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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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