12th Maths - Chapter 6: Applications of Vector Algebra (Part 1)
Topic: Triple Products and Geometrical Meanings
1. The value of the scalar triple product [a, b, c] represents the volume of a:
(B) Parallelepiped
(C) Sphere
(D) Pyramid
2. If three vectors a, b, and c are coplanar, then their scalar triple product [a, b, c] is:
(B) -1
(C) 0
(D) Any non-zero constant
3. The vector triple product a × (b × c) is equal to:
(B) (a.b)c - (a.c)b
(C) (b.c)a - (a.c)b
(D) a.(b × c)
4. If [a+b, b+c, c+a] is expanded, it is equal to:
(B) 2[a, b, c]
(C) 0
(D) [a, b, c]²
5. The volume of a tetrahedron with coterminous edges a, b, and c is:
(B) (1/3)[a, b, c]
(C) (1/6)[a, b, c]
(D) (1/2)[a, b, c]
6. The value of î . (ĵ × k̂) + ĵ . (k̂ × î) + k̂ . (î × ĵ) is:
(B) 1
(C) 3
(D) -3
7. The work done by a force F in displacing a particle from A to B is given by:
(B) F . AB
(C) |F| |AB|
(D) F / |AB|
8. The torque (moment) of a force F acting at a point P about a point O is:
(B) r × F
(C) F × r
(D) |r| |F|
9. For any three vectors a, b, c, the product (a × b) . c is called:
(B) Scalar Triple Product
(C) Projection Vector
(D) Cross Product
10. If |a × b| = a . b, then the angle between a and b is:
(B) π/2
(C) π/4
(D) π
12th Maths - Chapter 6: Applications of Vector Algebra (Part 2)
Topic: Equations of Straight Lines in 3D
11. The vector equation of a straight line passing through a fixed point with position vector 'a' and parallel to a vector 'b' is:
(B) r = b + ta
(C) r = a × tb
(D) r . b = a
12. The Cartesian equation of a line passing through (x₁, y₁, z₁) with direction ratios (l, m, n) is:
(B) (x-x₁)/l = (y-y₁)/m = (z-z₁)/n
(C) x/l = y/m = z/n
(D) l(x-x₁) + m(y-y₁) + n(z-z₁) = 0
13. Two lines in space which are neither parallel nor intersecting are called:
(B) Perpendicular lines
(C) Skew lines
(D) Coplanar lines
14. The shortest distance between two parallel lines r = a₁ + tb and r = a₂ + sb is:
(B) |(a₂ - a₁) × b| / |b|
(C) |[a₁, a₂, b]|
(D) 0
15. If the direction cosines of a line are (l, m, n), then l² + m² + n² is:
(B) -1
(C) 1
(D) 2
16. The condition for two lines with direction ratios (b₁, b₂, b₃) and (d₁, d₂, d₃) to be perpendicular is:
(B) b₁d₁ + b₂d₂ + b₃d₃ = 0
(C) b₁d₁ + b₂d₂ + b₃d₃ = 1
(D) b₁² + b₂² + b₃² = d₁² + d₂² + d₃²
17. The vector equation of a line passing through two points with position vectors 'a' and 'b' is:
(B) r = a + tb
(C) r = (1-t)a - tb
(D) r = a × b
18. If the shortest distance between two lines is zero, then the lines are:
(B) Skew
(C) Intersecting
(D) Perpendicular
19. The angle θ between two lines with direction vectors b and d is given by cos θ = :
(B) (b . d) / (|b||d|)
(C) |b + d|
(D) |b - d|
20. The acute angle between the line r = (i+2j-k) + t(2i+j+2k) and the x-axis is:
(B) cos⁻¹(1/3)
(C) sin⁻¹(2/3)
(D) π/4
12th Maths - Chapter 6: Applications of Vector Algebra (Part 3)
Topic: Vector and Cartesian Equations of Planes
21. The vector equation of a plane in normal form is:
(B) r × n = p
(C) r . n = 0
(D) r + n = p
22. The Cartesian equation of a plane passing through (x₁, y₁, z₁) and perpendicular to a line with direction ratios (a, b, c) is:
(B) a(x-x₁) + b(y-y₁) + c(z-z₁) = 0
(C) x/a + y/b + z/c = 1
(D) ax₁ + by₁ + cz₁ = d
23. The intercept form of the equation of a plane is:
(B) x/a + y/b + z/c = 1
(C) r . n = p
(D) lx + my + nz = p
24. The distance from the point (x₁, y₁, z₁) to the plane ax + by + cz + d = 0 is:
(B) |ax₁ + by₁ + cz₁|
(C) √(x₁² + y₁² + z₁²)
(D) (ax₁ + by₁ + cz₁) / d
25. The angle between two planes r . n₁ = p₁ and r . n₂ = p₂ is given by cos θ = :
(B) |n₁ × n₂|
(C) (p₁ - p₂) / |n₁ - n₂|
(D) n₁ . n₂
26. If a plane passes through the origin, its equation is of the form:
(B) ax + by + cz = 0
(C) x + y + z = 1
(D) r . n = p
27. The equation of a plane passing through three non-collinear points with position vectors a, b, and c is:
(B) r . (a + b + c) = 0
(C) [r, a, b] = c
(D) r = a + b + c
28. The distance between two parallel planes ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0 is:
(B) |d₁ + d₂|
(C) |d₁ - d₂|
(D) 0
29. The angle between a line r = a + tb and a plane r . n = p is given by sin θ = :
(B) (b × n) / (|b||n|)
(C) b . n
(D) cos⁻¹(b . n)
30. Two planes are perpendicular if their normal vectors n₁ and n₂ satisfy:
(B) n₁ × n₂ = 0
(C) n₁ = n₂
(D) n₁ . n₂ = 1
12th Maths - Chapter 6: Applications of Vector Algebra (Part 4)
Topic: Advanced Plane Geometry and Intersections
31. The equation of the plane passing through the line of intersection of two planes P₁ and P₂ is:
(B) P₁ - P₂ = 0
(C) P₁P₂ = λ
(D) P₁ / P₂ = λ
32. The coordinates of the foot of the perpendicular from the origin to the plane r . n = p are:
(B) p n
(C) (p/|n|²) n
(D) n / p
33. If a line is parallel to a plane, the dot product of the line's direction vector b and the plane's normal n is:
(B) -1
(C) 0
(D) Infinite
34. The vector equation of the yz-plane is:
(B) r . ĵ = 0
(C) r . k̂ = 0
(D) r . (ĵ+k̂) = 0
35. The angle between the plane 2x - y + z = 6 and the line (x-1)/1 = (y+2)/-2 = (z-3)/2 is:
(B) cos⁻¹(2/3√6)
(C) sin⁻¹(6/3√6)
(D) 45°
36. The length of the perpendicular from the origin to the plane x + 2y - 2z = 9 is:
(B) 3
(C) 1
(D) 4.5
37. If a plane has intercepts a, b, c on the axes, then the distance of the origin from the plane is 'p'. Which relation is true?
(B) 1/p² = 1/a² + 1/b² + 1/c²
(C) p = a + b + c
(D) 1/p = 1/a + 1/b + 1/c
38. The point of intersection of the line r = (j-k) + t(i+2j+k) and the plane r . (i+j+k) = 5 is:
(B) (1, 3, 1)
(C) (5, 5, 5)
(D) (0, 1, -1)
39. The vector equation of a plane passing through a point 'a' and parallel to two vectors 'b' and 'c' is:
(B) [r, b, c] = a
(C) r . (b × c) = 0
(D) r = s(a+b) + tc
40. A unit vector normal to the plane 2x + y + 2z = 5 is:
(B) (2/3)i + (1/3)j + (2/3)k
(C) i + j + k
(D) (1/3)i + (1/3)j + (1/3)k
12th Maths - Chapter 6: Applications of Vector Algebra (Part 5)
Topic: Coplanarity and Shortest Distance
41. If the lines (x-x₁)/l₁ = (y-y₁)/m₁ = (z-z₁)/n₁ and (x-x₂)/l₂ = (y-y₂)/m₂ = (z-z₂)/n₂ are coplanar, then:
(B) l₁l₂ + m₁m₂ + n₁n₂ = 0
(C) x₁x₂ + y₁y₂ + z₁z₂ = 0
(D) l₁/l₂ = m₁/m₂ = n₁/n₂
42. The shortest distance between the skew lines r = a + sb and r = c + td is:
(B) |[c-a, b, d]|
(C) |b × d| / |c-a|
(D) (c-a) . (b × d)
43. If a, b, c are three non-coplanar vectors, then [a × b, b × c, c × a] is equal to:
(B) 2[a, b, c]
(C) [a, b, c]²
(D) 0
44. The vector a × (b × c) is a vector lying in the plane containing:
(B) b and c
(C) a and c
(D) a, b and c
45. The shortest distance between the parallel lines r = (2i-j-k) + t(i+2j-2k) and r = (i-j+k) + s(i+2j-2k) is:
(B) √101 / 3
(C) 3 / √101
(D) √101
46. If the volume of a parallelepiped with edges i+j, i+2j, and i+j+λk is 1 unit, then λ is:
(B) 2
(C) 0
(D) -1
47. The projection of a vector 'a' on 'b' is given by:
(B) (a . b) / |a|
(C) a × b
(D) |a| cos θ b̂
48. The distance from the origin to the plane 3x + 4y + 12z = 26 is:
(B) 13
(C) 2
(D) 1
49. If a . (b × c) = 0, it implies that the vectors are:
(B) Unit vectors
(C) Coplanar
(D) Orthogonal
50. The vector triple product (a × b) × c is equal to:
(B) (a.b)c - (a.c)b
(C) (b.c)a - (a.b)c
(D) 0
12th Maths - Chapter 6: Applications of Vector Algebra (Part 6)
Topic: Vector Identities and Geometrical Applications
51. If a, b, c are three vectors such that a + b + c = 0 and |a|=3, |b|=5, |c|=7, the angle between a and b is:
(B) 45°
(C) 60°
(D) 90°
52. The scalar triple product [a-b, b-c, c-a] is always equal to:
(B) 2[a, b, c]
(C) 0
(D) [a, b, c]²
53. The distance of the point (2, 3, 4) from the x-axis is:
(B) 5
(C) √13
(D) √25 = 5
54. Lagrange's identity states that (a × b)² is equal to:
(B) a²b² + (a.b)²
(C) (a.b)²
(D) a² + b²
55. If the lines (x-2)/1 = (y-3)/1 = (z-4)/-k and (x-1)/k = (y-4)/2 = (z-5)/1 are coplanar, then k can have:
(B) Exactly two values
(C) Exactly three values
(D) No value
56. The equation r = (1-s-t)a + sb + tc represents a plane passing through:
(B) Points a, b, and c
(C) Only point a
(D) Parallel to a, b, and c
57. The sine of the angle between the straight line r = a + tb and the plane r . n = q is:
(B) |b × n| / (|b||n|)
(C) b . n
(D) 0
58. If a line makes angles α, β, γ with the coordinate axes, then sin²α + sin²β + sin²γ is:
(B) 2
(C) 3
(D) 0
59. The vector area of a triangle with vertices having position vectors a, b, c is:
(B) a × b + b × c + c × a
(C) 1/2 [a, b, c]
(D) 1/6 [a, b, c]
60. If |a| = 2, |b| = 7 and a × b = 3i + 2j + 6k, the angle between a and b is:
(B) 45°
(C) 60°
(D) sin⁻¹(1/2) = 30°
12th Maths - Chapter 6: Applications of Vector Algebra (Part 7)
Topic: Points, Lines, and Plane Intersections
61. The projection of the line segment joining (1, 2, 3) and (4, 5, 6) on the x-axis is:
(B) 4
(C) 5
(D) √27
62. The coordinates of the point where the line r = (i+j+k) + t(i-j+k) meets the xy-plane are:
(B) (2, 0, 0)
(C) (0, 0, 0)
(D) (1, 1, 0)
63. The image of the point (1, 2, 3) in the plane z = 0 is:
(B) (1, 2, -3)
(C) (-1, -2, 3)
(D) (1, 2, 0)
64. The perpendicular distance from the point (a, b, c) to the x-axis is:
(B) √(b² + c²)
(C) √(a² + b²)
(D) √(a² + c²)
65. If the angle between the planes r . (2i-j+k) = 3 and r . (i+j+2k) = 4 is θ, then cos θ = :
(B) 1/2
(C) 1/3
(D) 2/3
66. The equation of a plane parallel to the xy-plane and passing through (1, 4, 9) is:
(B) y = 4
(C) z = 9
(D) x+y+z = 14
67. The direction cosines of a line which is equally inclined to the axes are:
(B) (1/√2, 1/√2, 0)
(C) (1/√3, 1/√3, 1/√3)
(D) (1/3, 1/3, 1/3)
68. If [a, b, c] = 2, then the volume of the parallelepiped with vectors a+b, b+c, c+a as edges is:
(B) 4
(C) 8
(D) 16
69. The number of vectors of unit length perpendicular to both vectors a and b is:
(B) 2
(C) 3
(D) Infinite
70. The vector triple product a × (b × c) is perpendicular to:
(B) b
(C) c
(D) b × c
12th Maths - Chapter 6: Applications of Vector Algebra (Part 8)
Topic: Quadruple Products and Plane Properties
71. The scalar quadruple product (a × b) . (c × d) is equal to:
(B) (a.b)(c.d) - (a.c)(b.d)
(C) [a, b, c][d]
(D) 0
72. The vector quadruple product (a × b) × (c × d) can be expressed as:
(B) [c, d, a]b - [c, d, b]a
(C) Both A and B
(D) None of these
73. The distance between the origin and the plane r . (i + 2j - 2k) = 6 is:
(B) 3
(C) 6
(D) 1
74. If the line (x-2)/3 = (y-1)/-5 = (z+2)/2 lies in the plane x + 3y - αz + β = 0, then (α, β) is:
(B) (6, -7)
(C) (-6, -7)
(D) (6, 7)
75. The area of a parallelogram with diagonals d₁ and d₂ is:
(B) 1/2 |d₁ × d₂|
(C) d₁ . d₂
(D) 1/2 |d₁ . d₂|
76. If a × (b × c) = (a × b) × c, then:
(B) a and b are perpendicular
(C) b and c are collinear
(D) a and c are perpendicular
77. The equation of the plane passing through (2, -1, 4) and parallel to the plane x + y + z = 10 is:
(B) x + y + z = 10
(C) x + y + z = 15
(D) 2x - y + 4z = 10
78. The angle between the planes 2x + y - z = 9 and x + 2y + z = 7 is:
(B) 45°
(C) 60°
(D) 90°
79. The vector equation of a plane with intercepts 2, 3, 4 on the axes is:
(B) r . (2i + 3j + 4k) = 1
(C) r . (i + j + k) = 9
(D) r . (3i + 4j + 6k) = 12
80. If [a, b, c] = 0, then the vectors a, b, c are:
(B) Coplanar
(C) Non-coplanar
(D) Collinear
12th Maths - Chapter 6: Applications of Vector Algebra (Part 9)
Topic: Systems of Planes and Line-Plane Distance
81. The distance of the point (1, 1, 1) from the plane x + y + z = 6 is:
(B) 2√3
(C) 1/√3
(D) 3
82. If the planes r . n₁ = q₁ and r . n₂ = q₂ are parallel, then:
(B) n₁ × n₂ = 0
(C) n₁ + n₂ = 0
(D) n₁ = q₁
83. The length of the projection of vector i + j + k on the vector i + j is:
(B) √3
(C) 2
(D) 2/√2 = √2
84. The coordinates of the foot of the perpendicular from (1, 0, 0) to the plane x + y + z = 1 are:
(B) (0, 1, 0)
(C) (1/3, 1/3, 1/3)
(D) (0, 0, 1)
85. If a line makes an angle θ with the x, y, and z axes, then the value of cos 2θ is:
(B) 1/3
(C) 2/3
(D) 0
86. The equation of the plane containing the line r = a + tb and perpendicular to the plane r . n = q is:
(B) r . (b × n) = 0
(C) [r, a, b] = 0
(D) r . n = a . b
87. If [a, b, c] = 3, then the value of a . (c × b) is:
(B) -3
(C) 0
(D) 6
88. The angle between the line r = ti and the plane r . (i + j) = 5 is:
(B) 45°
(C) 60°
(D) 90°
89. The vector equation of a line passing through (2, 3, 1) and parallel to the z-axis is:
(B) r = (2i + 3j + k) + ti
(C) r = tk
(D) r = (2i + 3j) + tk
90. If |a + b| = |a - b|, then the angle between a and b is:
(B) 45°
(C) 90°
(D) 180°
12th Maths - Chapter 6: Applications of Vector Algebra (Final Part)
Topic: Tetrahedron Volume and Advanced Properties
91. The volume of a tetrahedron with coterminous edges a, b, and c is:
(B) 1/2 [a, b, c]
(C) 1/3 [a, b, c]
(D) 1/6 [a, b, c]
92. If a' = (b × c) / [a, b, c], then a . a' is equal to:
(B) 1
(C) [a, b, c]
(D) |a|²
93. The work done by a force F = i + 2j + 3k in displacing a particle from (1, 1, 1) to (2, 3, 5) is:
(B) 15 units
(C) 17 units
(D) 20 units
94. The torque of a force F = 3i + j + 2k acting at the point (1, -1, 2) about the origin is:
(B) i + j + k
(C) 2i - j + k
(D) 3i + 2k
95. If the angle between a and b is π/3, then the angle between 2a and -3b is:
(B) 2π/3
(C) π/6
(D) 5π/6
96. The shortest distance between the line r = a + tb and the origin is:
(B) |a . b| / |b|
(C) |a|
(D) |b|
97. If a, b, c are three unit vectors such that a + b + c = 0, then a.b + b.c + c.a is:
(B) -3/2
(C) 0
(D) 1
98. The vector (a × b) . c = 0 means that:
(B) a is parallel to b
(C) a is perpendicular to c
(D) b is parallel to c
99. The equation of a plane passing through (1, 2, 3) and parallel to the xy-plane is:
(B) y = 2
(C) z = 3
(D) x + y + z = 6
100. If |a × b|² + (a . b)² = 144 and |a| = 4, then |b| is:
(B) 4
(C) 12
(D) 9
End of Chapter 6 Multiple Choice Questions.