12th Math chapter 10: full solution, notes, MCQ, & free PDF | Bihar board

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12th Math chapter 10 Bihar board with NCERT-Details & Highlights

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12th Math chapter 10 all objective solutions – download PDF

Question 1. If (12,13,n) are the direction cosines of a line, then the value of n is

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    • (a) 236
      (b) 236
      (c) 23
      (d) 32

Answer: (a) 236

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.

Bihar Board 12th Maths Objective Answers Chapter 10 Vector Algebra Q2

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 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 7. 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 8. The vector i^+xj^+3k^ is rotated through an angle θ and doubled in magnitude, then it becomes 4i^+(4x2)i^+2k^. The value of x is

    • (a) {23,2}
      (b) {13,2}
      (c) {23,0}
      (d) {2, 7}

Answer: (a) {23,2}

Question 9. 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

Bihar Board most mcq solution 12th math chapter 10

Question 10. 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 11. 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 12. 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 13. 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 14. 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 15. 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 16. 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 17. 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 18. 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 19. 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

12th math chapter 10 bihar board mcq solution

Question 20. 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 21. 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 22. 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 23. 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 24. 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 25. 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 26. 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 27. 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 28. If a, b, c are three mutually perpendicular vectors of equal magnitude, find the angle between a and a + b + c.

    • (a) cos1(1/3–√)
      (b) cos1(1/22–√)
      (c) cos1(1/33–√)
      (d) cos1(1/23–√)

Answer: (a) cos1(1/3–√)

Question 29. 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

Important mcq notes 12th math chapter 10

Question 30. 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 31. 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 32. The component of i in the direction of the vector i^+j^+2k^ is

    • (a) √6
      (b) 6
      (c) 6√6
      (d) 66

Answer: (d) 66

Question 33. Find the projection of b + c on a where a = i^+2j^+k^, b = i^+3j^+k^ and c = i^+k^.

    • (a) 53
      (b) 2√2
      (c) 32
      (d) 106

Answer: (d) 106

Question 34. 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) 62
      (d) √6

Answer: (a) 4√6

Question 35. 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 36. 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 37.  |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 38. If |a × b| = 4 and |a.b| = 2, then |a||b|2 is equal to

    • (a) 2
      (b) 6
      (c) 8
      (d) 20

Answer:(d) 20

Question 39.
Bihar Board 12th Maths Objective Answers Chapter 10 Vector Algebra Q47
Answer:(c) i^

BSEB 12th math chapter 10 objective solution

Question 40.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 41. If |a|= 5, |b|= 13 and |a × b|= 25, find a.b

    • (a) ±10
      (b) ±40
      (c) ±60
      (d) ±25

Answer: (c) ±60

Question 42. Find the value of λ so that the vectors 2i4j^+k^ and 4i8j^+λk^ are parallel.

    • (a) -1
      (b) 3
      (c) -4
      (d) 2

Answer: (d) 2

Question 43. 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 44. 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 45.
Bihar Board 12th Maths Objective 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 π3 with i^×j^, then minimum and maximum values of (i^×j^)c respectively are

    • (a) 0, 32
      (b) 32,32
      (c) -1, 32
      (d) None of these

Answer: (b) 32,32

Question 49. 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

most mcq solution 12th math chapter 10

Question 50. 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 51. The volume of the parallelopiped whose edges are represented by 12i^+αk^,3jk^ and 2i^+j15k^ is 546 cu. units. Then α =

    • (a) 3
      (b) 2
      (c) -3
      (d) -2

Answer: (c) -3

Question 52.
Bihar Board 12th Maths Objective Answers Chapter 10 Vector Algebra Q62
Answer: (a) -2

Question 53. 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 54. 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 55. 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 56. 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 57. 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 58. 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 59. 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 60.  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 61. 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 62. 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]


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