Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra

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Vector Algebra Class 12 Maths MCQs Pdf

Question 1.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q41
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q44
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q47
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q19
Answer:
(d) \(\frac{1}{\sqrt{6}}(2 \hat{i}-\hat{j}+\hat{k})\)

Question 15.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q20
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q23
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q25
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.

Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q2

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
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q6
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q8
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q53
Answer:
(b) a

Question 45.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q54
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.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q61
Answer:
(d) None of these

Question 53.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q62
Answer:
(a) -2

Question 54.
Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra Q63
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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