Class 11 Mathematics 3D Plane MCQs Set 01

LEVEL - I

Cartesian equation of a plane:

Question. The plane which passes through the point (-1, 0, -6) and perpendicular to the line whose d.r's are(6, 20, -1) also passes through the point
(a) (1, 1, -26)
(b) (0, 0, 0)
(c) (2, 1, -32)
(d) (1, 1, 1)
Answer: (b) (0, 0, 0)

Question. Equation of the plane through the mid-point of the join of A(4, 5, -10) and B(-1, 2, 1) and perpendicular to AB is
(a) \( 5x + 3y - 11z + \frac{135}{2} = 0 \)
(b) \( 5x + 3y - 11z = \frac{135}{2} \)
(c) \( 5x + 3y + 11z = 135 \)
(d) \( 5x + 3y - 11z + \frac{185}{2} = 0 \)
Answer: (b) \( 5x + 3y - 11z = \frac{135}{2} \)

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

Question. The equation of the plane passing through a point with position vector \( \hat{i} + 2\hat{j} + 3\hat{k} \) and parallel to the plane \( \vec{r} \cdot (3\hat{i} + 4\hat{j} + 5\hat{k}) = 0 \) is
(a) \( 3x + 4y - 5z + 26 = 0 \)
(b) \( 3x + 4y + 5z - 26 = 0 \)
(c) \( 3x - 4y + 5z - 26 = 0 \)
(d) \( 3x + 4y - 5z - 26 = 0 \)
Answer: (b) \( 3x + 4y + 5z - 26 = 0 \)

Question. Distance between two parallel planes \( 2x + y + 2z = 8 \) and \( 4x + 2y + 4z + 5 = 0 \) is
(a) \( \frac{3}{2} \)
(b) \( \frac{9}{2} \)
(c) \( \frac{7}{2} \)
(d) \( \frac{5}{2} \)
Answer: (c) \( \frac{7}{2} \)

Foot and image:

Question. If the foot of perpendicular from (0, 0, 0) to a plane is (1, 2, 2) then the equation of the plane is
(a) \( -x + 2y + 8z - 9 = 0 \)
(b) \( x + 2y + 2z - 9 = 0 \)
(c) \( x + y + z - 5 = 0 \)
(d) \( x + y + z = 23 + 1 = 0 \)
Answer: (b) \( x + 2y + 2z - 9 = 0 \)

Question. The foot of the perpendicular from (7, 14, 5) to \( 2x + 4y - z = 2 \) is
(a) \( (-1, 1, 0) \)
(b) \( (1, 2, 8) \)
(c) \( (2, 1, -2) \)
(d) \( (1, 2, 3) \)
Answer: (b) \( (1, 2, 8) \)

Question. The image of the point \( (-1, 3, 4) \) in the plane \( x - 2y = 0 \) is
(a) \( (-\frac{17}{3}, -\frac{19}{3}, 1) \)
(b) \( (\frac{9}{5}, -\frac{13}{5}, 4) \)
(c) \( (-\frac{17}{3}, -\frac{19}{3}, 4) \)
(d) \( (15, 11, 4) \)
Answer: (b) \( (\frac{9}{5}, -\frac{13}{5}, 4) \)

Ratio formula:

Question. If the points (1, -1, 1) and (-2, 0, 5) with respect to the plane \( 2x + 3y - z + 7 = 0 \) lie on
(a) opposite sides
(b) same side
(c) on the plane
(d) in side the plane
Answer: (a) opposite sides

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

Normal form of a plane:

Question. For the plane \( \Pi \equiv 2x + 3y + 5z + 10 = 0 \), the point \( (2, 3, -5) \) lie in the
(a) Opposite to the origin side
(b) Origin side
(c) Plane
(d) can not say
Answer: (b) Origin side

Question. The normal form of \( 2x - 2y + z = 5 \) is
(a) \( 12x - 4y + 3z = 39 \)
(b) \( -\frac{6}{7}x + \frac{2}{7}y + \frac{3}{7}z = 1 \)
(c) \( \frac{12}{13}x - \frac{4}{13}y + \frac{3}{13}z = 3 \)
(d) \( \frac{2}{3}x - \frac{2}{3}y + \frac{1}{3}z = \frac{5}{3} \)
Answer: (d) \( \frac{2}{3}x - \frac{2}{3}y + \frac{1}{3}z = \frac{5}{3} \)

Perpendicular distance:

Question. A plane passes through \( (2, 3, -1) \) and is perpendicular to the line having dr’s \( (3, -4, 7) \). The perpendicular distance from the origin to this plane is
(a) \( \frac{3}{\sqrt{74}} \)
(b) \( \frac{5}{\sqrt{74}} \)
(c) \( \frac{6}{\sqrt{74}} \)
(d) \( \frac{13}{\sqrt{74}} \)
Answer: (d) \( \frac{13}{\sqrt{74}} \)

Question. An equation of a plane parallel to the plane \( x - 2y + 2z - 5 = 0 \) and at a unit distance from the origin is
(a) \( x - 2y + 2z - 7 = 0 \)
(b) \( x - 2y + 2z + 5 = 0 \)
(c) \( x - 2y + 2z - 3 = 0 \)
(d) \( x - 2y + 2z + 1 = 0 \)
Answer: (c) \( x - 2y + 2z - 3 = 0 \)

Intercept form of a plane:

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) \( 2x + y - z = 1 \)
(d) \( \frac{x}{a} + \frac{y}{a} + \frac{z}{a} = 1 \)
Answer: (b) \( x + y + z = 1 \)

Question. If the plane \( 7x + 11y + 13z = 3003 \) meets the coordinate axes in A, B, C then the centroid of the \( \Delta ABC \) is
(a) \( (143, 91, 77) \)
(b) \( (143, 77, 91) \)
(c) \( (91, 143, 77) \)
(d) \( (143, 66, 91) \)
Answer: (a) \( (143, 91, 77) \)

Question. If the areas of triangles formed by a plane with the positive \( X, Y; Y, Z; Z, X \) axes respectively are 12, 9, 6 sq. unit respectively then the equation of the plane is
(a) \( \frac{x}{4} + \frac{y}{6} + \frac{z}{3} = 1 \)
(b) \( \frac{x}{6} + \frac{y}{3} + \frac{z}{4} = 1 \)
(c) \( \frac{x}{3} + \frac{y}{4} + \frac{z}{6} = 1 \)
(d) \( \frac{x}{3} + \frac{y}{6} + \frac{z}{4} = 1 \)
Answer: (a) \( \frac{x}{4} + \frac{y}{6} + \frac{z}{3} = 1 \)

Angle between two planes:

Question. If the planes \( x + y + kz = 2 \) and \( 2x + y - 2z + 3 = 0 \) are at right angles, then the values of k is
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{2} \)
(c) -2
(d) 2
Answer: (d) 2

Question. The angle between the planes \( 2x + y + z = 3, x - y + 2z = 5 \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{3\pi}{4} \)
(d) \( \frac{\pi}{3} \)
Answer: (d) \( \frac{\pi}{3} \)

Question. If the planes \( 2x + 3y - z + 5 = 0 \), \( x + 2y - kz + 7 = 0 \) are perpendicular then k =
(a) 4
(b) 6
(c) 8
(d) -8
Answer: (d) -8

LEVEL - II

Cartesian equation of a plane:

Question. The product of the d.r's of a line perpendicular to the plane passing through the points (4, 0, 0), (0, 2, 0) and (1, 0, 1) is
(a) 6
(b) 2
(c) 0
(d) 1
Answer: (a) 6

Question. A plane which passes through the point (3, 2, 0) and the line \( \frac{x-4}{1} = \frac{y-7}{5} = \frac{z-4}{4} \) is
(a) \( x - y + z = 1 \)
(b) \( x + y + z = 5 \)
(c) \( x - y - z = 21 \)
(d) \( 2x - y + 5z = 0 \)
Answer: (a) \( x - y + z = 1 \)

Question. The equation of the plane parallel to the plane \( 2x + 3y + 4z + 5 = 0 \) and passing through the point (1, 1, 1) is
(a) \( 2x + 3y + 4z - 9 = 0 \)
(b) \( 2x + 3y + 4z + 9 = 0 \)
(c) \( 2x + 3y + 4z + 7 = 0 \)
(d) \( 2x + 3y + 4z - 7 = 0 \)
Answer: (a) \( 2x + 3y + 4z - 9 = 0 \)

Question. The equation of the plane passing through the point (3, -6, 9) and perpendicular to the x-axis is
(a) \( x + 2 = 0 \)
(b) \( y - 3 = 0 \)
(c) \( z - 7 = 0 \)
(d) \( x - 3 = 0 \)
Answer: (d) \( x - 3 = 0 \)

Question. Distance between two parallel planes \( 7x + 4y - 4z + 3 = 0 \) and \( 14x + 8y - 8z - 12 = 0 \) is
(a) \( \frac{15}{9} \)
(b) 1
(c) \( \frac{9}{15} \)
(d) \( \frac{1}{2} \)
Answer: (b) 1

Foot and image:

Question. If the foot of the perpendicular from \( (0, 0, 0) \) to a plane is \( (1, 2, 3) \), then the equation of the plane is
(a) \( 2x + 3y + 14z = 0 \)
(b) \( x + 2y + 3z = 14 \)
(c) \( x + 2y + 3z + 14 = 0 \)
(d) \( x + 2y - 3z = 14 \)
Answer: (b) \( x + 2y + 3z = 14 \)

Question. The image of the point \( (3, 2, 1) \) in the plane \( 2x - y + 3z = 7 \) is
(a) \( (1, 2, 3) \)
(b) \( (2, 3, 1) \)
(c) \( (3, 2, 1) \)
(d) \( (2, 1, 3) \)
Answer: (c) \( (3, 2, 1) \)

Question. If \( (2, 3, -1) \) is the foot of the perpendicular from \( (4, 2, 1) \) to a plane, the equation of the plane is
(a) \( 2x - y - 2z - 3 = 0 \)
(b) \( 2x + y - 2z - 9 = 0 \)
(c) \( 2x + y + z - 5 = 0 \)
(d) \( 2x - y + 2z + 1 = 0 \)
Answer: (d) \( 2x - y + 2z + 1 = 0 \)

Question. If the points \( (1, 1, p) \) and \( (-3, 0, 1) \) be equidistant from the plane \( \vec{r} \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) + 13 = 0 \) then the value of p =
(a) \( -\frac{1}{3} \)
(b) 6
(c) 3
(d) \( \frac{1}{3} \)
Answer: (a) \( -\frac{1}{3} \)

Ratio formula:

Question. The ratio in which the line joining \( (2, -4, 3) \) and \( (-4, 5, -6) \) is divided by the plane \( 3x + 2y + z - 4 = 0 \) is
(a) 2 : 1
(b) 4 : 3
(c) -1 : 4
(d) 2 : 3
Answer: (c) -1 : 4

Question. For the plane \( \Pi \equiv 4x - 3y + 2z - 3 = 0 \), the points \( A(-2, 1, 2), B(3, 1, -2) \)
(a) Lie on the same side of \( \Pi = 0 \)
(b) Lie on the opposite sides of \( \Pi = 0 \)
(c) Lie on the normal to \( \Pi = 0 \)
(d) Lie on \( \Pi = 0 \)
Answer: (b) Lie on the opposite sides of \( \Pi = 0 \)

Normal form of a plane:

Question. The d.c's of the normal to the plane \( 2x - y + 2z + 5 = 0 \) are
(a) (3, -2, 6)
(b) \( (\frac{2}{7}, \frac{3}{7}, \frac{6}{7}) \)
(c) \( (\frac{3}{7}, -\frac{2}{7}, \frac{6}{7}) \)
(d) \( (\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}) \)
Answer: (d) \( (\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}) \)

Perpendicular distance:

Question. The perpendicular distance from the origin to the plane \( x - 2y + 2z - 9 = 0 \) is
(a) 3
(b) 6
(c) 4
(d) 2
Answer: (a) 3

Question. The length of the perpendicular from the point \( (7, 14, 5) \) to the plane \( 2x + 4y - z = 2 \) is
(a) \( 21\sqrt{3} \)
(b) \( 3\sqrt{21} \)
(c) \( \sqrt{321} \)
(d) \( 5\sqrt{2} \)
Answer: (b) \( 3\sqrt{21} \)

Intercept form of a plane:

Question. 5, 7 are the intercepts of a plane on the Y-axis, Z-axis respectively, if the plane is parallel to the X-axis then the equation of that plane is
(a) \( 5y + 7z = 35 \)
(b) \( 7y + 5z = 1 \)
(c) \( \frac{y}{5} + \frac{z}{7} = 35 \)
(d) \( 7y + 5z = 35 \)
Answer: (d) \( 7y + 5z = 35 \)

Question. The intercepts of the plane \( 2x - 3y + 5z - 30 = 0 \) are
(a) 15, -10, 6
(b) 5, 10, 6
(c) \( \frac{1}{8}, -\frac{1}{6}, \frac{1}{4} \)
(d) 3, -4, 6
Answer: (a) 15, -10, 6

Question. The area of the triangle formed by \( \frac{x}{4} + \frac{y}{3} - \frac{z}{2} = 1 \) with X-axis and Y-axis is
(a) 2
(b) 3
(c) 6
(d) 12
Answer: (c) 6

Angle between two planes:

Question. If the planes \( 2x + 3y + 4z + 7 = 0 \) and \( 4x + ky + 8z + 1 = 0 \) are parallel then k =
(a) 2
(b) 4
(c) 5
(d) 6
Answer: (d) 6

Question. The planes \( 2x - 4y + 5z = 5 \) and \( 5x - 2.5y + 10z = 6 \) are
(a) Perpendicular
(b) parallel
(c) intersect Y-axis
(d) passes through (0, 0, 5/4)
Answer: (b) parallel

Question. If \( \lambda x + y + z = 4, 4x + 4\lambda y + 10z - 14 = 0 \) represent the same plane then the value of \( \lambda = \)
(a) 1
(b) 2
(c) 0
(d) 3
Answer: (b) 2