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 Bihar Board 12th Maths Objective Answers Chapter 10 Vector Algebra

Bihar Board 12th Maths Objective Answers Chapter 10 Vector Algebra

Question 1.
If (12,13,n) are the direction cosines of a line, then the value of n is
(a) 23√6
(b) 236
(c) 23
(d) 32
Answer:
(a) 23√6

Question 2.
Find the magnitude of vector 3i^+2j^+12k^.
(a) √157
(b) 4√11
(c) √213
(d) 9√3
Answer:
(a) √157

Direction (3 – 5): Study the given parallelogram and answer the following questions.

Question 3.
Which of the following represents equal vectors?
(a) a, c
(b) b, d
(c) b, c
(d) m, d
Answer:
(b) b, d

Question 4.
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 5.
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 6.
The unit vector in the direction of the sum of vectors

Answer:
(a) 152√(3i^+4j^+5k^)

Question 7.
The vectors 3i^+5j^+2k^,2i^−3j^−5k^ and 5i^+2j^−3k^ form the sides of
(a) Isosceles triangle
(b) Right triangle
(c) Scalene triangle
(d) Equilaterala triangle
Answer:
(d) Equilaterala triangle

Question 8.

Answer:
(d) α = ±1, β = 1

Question 9.
The vectors a=xi^−2j^+5k^ and b=i^+yj^−zk^ 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 10.
The vector i^+xj^+3k^ is rotated through an angle θ and doubled in magnitude, then it becomes 4i^+(4x−2)i^+2k^. The value of x is
(a) {−23,2}
(b) {13,2}
(c) {23,0}
(d) {2, 7}
Answer:
(a) {−23,2}

Question 11.
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:
(c) Collinear

Question 12.
The points with position vectors 60i^+3j^,40i^−8j^ and ai^−5j^ are collinear if
(a) a = -40
(b) a = 40
(c) a = 20
(d) None of these
Answer:
(a) a = -40

Question 13.
The position vectors of the points A, B, C are (2i^+j^−k^),(3i^−2j^+k^) and (i^+4j^−3k^) respectively. These points
(a) form an isosceles triangle
(b) form a right angled triangle
(c) are collinear
(d) form a scalene triangle
Answer:
(a) form an isosceles triangle

Question 14.
The figure formed by the four points i^+j^−k^, 2i^+3j^,5j^−2k^ and k^−j^ is
(a) trapezium
(b) rectangle
(c) parallelogram
(d) None of these
Answer:
(d) None of these

Question 15.
If x coordinate of a point P of a line joining the points Q(2, 2, 1) and R(5, 2, -2) is 4, then the z coordinate of P is
(a) -2
(b) -1
(c) 1
(d) 2
Answer:
(b) -1

Question 16.
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) i^+j^
(b) i^−j^
(c) −i^+j^
(d) −i^−j^
Answer:
(c) −i^+j^

Question 17.
The vectors AB = 3i^+4k^ and AC = AC=5i^−2j^+4k^ 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 18.
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 19.

Answer:
(d) 16√(2i^−j^+k^)

Question 20.

Answer:
(c) π≥θ>2π3

Question 21.
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 22.
If |a| = |b| = 1 and |a + b| = √3, then the value of (3a – 4b).(2a + 5b) is
(a) -21
(b) −212
(c) 21
(d) 212
Answer:
(b) −212

Question 23.

Answer:
(c) 12√(i^+j^)

Question 24.
If |a – b| = |a| = |b| = 1, then the angle between a and b is
(a) π3
(b) 3π4
(c) π2
(d) 0
Answer:
(a) π3

Question 25.

Answer:
(d) |a|2

Question 26.
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 27.
If |a + b| = |a – b|, then angle between a and b is (a ≠ 0, b ≠ 0)
(a) π3
(b) π6
(c) π4
(d) π2
Answer:
(d) π2

Question 28.
If a and b are two unit vectors inclined to x-axis at angles 30° and 120° respectively, then |a + b| equals
(a) 23−−√
(b) √2
(c) √3
(d) 2
Answer:
(d) 2

Question 29.
If the angle between i^+k^ and i^+j^+ak^ is π3, 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 30.
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 π4, is
(a) 15
(b) √113
(c) √593
(d) √369
Answer:
(c) √593

Question 31.
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 32.
Find the value of λ so that the vectors 2i^−4j^+k^ and 4i^−8j^+λk^ are perpendicular.
(a) -15
(b) 10
(c) -40
(d) 20
Answer:
(c) -40

Question 33.
The dot product of a vector with the vectors i^+j^−3k^,i^+3j^−2k^ and 2i^+j^+4k^ are 0, 5 and 8 respectively. Find the vector.
(a) i^+2j^+k^
(b) −i^+3j^−2k^
(c) i^+2j^+3k^
(d) i^−3j^−3k^
Answer:
(a) i^+2j^+k^

Question 34.
If a, b, c are three mutually perpendicular vectors of equal magnitude, find the angle between a and a + b + c.
(a) cos−1(1/3–√)
(b) cos−1(1/22–√)
(c) cos−1(1/33–√)
(d) cos−1(1/23–√)
Answer:
(a) cos−1(1/3–√)

Question 35.
Find the angle between the vectors a + b and a – b if a = 2i^−j^+3k^ and b = b=3i^+j^−2k^
(a) π6
(b) π3
(c) π2
(d) 0
Answer:
(c) π2

Question 36.
If a = 2i^+j^+2k^ and b = 5i^−3j^+k^, then the projection of b on a is
(a) 3
(b) 4
(c) 5
(d) 6
Answer:
(a) 3

Question 37.
Let a=i^+2j^+k^,b=i^−j^+k^,c=i^+j^−k^. A vector coplanar to a and b has a projection along c of magnitude 13√, then the vector is
(a) 4i^−j^+4k^
(b) 4i^+j^−4k^
(c) 2i^+j^+k^
(d) None of these
Answer:
(a) 4i^−j^+4k^

Question 38.
The component of i in the direction of the vector i^+j^+2k^ is
(a) √6
(b) 6
(c) 6√6
(d) 6√6
Answer:
(d) 6√6

Question 39.
Find the projection of b + c on a where a = i^+2j^+k^, b = i^+3j^+k^ and c = i^+k^.
(a) 5√3
(b) 2√2
(c) 3√2
(d) 10√6
Answer:
(d) 10√6

Question 40.
If a = i^+j^+k^, b = i^+3j^+5k^ and c = 7i^+9j^+11k^, then the area of parallelogram having diagonals a + b and b + c is
(a) 4√6
(b) 1221−−√
(c) 6√2
(d) √6
Answer:
(a) 4√6

Question 41.

Answer:
(c) 3i^−2j^+6k^7

Question 42.
The area of parallelogram whose adjacent sides are i^−2j^+3k^ and 2i^+j^−4k^ is
(a) 10√6
(b) 5√6
(c) 10√3
(d) 5√3
Answer:
(b) 5√6

Question 43.
If AB × AC = 2i^−4j^+4k^, 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 44.

Answer:
(a) 53√3(i^+j^+k^)

Question 45.
|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 46.
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 47.

Answer:
(c) i^

Question 48.
The two vectors a = 2i^+j^+3k^ 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 49.
If |a|= 5, |b|= 13 and |a × b|= 25, find a.b
(a) ±10
(b) ±40
(c) ±60
(d) ±25
Answer:
(c) ±60

Question 50.
Find the value of λ so that the vectors 2i−4j^+k^ and 4i−8j^+λk^ are parallel.
(a) -1
(b) 3
(c) -4
(d) 2
Answer:
(d) 2

Question 51.
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 52.
If a is perpendicular to b and c, |a| = 2, |b| = 3, |c| = 4 and the angle between b and c is 2π3, |abc| is equal to
(a) 4√3
(b) 6√3
(c) 12√3
(d) 18√3
Answer:
(c) 12√3

Question 53.

Answer:
(b) a

Question 54.

Answer:
(a) neither x nor y

Question 55.
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 56.
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 57.
If unit vector c makes an angle π3 with i^×j^, then minimum and maximum values of (i^×j^)⋅c respectively are
(a) 0, 3√2
(b) −3√2,3√2
(c) -1, 3√2
(d) None of these
Answer:
(b) −3√2,3√2

Question 58.
The volume of the tetrahedron whose conterminous edges are j^+k^,i^+k^,i+j^ is
(a) 16 cu. unit
(b) 13 cu. unit
(c) 12 cu. unit
(d) 23 cu. unit
Answer:
(b) 13 cu. unit

Question 59.
If the vectors 2i^−3j^,i+j^−k^ and 3i^−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 60.
The volume of the parallelopiped whose edges are represented by −12i^+αk^,3j−k^ and 2i^+j−15k^ is 546 cu. units. Then α =
(a) 3
(b) 2
(c) -3
(d) -2
Answer:
(c) -3

Question 61.

Answer:
(d) None of these

Question 62.

Answer:
(a) -2

Question 63.

Answer:
(a) all values of x

Question 64.
If the vectors i^−2j^+3k^,−2i^+3j^−4k^,λi^−j^+2k^ are coplanar, then the value of λ is equal to
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(a) 0

Question 65.
Find the value of λ if the vectors, a = 2i^−j^+k^, b = i^+2j^−3k^ and c = 3i^−λj^+5k^ are coplanar.
(a) 4
(b) -2
(c) -6
(d) 5
Answer:
(a) 4

Question 66.
Find λ if the vectors i^−j^+k^,3i^+j^+2k^ and i^+λj^−k^ are coplanar.
(a) 5
(b) 12
(c) 15
(d) 8
Answer:
(c) 15

Question 67.
The vector in the direction of the vector i^−2j^+2k^ that has magnitude 9 is

Answer:
(c) 3(i^−2j^+2k^)

Question 68.
The position vector of the point which divides the join of points 2a – 3b and a + b in the ratio 3 : 1 is

Answer:
(d) 5a4

Question 69.
The angle between two vectors a and b with magnitudes √3 and 4, respectively and a.b = 2√3 is
(a) π6
(b) π3
(c) π2
(d) 5π2
Answer:
(b) π3

Question 70.
Find the value of λ such that the vectors a = 2i^+λj^+k^ and b = i^+2j^+3k^ are orthogonal.
(a) 0
(b) 1
(c) 32
(d) −52
Answer:
(d) −52

Question 71.
The value of λ for which the vectors 3i^−6j^+k^ and 2i^−4j^+λk^ are parallel is
(a) 23
(b) 32
(c) 52
(d) 25
Answer:
(a) 23

Question 72.
The vectors from origin to the points A and B are a = 2i^−3j^+2k^ and b = 2i^+3j^+k^, respectively then the area of triangle OAB is
(a) 340
(b) √25
(c) √229
(d) 12 √229
Answer:
(d) 12 √229

Question 73.
The vectors λi^+j^+2k^,i^+λj^−k^ and 2i^−j^+λk^ are coplanar if
(a) λ = -2
(b) λ = 0
(c) λ = 1
(d) λ = -1
Answer:
(a) λ = -2

Question 74.
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) −32
(d) None of these
Answer:
(c) −32

Question 75.
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 76.
The number of vectors of unit length perpendicular to the vectors a = 2i^+j^+2k^ and b = j^+k^ is
(a) one
(b) two
(c) three
(d) infinite
Answer:
(b) two