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NCERT Solutions for class 12 Maths | Chapter 11 - Three Dimensional Geometry

(1) The equation of the plane passing through three non- collinear points with position vectors a, b, c is
[A] r.(b × c + c × a + a × b) = 0
[B] r.(b × c + c × a + a × b) = [abc]
[C] r.(a × (b + c)) = [abc]
[D] r.(a + b + c) = 0
Answer: r.(b × c + c × a + a × b) = [abc]
(2) Find the length of perpendicular from the origin to the plane
[A] 5/13
[B] 5/√13
[C] 5/23
[D] 5/√13
Answer: 5/13

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(3) Find the vector equation of the plane which is at a distance of 8 units from the origin and which is normal to the vector
[A]
[B]
[C]
[D] None of these
Answer:
(4) The coordinate of the foot of perpendicular drawn from origin to the plane 2x – 3y + 4z – 6 = 0 is
[A]
[B]
[C] (12, -18, 24)
[D]
Answer:
(5) The direction cosines of the unit vector perpendicular to the plane passing through the origin are
[A] 6/7,3/7,2/7
[B] 6, 3, 2
[C] −6/7,3/7,2/7
[D] -6, 3, 2
Answer: −6/7,3/7,2/7
(6) The angle between the lines x = 1, y = 2 and y = -1, z = 0 is
[A] 90°
[B] 30°
[C] 60°
[D] 0°
Answer: 90°
(7) The angle between the line 2x = 3y = -z and 6x = -y = -4z is
[A] 30°
[B] 45°
[C] 90°
[D] 0°
Answer: 90°
(8) The vector equation of the line through the points A(3, 4, -7) and B(1, -1, 6) is
[A]
[B]
[C]
[D]
Answer:
(9) The equation of the line joining the points (-3, 4, 11) and (1, -2, 7) is
[A]
[B]
[C]
[D]
Answer:
(10) The point A(1, 2, 3), B(-1, -2, -1) and C(2, 3, 2) are three vertices of a parallelogram ABCD. Find the equation of CD.
[A]
[B]
[C]
[D]
Answer:
(11) Find the coordinatets of the point where the line through the points (5, 1, 6) and (3, 4, 1) crosses the yz-plane.
[A] (0,−17/2,13/2)
[B] (0,17/2,−13/2)
[C] (10,19/2,13/2)
[D] (0, 17, 13)
Answer: (0,17/2,−13/2)
(12) The coordinates of a point on the line at a distance of 6√12 from the point (1, 2, 3) is
[A] (56, 43, 111)
[B] (56/17,43/17,111/17)
[C] (2, 1, 3)
[D] (-2, -1, -3)
Answer: (56/17,43/17,111/17)
(13) The equation of the straight line passing through the point (a, b, c) and parallel to Z-axis is
[A]
[B]
[C]
[D]
Answer:
(14) The certesian equation of the line l when it passes through the point (x1, y1, z1) and parallel to the vector

[A] x – x1 = y – y1 = z – z1
[B] x + x1 = y + y1 = z + z1
[C]
[D]
Answer:
(15) If l, m and n are the direction cosines of line l, then the equation of the line (l) passing through (x1, y1, z1) is
[A]
[B]
[C]
[D] None of these
Answer:
(16) The equation of line passing through the point (-3, 2, -4) and equally inclined to the axes are
[A] x – 3 = y + 2 = z – 4
[B] x + 3 = y – 2 = z + 4
[C]
[D] None of these
Answer: x + 3 = y – 2 = z + 4
(17) Find the direction cosines of the line joining A(0, 7, 10) and B(-1, 6, 6).
[A]
[B]
[C]
[D] None of these
Answer:
(18) The coordinates of a point P are (3, 12, 4) w.r.t. origin O, then the direction cosines of OP are
[A] 2, 12 , 14
[B] 1/4, 1/3, 1/2
[C]
[D] 3/13, 12/13, 4/13
Answer: 3/13, 12/13, 4/13
(19) If a line makes an angle θ1, θ2, θ3 with the axis respectively, then cos 2θ1 + cos 2θ2 + cos 2θ3 =
[A] -4
[B] -2
[C] -3
[D] -1
Answer: -1
(20) A line makes angles α, β and γ with the co-ordinate axes. If α + β = 90°, then γ is equal to
[A] 0°
[B] 90°
[C] 180°
[D] None of these
Answer: 90°

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