, then the value of λ is equal to
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(a) 0
Question 56.
Find the value of λ if the vectors, a = \(2 \hat{i}-\hat{j}+\hat{k}\), b = \(\hat{i}+2 \hat{j}-3 \hat{k}\) and c = \(3 \hat{i}-\lambda \hat{j}+5 \hat{k}\) are coplanar.
(a) 4
(b) -2
(c) -6
(d) 5
Answer:
(a) 4
Question 57.
If a, b, c are unit vectors, then |a – b| + |b – c| + |c – a| does not exceed
(a) 4
(b) 9
(c) 8
(d) 6
Answer:
(b) 9
Question 58.
Find the value of λ so that the vectors \(2 \hat{i}-4 \hat{j}+\hat{k}\) and \(4 \hat{i}-8 \hat{j}+\lambda \hat{k}\) are perpendicular.
(a) -15
(b) 10
(c) -40
(d) 20
Answer:
(c) -40
Question 59.
The dot product of a vector with the vectors \(\hat{i}+\hat{j}-3 \hat{k}, \hat{i}+3 \hat{j}-2 \hat{k}\) and \(2 \hat{i}+\hat{j}+4 \hat{k}\) are 0, 5 and 8 respectively. Find the vector.
(a) \(\hat{i}+2 \hat{j}+\hat{k}\)
(b) \(-\hat{i}+3 \hat{j}-2 \hat{k}\)
(c) \(\hat{i}+2 \hat{j}+3 \hat{k}\)
(d) \(\hat{i}-3 \hat{j}-3 \hat{k}\)
Answer:
(a) \(\hat{i}+2 \hat{j}+\hat{k}\)
Question 60.
If a, b, c are three mutually perpendicular vectors of equal magnitude, find the angle between a and a + b + c.
(a) \(\cos ^{-1}(1 / \sqrt{3})\)
(b) \(\cos ^{-1}(1 / 2 \sqrt{2})\)
(c) \(\cos ^{-1}(1 / 3 \sqrt{3})\)
(d) \(\cos ^{-1}(1 / 2 \sqrt{3})\)
Answer:
(a) \(\cos ^{-1}(1 / \sqrt{3})\)
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