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

(1) The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
[A] (-3, 5, 2)
[B] (3, 5, 2)
[C] (3, -5, 2)
[D] (3, 5, -2)
Answer: (-3, 5, 2)
(2) If the end points of a diagonal of a square are (1, -2, 3) and (2, -3, 5) then the length of the side of square is
[A] √3 unit
[B] 2√3 unit
[C] 3√3 unit
[D] 4√3 unit
Answer: √3 unit

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(3) If P(x, y, z) is a point on the line segment joining Q(2, 2, 4) and R(3, 5, 6) such that the projections of OP on the axes are 13/5, 19/5, 26/5 respectively, then P divides QR in the ration
[A] 1 : 2
[B] 3 : 2
[C] 2 : 3
[D] 1 : 3
Answer: 3 : 2
(4) The coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ plane is
[A] (0, 17/2, 13/2)
[B] (0, -17/2, -13/2)
[C] (0, 17/2, -13/2)
[D] None of these
Answer: (0, 17/2, -13/2)
(5) If α, β, γ are the angles made by a half ray of a line respectively with positive directions of X-axis Y-axis and Z-axis, then sin² α + sin² β + sin² γ =
[A] 1
[B] 0
[C] -1
[D] None of these
Answer: None of these
(6) The plane 2x – (1 + a)y + 3az = 0 passes through the intersection of the planes
[A] 2x – y = 0 and y + 3z = 0
[B] 2x – y = 0 and y – 3z = 0
[C] 2x + 3z = 0 and y = 0
[D] 2x – 3z = 0 and y = 0
Answer: 2x – 3z = 0 and y = 0
(7) The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
[A] (-3, 5, 2)
[B] (3, 5, 2)
[C] (3, -5, 2)
[D] (3, 5, -2)
Answer: (-3, 5, 2)
(8) The equation of plane passing through the point i + j + k and parallel to the plane r . (2i – j + 2k) = 5 is
[A] r . (2i – j + 2k) = 2
[B] r . (2i – j + 2k) = 3
[C] r . (2i – j + 2k) = 4
[D] r . (2i – j + 2k) = 5
Answer: r . (2i – j + 2k) = 3
(9) In a three dimensional space, the equation 3x – 4y = 0 represents
[A] a plane containing Y axis
[B] none of these
[C] a plane containing Z axis
[D] a plane containing X axis
Answer: a plane containing Z axis
(10) Three planes x + y = 0, y + z = 0, and x + z = 0
[A] none of these
[B] meet in a line
[C] meet in a unique point
[D] meet taken two at a time in parallel lines
Answer: meet in a unique point
(11) The points on the y- axis which are at a distance of 3 units from the point (2, 3, -1) is
[A] either (0, -1, 0) or (0, -7, 0)
[B] either (0, 1, 0) or (0, 7, 0)
[C] either (0, 1, 0) or (0, -7, 0)
[D] either (0, -1, 0) or (0, 7, 0)
Answer: either (0, -1, 0) or (0, 7, 0)
(12) The cartesian equation of the line is 3x + 1 = 6y – 2 = 1 – z then its direction ratio are
[A] 1/3, 1/6, 1
[B] -1/3, 1/6, 1
[C] 1/3, -1/6, 1
[D] 1/3, 1/6, -1
Answer: 1/3, 1/6, 1
(13) The equation of the set of point P, the sum of whose distance from A(4, 0, 0) and B(-4, 0, 0) is equal to 10 is
[A] 9x² + 25y² + 25z² + 225 = 0
[B] 9x² + 25y² + 25z² – 225 = 0
[C] 9x² + 25y² – 25z² – 225 = 0
[D] 9x² – 25y² – 25z² – 225 = 0
Answer: 9x² + 25y² + 25z² – 225 = 0
(14) A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
[A] 2√3(i – j + k)
[B] 2√3(-i + j + k)
[C] 2√3(i + j – k)
[D] 2√3(i + j + k)
Answer: 2√3(i + j + k)
(15) The maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is
[A] 3
[B] 2
[C] 4
[D] 5
Answer: 5
(16) The coordinate of foot of perpendicular drawn from the point A(1, 0, 3) to the join of the point B(4, 7, 1) and C(3, 5, 3) are
[A] (5/3, 7/3, 17/3)
[B] (5, 7, 17)
[C] (5/3, -7/3, 17/3)
[D] (5/7, -7/3, -17/3)
Answer: (5/3, 7/3, 17/3)
(17) A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
[A] 2√3(i – j + k)
[B] 2√3(-i + j + k)
[C] 2√3(i + j – k)
[D] 2√3(i + j + k)
Answer: 2√3(i + j + k)
(18) The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a
[A] Straight line
[B] Plane
[C] Sphere
[D] None of these
Answer: Plane
(19) The maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is
[A] 3
[B] 4
[C] 5
[D] Can not be find
Answer: 5
(20) The angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is
[A] π/2
[B] π/3
[C] π/4
[D] π/6
Answer: π/3

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