Class 11 Mathematics Introduction To Three-Dimensional Geometry MCQs Set 08

Question. The equation of the plane passing through (1, 1, 1) and (1, –1, –1) and perpendicular to \( 2x + y + 2z + 5 = 0 \) is
(a) \( 2x + 5y + 2z - 8 = 0 \)
(b) \( x + y - z - 1 = 0 \)
(c) \( 2x + 5y + z + 4 = 0 \)
(d) \( x - y + z - 1 = 0 \)
Answer: (d)

Question. The equation of the plane through the intersection of the planes \( x + y + z = 1 \) and \( 2x + 3y - z + 4 = 0 \) and parallel to x-axis is
(a) \( y - 3z + 6 = 0 \)
(b) \( 3y - z + 6 = 0 \)
(c) \( y + 3z + 6 = 0 \)
(d) \( 3y - 2z + 6 = 0 \)
Answer: (a)

Question. If O is the origin and A is the point (a, b, c), then the equation of the plane through A and at right angles to OA is
(a) \( a(x-a) - b(y-b) - c(z-c) = 0 \)
(b) \( a(x+a) + b(y+b) + c(z+c) = 0 \)
(c) \( a(x-a) + b(y-b) + c(z-c) = 0 \)
(d) None of these
Answer: (c)

Question. The equation of the plane through the point (1, 2, 3) and parallel to the plane \( x + 2y + 5z = 0 \) is
(a) \( (x-1) + 2(y-2) + 5(z-3) = 0 \)
(b) \( x + 2y + 5z = 14 \)
(c) \( x + 2y + 5z = 6 \)
(d) None of these
Answer: (a)

Question. The equation of the plane passing through the intersection of the planes \( x + y + z = 6 \) and \( 2x + 3y + 4z + 5 = 0 \) and the point (1, 1, 1), is
(a) \( 20x + 23y + 26z - 69 = 0 \)
(b) \( 20x + 23y + 26z + 69 = 0 \)
(c) \( 23x + 20y + 26z + 69 = 0 \)
(d) None of these
Answer: (a)

Question. The equation of the plane passing through the intersection of the planes \( x + 2y + 3z + 4 = 0 \) and \( 4x + 3y + 2z + 1 = 0 \) and the origin is
(a) \( 3x + 2y + z + 1 = 0 \)
(b) \( 3x + 2y + z = 0 \)
(c) \( 2x + 3y + z = 0 \)
(d) \( x + y + z = 0 \)
Answer: (b)

Question. If the plane \( x - 2y + 3z = 0 \) is rotated through a right angle about its line of intersection with the plane \( 2x + 3y - 4z - 5 = 0 \), then the equation of plane in its new position is
(a) \( 28x - 17y + 9z = 0 \)
(b) \( 22x + 5y - 4z - 35 = 0 \)
(c) \( 25x + 17y - 52z - 25 = 0 \)
(d) \( x + 35y - 10z - 70 = 0 \)
Answer: (b)

Question. The equation of the plane passing through the point (–2, –2, 2) and containing the line joining the points (1, 1, 1) and (1, –1, 2) is
(a) \( x + 2y - 3z + 4 = 0 \)
(b) \( 3x - 4y + 1 = 0 \)
(c) \( 5x + 2y - 3z - 17 = 0 \)
(d) \( x - 3y - 6z + 8 = 0 \)
Answer: (d)

Question. The equation of the plane containing the line \( 2x + z - 4 = 0, 2y + z = 0 \) and passing through the point (2, 1, –1) is
(a) \( x + y + z + 2 = 0 \)
(b) \( x + y - z - 4 = 0 \)
(c) \( x - y - z - 2 = 0 \)
(d) \( x + y + z - 2 = 0 \)
Answer: (d)

Question. In three dimensional space, the equation \( 3y + 4z = 0 \) represents
(a) A plane containing x-axis
(b) A plane containing y-axis
(c) A plane containing z-axis
(d) A line with direction numbers 0, 3, 4
Answer: (a)

Question. Direction ratios of the normal to the plane passing through the point (2, 1, 3) and the point of intersection of the planes \( x + 2y + z = 3 \) and \( 2x - y - z = 5 \) are
(a) 13, 6, 1
(b) 5, 7, 3
(c) 4, 3, 2
(d) None of these
Answer: (a)

Question. The plane of intersection of \( x^2 + y^2 + z^2 + 2x + 2y + 2z + 0 = 0 \) and \( 4x^2 + 4y^2 + 4z^2 + 4x + 4y + 4z - 14 = 0 \) is
(a) \( 4x + 4y + 4z + 9 = 0 \)
(b) \( x + y + z + 9 = 0 \)
(c) \( 4x + 4y + 4z + 14 = 0 \)
(d) They do not intersect
Answer: (c)

Question. If the planes \( x + 2y + kz = 0 \) and \( 2x + y - 2z = 0 \) are at right angles, then the value of k is
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{2} \)
(c) –2
(d) 2
Answer: (d)

Question. The value of k for which the planes \( 3x - 6y - 2z = 7 \) and \( 2x + y - kz = 5 \) are perpendicular to each other, is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a)

Question. If the given planes \( ax + by + cz + d = 0 \) and \( a'x + b'y + c'z + d' = 0 \) be mutually perpendicular, then
(a) \( \frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} \)
(b) \( \frac{a}{a'} + \frac{b}{b'} + \frac{c}{c'} = 0 \)
(c) \( aa' + bb' + cc' + dd' = 0 \)
(d) \( aa' + bb' + cc' = 0 \)
Answer: (d)

Question. The angle between two planes is equal to
(a) The angle between the tangents to them from any point
(b) The angle between the normals to them from any point
(c) The angle between the lines parallel to the planes from any point
(d) None of these
Answer: (b)

Question. If the planes \( 3x - 2y + 2z + 17 = 0 \) and \( 4x + 3y - kz = 25 \) are mutually perpendicular, then k =
(a) 3
(b) –3
(c) 9
(d) –6
Answer: (a)

Question. The angle between the planes \( 2x - y + z = 6 \) and \( x + y + 2z = 7 \) is
(a) 30°
(b) 45°
(c) 0°
(d) 60°
Answer: (d)

Question. The angle between the planes \( 3x - 4y + 5z = 0 \) and \( 2x - y - 2z = 5 \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{6} \)
(d) None of these
Answer: (b)

Question. If \( \theta \) is the angle between the planes \( 2x - y + 2z = 3, 6x - 2y + 3z = 5 \), then \( \cos \theta \) is equal to
(a) \( \frac{21}{20} \)
(b) \( \frac{11}{20} \)
(c) \( \frac{20}{21} \)
(d) \( \frac{12}{25} \)
Answer: (c)

Question. The value of \( aa' + bb' + cc' \) being negative, the origin will lie in the acute angle between the planes \( ax + by + cz + d = 0 \) and \( a'x + b'y + c'z + d' = 0 \), if
(a) \( a = a' = 0 \)
(b) d and d' are of same sign
(c) d and d' are of opposite sign
(d) None of these
Answer: (b)

Question. The equation of the plane which bisects the angle between the planes \( 3x - 6y + 2z + 5 = 0 \) and \( 4x - 12y + 3z - 3 = 0 \) which contains the origin is
(a) \( 33x - 13y + 32z + 45 = 0 \)
(b) \( 3x - y + z - 5 = 0 \)
(c) \( 33x + 13y + 32z + 45 = 0 \)
(d) None of these
Answer: (a)

Question. The equation of the bisector of the obtuse angle between the planes \( 3x - 4y + 12z + 26 = 0, 5x - 12y + 13z + 13 = 0 \) is
(a) \( 11x + 4y - 3z = 0 \)
(b) \( 14x - 8y + 13 = 0 \)
(c) \( x + y + z = 9 \)
(d) \( 13x - 7z + 18 = 0 \)
Answer: (b)

Question. The two points (1, 1, 1) and (–3, 0, 1) with respect to the plane \( 12x + 13y - 4z + 13 = 0 \) lie on
(a) Opposite side
(b) Same side
(c) On the plane
(d) None of these
Answer: (a)

Question. Distance between parallel planes \( 2x - 2y + z + 3 = 0 \) and \( 4x - 4y + 2z + 5 = 0 \) is
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{6} \)
(d) 2
Answer: (c)

Question. The distance between the planes \( x + 2y + 3z + 7 = 0 \) and \( 2x + 4y + 6z + 7 = 0 \) is
(a) \( \frac{\sqrt{7}}{2\sqrt{2}} \)
(b) \( \frac{7}{2} \)
(c) \( \frac{\sqrt{7}}{2} \)
(d) \( \frac{7}{2\sqrt{2}} \)
Answer: (a)

Question. Distance of the point (2, 3, 4) from the plane \( 3x - 6y + 2z + 11 = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (a)

Question. The distance of the plane \( 6x - 3y + 2z - 14 = 0 \) from the origin is
(a) 2
(b) 1
(c) 14
(d) 8
Answer: (a)

Question. The distance of the point (2, 3, –5) from the plane \( x + 2y - 2z = 9 \) is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (b)

Question. If the points (1, 1, k) and (–3, 0, 1) be equidistant from the plane \( 12x + 13y - 4z + 13 = 0 \), then k =
(a) 0
(b) 1
(c) 2
(d) None of these
Answer: (c)

Question. If the product of distances of the point (1, 1, 1) from the origin and the plane \( x - y + z + k = 0 \) be 5, then k =
(a) –2
(b) –3
(c) 4
(d) 7
Answer: (c)

Question. If two planes intersect, then the shortest distance between the planes is
(a) \( \cos 0^\circ \)
(b) \( \cos 90^\circ \)
(c) \( \sin 90^\circ \)
(d) 1
Answer: (b)

Question. The length of the perpendicular from the origin to the plane \( 3x + 4y + 12z = 52 \) is
(a) 3
(b) –4
(c) 5
(d) None of these
Answer: (d)

Question. If the length of perpendicular drawn from origin on a plane is 7 units and its direction ratios are \( -3, 2, 6 \), then that plane is
(a) \( -3x + 2y + 6z - 7 = 0 \)
(b) \( -3x + 2y + 6z - 49 = 0 \)
(c) \( 3x - 2y + 6z + 7 = 0 \)
(d) \( -3x + 2y - 6z - 49 = 0 \)
Answer: (b)

Question. If a plane cuts off intercepts –6, 3, 4 from the coordinate axes, then the length of the perpendicular from origin to the plane is
(a) \( \frac{1}{\sqrt{61}} \)
(b) \( \frac{13}{\sqrt{61}} \)
(c) \( \frac{12}{\sqrt{29}} \)
(d) \( \frac{5}{\sqrt{41}} \)
Answer: (c)

Question. If A (–1, 2, 3), B (1, 1, 1) and C (2, –1, 3) are points on a plane. A unit normal vector to the plane ABC is
(a) \( \pm \frac{2\mathbf{i} + 2\mathbf{j} + \mathbf{k}}{3} \)
(b) \( \pm \frac{2\mathbf{i} - 2\mathbf{j} + \mathbf{k}}{3} \)
(c) \( \pm \frac{2\mathbf{i} - 2\mathbf{j} - \mathbf{k}}{3} \)
(d) \( \frac{2\mathbf{i} + 2\mathbf{j} + \mathbf{k}}{-3} \)
Answer: (b)

Question. If the position vectors of three points A, B and C are respectively \( \mathbf{i} + \mathbf{j} + \mathbf{k}, 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k} \) and \( 7\mathbf{i} + 4\mathbf{j} + 9\mathbf{k} \), then the unit vector to the plane containing the triangle ABC is
(a) \( 31\mathbf{i} - 18\mathbf{j} - 9\mathbf{k} \)
(b) \( \frac{31\mathbf{i} - 38\mathbf{j} - 9\mathbf{k}}{\sqrt{2486}} \)
(c) \( \frac{31\mathbf{i} + 18\mathbf{j} + 9\mathbf{k}}{\sqrt{2486}} \)
(d) None of these
Answer: (b)

Question. The projection of point (a, b, c) in yz plane are
(a) (0, b, c)
(b) (a, 0, c)
(c) (a, b, 0)
(d) (a, 0, 0)
Answer: (a)