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

Plane

Question. The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
(a) \( a:b \)
(b) \( b:c \)
(c) \( c:a \)
(d) \( c:b \)
Answer: (d)

Question. The ratio in which the line joining (2, 4, 5) (3, 5, –4) is divided by the yz-plane is
(a) \( 3:2 \)
(b) \( 2:3 \)
(c) \( -2:3 \)
(d) \( 4:-3 \)
Answer: (c)

Question. xy-plane divides the line joining the points (2, 4, 5) and (–4, 3, –2) in the ratio
(a) \( 3:5 \)
(b) \( 5:2 \)
(c) \( 1:3 \)
(d) \( 3:4 \)
Answer: (b)

Question. The coordinates of the point where the line through P (3, 4, 1) and Q (5, 1, 6) crosses the xy-plane are
(a) \( (\frac{3}{5}, \frac{13}{5}, \frac{23}{5}) \)
(b) \( (\frac{13}{5}, \frac{23}{5}, \frac{3}{5}) \)
(c) \( (\frac{13}{5}, \frac{23}{5}, 0) \)
(d) \( (\frac{13}{5}, 0, 0) \)
Answer: (c)

Question. The plane XOZ divides the join of (1, –1, 5) and (2, 3, 4) in the ratio \( \lambda : 1 \), then \( \lambda \) is
(a) –3
(b) 3
(c) \( -\frac{1}{3} \)
(d) \( \frac{1}{3} \)
Answer: (d)

Question. XOZ plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
(a) \( 3:7 \)
(b) \( 2:7 \)
(c) \( -3:7 \)
(d) \( -2:7 \)
Answer: (c)

Question. The plane \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3 \) meets the coordinate axes in A, B, C. The centroid of the triangle ABC is
(a) \( (\frac{a}{3}, \frac{b}{3}, \frac{c}{3}) \)
(b) \( (a, b, c) \)
(c) \( (\frac{1}{a}, \frac{1}{b}, \frac{1}{c}) \)
(d) \( (a, b, c) \)
Answer: (b)

Question. The ratio in which the plane \( \mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}) = 17 \) divides the line joining the points \( -2\mathbf{i} + 4\mathbf{j} + 7\mathbf{k} \) and \( 3\mathbf{i} - 5\mathbf{j} + 8\mathbf{k} \) is
(a) \( 1:5 \)
(b) \( 1:10 \)
(c) \( 3:5 \)
(d) \( 3:10 \)
Answer: (d)

Question. If a plane cuts off intercepts \( OA = a, OB = b, OC = c \) from the coordinate axes, then the area of the triangle ABC =
(a) \( \frac{1}{2}\sqrt{b^2c^2 + c^2a^2 + a^2b^2} \)
(b) \( \frac{1}{2}(bc + ca + ab) \)
(c) \( \frac{1}{2}abc \)
(d) \( \frac{1}{2}\sqrt{(b-c)^2 + (c-a)^2 + (a-b)^2} \)
Answer: (a)

Question. The plane \( \frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1 \) cuts the axes in A, B, C, then the area of the \( \Delta ABC \) is
(a) \( \sqrt{29} \)
(b) \( \sqrt{41} \)
(c) \( \sqrt{61} \)
(d) None of these
Answer: (c)

Question. The volume of the tetrahedron included between the plane \( 3x - 4y + 2z - 12 = 0 \) and the three coordinate planes is
(a) \( 3\sqrt{29} \)
(b) \( 6\sqrt{29} \)
(c) 12
(d) None of these
Answer: (c)

Question. A point located in space moves in such a way that sum of its distances from xy-and yz plane is equal to distance from zx plane, the locus of the point is
(a) \( x - y + z = 2 \)
(b) \( x + y - z = 0 \)
(c) \( x + y - z = 2 \)
(d) \( x - y + z = 0 \)
Answer: (b)

Question. The equation of a plane parallel to x-axis is
(a) \( ax + by + cz + d = 0 \)
(b) \( ax + by + d = 0 \)
(c) \( by + cz + d = 0 \)
(d) \( ax + cz + d = 0 \)
Answer: (c)

Question. In the space the equation \( by + cz + d = 0 \) represents a plane perpendicular to the plane
(a) YOZ
(b) Z=k
(c) ZOX
(d) XOY
Answer: (a)

Question. The intercepts of the plane \( 5x - 3y + 6z = 60 \) on the coordinate axes are
(a) (10, 20, –10)
(b) (10, –20, 12)
(c) (12, –20, 10)
(d) (12, 20, –10)
Answer: (c)

Question. The coordinates of the points A and B are (2, 3, 4) and (–2, 5, –4) respectively. If a point P moves, so that \( PA^2 - PB^2 = k \) where k is constant, then the locus of P is
(a) A line
(b) A plane
(c) A sphere
(d) None of these
Answer: (b)

Question. In a three dimensional xyz space the equation \( x^2 - 5x + 6 = 0 \) represents
(a) Points
(b) Plane
(c) Curves
(d) Pair of straight line
Answer: (b)

Question. The equation of yz-plane is
(a) \( x = 0 \)
(b) \( y = 0 \)
(c) \( z = 0 \)
(d) \( x + y + z = 0 \)
Answer: (a)

Question. The intercepts of the plane \( 2x - 3y + 4z = 12 \) on the coordinate axes are given by
(a) 2, –3, 4
(b) 6, –4, –3
(c) 6, –4, 3
(d) 3, –2, 1.5
Answer: (c)

Question. The locus of the point (x, y, z,) for which \( z = k \), is
(a) A plane parallel to xy plane at a distance k from it
(b) A plane parallel to yz plane at a distance k from it
(c) A plane parallel to zx plane at a distance k from it
(d) A line parallel to z-axis at a distance k from it
Answer: (a)

Question. A point (x, y, z) moves parallel to x- axis. Which of the three variables x, y, z remains fixed
(a) x
(b) x and y
(c) y and z
(d) z and x
Answer: (c)

Question. If a, b, c are three non-coplanar vectors, then the vector equation \( \mathbf{r} = (1-p-q)\mathbf{a} + p\mathbf{b} + q\mathbf{c} \) represents a
(a) Straight line
(b) Plane
(c) Plane passing through the origin
(d) Sphere
Answer: (b)

Question. The direction cosines of the normal to the plane \( 3x + 4y + 12z = 52 \) will be
(a) 3, 4, 12
(b) –3, –4, –12
(c) \( \frac{3}{13}, \frac{4}{13}, \frac{12}{13} \)
(d) \( \frac{3}{\sqrt{13}}, \frac{4}{\sqrt{13}}, \frac{12}{\sqrt{13}} \)
Answer: (c)

Question. The direction cosines of the normal to the plane \( x + 2y - 3z + 4 = 0 \) are
(a) \( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{-3}{\sqrt{14}} \)
(b) \( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \)
(c) \( \frac{-1}{\sqrt{14}}, \frac{-2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \)
(d) \( \frac{1}{\sqrt{14}}, \frac{-2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \)
Answer: (a)

Question. Normal form of the plane \( 2x + 6y + 3z = 1 \) is
(a) \( \frac{2}{7}x + \frac{6}{7}y + \frac{3}{7}z = 1 \)
(b) \( \frac{2}{7}x + \frac{6}{7}y + \frac{3}{7}z = \frac{1}{7} \)
(c) \( \frac{2}{7}x + \frac{6}{7}y + \frac{3}{7}z = 0 \)
(d) None of these
Answer: (b)

Question. The equation of a plane which cuts equal intercepts of unit length on the axes, is
(a) \( x + y + z = 0 \)
(b) \( x + y + z = 1 \)
(c) \( x + y - z = 1 \)
(d) \( \frac{x}{a} + \frac{y}{a} + \frac{z}{a} = 1 \)
Answer: (b)

Question. The equation of the plane which is parallel to y- axis and cuts off intercepts of length 2 and 3 from x-axis and z axis is
(a) \( 3x + 2z = 1 \)
(b) \( 3x + 2z = 6 \)
(c) \( 2x + 3z = 6 \)
(d) \( 3x + 2z = 0 \)
Answer: (b)

Question. A plane \( \pi \) makes intercepts 3 and 4 respectively on z-axis and x-axis. If \( \pi \) is parallel to y- axis, then its equation is
(a) \( 3x + 4z = 12 \)
(b) \( 4x + 3z = 12 \)
(c) \( 3y + 4z = 12 \)
(d) \( 3z + 4y = 12 \)
Answer: (a)

Question. The equation of the plane through the three points (1, 1, 1), (1, –1, 1), and (–7, –3, –5), is
(a) \( 3x - 4z + 1 = 0 \)
(b) \( 3x - 4y + 1 = 0 \)
(c) \( 3x + 4y + 1 = 0 \)
(d) None of these
Answer: (a)

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

Question. The equation of the plane through (2, 3, 4) and parallel to the plane \( x + 2y + 4z = 5 \) is
(a) \( x + 2y + 4z = 10 \)
(b) \( x + 2y + 4z = 3 \)
(c) \( x + y + 2z = 2 \)
(d) \( x + 2y + 4z = 24 \)
Answer: (d)

Question. The equation of the plane passing through the points (1, –3, –2) and perpendicular to planes \( x + 2y + 2z = 5 \) and \( 3x + 3y + 2z = 8 \), is
(a) \( 2x - 4y + 3z - 8 = 0 \)
(b) \( 2x - 4y - 3z + 8 = 0 \)
(c) \( 2x + 4y + 3z + 8 = 0 \)
(d) None of these
Answer: (a)

Question. The line drawn from (4, –1, 2) to the point (–3, 2, 3) meets a plane at right angles at the point (–10, 5, 4), then the equation of plane is
(a) \( 7x - 3y - z + 89 = 0 \)
(b) \( 7x + 3y + z + 89 = 0 \)
(c) \( 7x - 3y + z + 89 = 0 \)
(d) None of these
Answer: (a)

Question. \( x + y + z + 2 = 0 \) together with \( x + y + z + 3 = 0 \) represents in space
(a) A line
(b) A point
(c) A plane
(d) None of these
Answer: (d)

Question. The equation of the plane which contains the line of intersection of the planes \( 2x + 3y - z + 4 = 0 \) and \( x + y + z - 2 = 0 \) and which is perpendicular to the plane \( 3x + 5y - 6z + 8 = 0 \), is
(a) \( 33x + 50y + 45z - 41 = 0 \)
(b) \( 33x + 45y + 50z + 41 = 0 \)
(c) \( 45x + 45y + 50z - 41 = 0 \)
(d) \( 33x + 45y + 50z - 41 = 0 \)
Answer: (d)

Question. The equation of the planes passing through the line of intersection of the planes \( x - 3y + 4z = 0 \) and \( 2x - y + z + 3 = 0 \), whose distance from the origin is 1, are
(a) \( x - 2y + 2z - 3 = 0, 2x + y + 2z + 3 = 0 \)
(b) \( x - 2y + 2z - 3 = 0, 2x + y + 2z + 3 = 0 \)
(c) \( x + 2y - 2z - 3 = 0, 2x - y - 2z + 3 = 0 \)
(d) None of these
Answer: (a)

Question. The equation of the plane which passes through the point (2, 1, 4) and parallel to the plane \( 2x + 3y + 5z + 6 = 0 \) is
(a) \( 2x + 3y + 5z + 27 = 0 \)
(b) \( 2x + 3y + 5z - 27 = 0 \)
(c) \( 2x + y + 4z - 27 = 0 \)
(d) \( 2x + y + 4z + 27 = 0 \)
Answer: (b)

Question. The equation of a plane which passes through (2, –3, 1) and is normal to the line joining the points (3, 4, –1) and (2, –1, 5) is given by
(a) \( x + 5y - 6z + 19 = 0 \)
(b) \( x - 5y + 6z - 19 = 0 \)
(c) \( x + 5y + 6z + 19 = 0 \)
(d) \( x - 5y - 6z - 19 = 0 \)
Answer: (a)

Question. The coordinates of the point in which the line joining the points (3, 5, –7) and (–2, 1, 8) is intersected by the plane yz are given by
(a) \( (0, \frac{13}{5}, 2) \)
(b) \( (0, -\frac{13}{5}, -2) \)
(c) \( (0, -\frac{13}{5}, \frac{2}{5}) \)
(d) \( (0, \frac{13}{5}, \frac{2}{5}) \)
Answer: (a)

Question. If P be the point (2, 6, 3), then the equation of the plane through P at right angle to OP, O being the origin, is
(a) \( 2x + 6y + 3z = 7 \)
(b) \( 2x - 6y + 3z = 7 \)
(c) \( 2x + 6y - 3z = 49 \)
(d) \( 2x + 6y + 3z = 49 \)
Answer: (d)

Question. The equation of the plane containing the line of intersection of the planes \( 2x - y = 0 \) and \( 3z - y = 0 \) the perpendicular to the plane \( 4x + 5y - 3z + 8 = 0 \) is
(a) \( 28x - 17y + 9z = 0 \)
(b) \( 28x + 17y + 9z = 0 \)
(c) \( 28x - 17y - 9z = 0 \)
(d) \( 7x - 3y + z = 0 \)
Answer: (a)