3D Geometry MCQs Part II

1) The angle between the lines x = 1, y = 2 and y = -1, z = 0 is
a)
b)
c)
d)

2) The line
a) coincident
b) parallel
c) skew
d) perpendicular

3) The vector equation of a line which passes through a point whose
     position vector is
and parallel to a vector
is

a)
b)
c)
d)

4) The co-ordinates of the point of intersection of the line

with the plane 3x + 4y +5z = 5 are

a) (-1, 2, -3)
b) (1, 3, -2)
c) (-1, -2, 3)
d) none of these

5) The equation of the sphere concentric with
and passing through (1,2,-1) is

a)
b)
c)
d)

6) The foot of the perpendicular drawn from the point A(2,0,4) to the join
of the points B(3,6,0) and C(3,4,2,) is

a) (3,1,5)
b) (3, 0, 5)
c) (3,1,0)
d) None of these

7) If r makes angles
with x- axis., y- axis, z- axis respectively the direction cosines of r=

a)
b)
c)
d)

8) The points (2,0,1), ( 3,2,-1) and ( 1,1,-3) form
a) An isosceles triangle but not right angled
b) An isosceles right angled triangle
c) An equilateral triangle
d) None of these.

9) The points (4,1,0), (3,-1,2), (5,0,4) and (6,2,2 ) are given. Which
of the following is more correct?

a) the points form the vertices of a parallelogram
b) the points form the vertices of a rhombus
c) the points form the vertices of a rectangle
d) the points form the vertices of a square.

10) The ratio in which the line segment joining the points ( 2,4,5)
and ( 3,5,-4) is divided by the yz-plane is

a) 2:3 internal
b) 3:4 internal
c) 2:3 external
d) 3:4 external

11) The angle between the lines 2x+3y+z-4=x+y-2z-3=0 and 5x+8y-7z
= 10x-2y-2z=0 is

a)
b)
c)
d) None of these

12) The point of intersection of the lines,
is

a) (2,1,-3)
b) (1,-3,2)
c) (-3,2,1)
d) None of these

13) The shortest distance between the lines
is

a)
b)
c)
d)

14) The equation of the plane passing through the points
( +2,6,-6), (+3,10,-9) and (+5,0,+6) is

a) 10x+7y+6z-14=0
b) 10x-7y-6z-14=0
c) 10x-7y+6z+14=0
d) None of these

15) The image of the point P(1,2,3) in the plane 2x+y+z-3=0
is

a)
b)
c)
d)

16) The equation of the plane through the point (1,2,-3)
and perpendicular to the planes 2x+y+z=5 and 3x+y+2z=6
is

a) x + y +z – 2 =0
b) x +y – z -2 =0
c) x+y+z+2 =0
d) x-y-z-2 =0

17) The equation of the plane through the point (1,2,3)
and parallel to the plane 2x-3y+5z =2 is

a) 2x-3y+5z =4
b)
c) 2x-3y+5z =10
d) 2x-3y+5z =11

18) A Point P (x,y,z) is such b that 2PA=PB where A and B are the points (1,2,3)
and (1,-1,2) respectively. The equation of the locus of the point P is

a)
b)
c)
d)

19) The distance of the point of intersection of the line
and the plane x+y+z=17 from the point ( 3,4,5) is

a)
b)
c)
d) 3

20) The equation of the sphere on the line joining the points (2,3,5)
and

(4,9,-3) as diameter is



a)
b)
c)
d) (x-2)(x-4)+(y-3)(y-9)=0

21) The locus of the point P(x,y,z) which moves in such a way that x = a and
y = b is a

a) plane parallel to xy-plane
b) line parallel to x-axis
c) line parallel to y-axis
d) line parallel to z-axis

22) The xy-plane divides the line joining the points (-1,3,4) and (2,-5,6)
a) internally in the ratio 2:3
b) externally in the ratio 2:3
c) internally in the ratio 3:2
d) externally in the ratio 3:2

23) The direction cosines of a line which make equal angles with the axes is
a)
b) 1,1,1
c)
d) none of these

24) The projection of a directed
line segment on the co-ordinate axes are 12,4,3. The d.c.`s of the
line are

a)
b)
c)
d) none of these

25) The co-ordinates of the
foot of the perpendicular from the point A (1,8,4) to the line joining B (0,-1,3)
and C(2,-3,-1) is

a)
b)
c)
d) none of these

26) The point of intersection
of the line

a)
b)
c)
d) none of these

27) The length of shortest
distance between the two lines


a) 7
b) 9
c) 13
d) 8

28) The
area of the triangle whose vertices are (0,0,0), (3,4,7) and (5,2,6) is

a)
b)
c)
d) none of these

29) The angle between the
line whose direction cosines are given by the equations,

a)
b)
c)
d)

30) 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

31) The distance of the
point (1,-2,3) from the plane , x-y+z = 5 measured parallel to

a) 1
b) 2
c) 3
d) 4

32) The equation of the
plane through the points (1, 0, -1) and (3,2,2) and parallel to the line

a) 4x+y+2z = 6
b) 4x – y – 2z = 6
c) 4x – y + 2z = 6
d) none of these

33) A plane meets the co-ordinate
axes in A,B, C such that the centroid of the triangle ABC is the point (a ,
b, c). Then the equation of the planes is

a)
b)
c) ax + by + cz = 3
d) none of these

34) The
equation of the plane through the line of intersection of planes


and parallel to the line y = 0, z = 0 is

a)
b)
c)
d) none of these

35) If a line makes angleswith
the axes respectively, then,

a) -2
b) -1
c) 1
d) 2

36) The image of the point
P (1,3,4) in the plane 2x-y+z+3=0 is

a) (3,5,-2)
b) (-3,5,2)
c) (3,-5,2)
d) (3,5,2)

37) The line,
intersects the curve
if c=

a)
b)
c)
d) None of these

38) A plane passes through
a fixed point ( a,b,c ) . the locus of the foot of the perpendicular to it from
the origin is a sphere of radius

a)
b)
c)
d) None of these

39) The
shortest distance between the z-axis and the line, x+y+2z-3 = 0,
2x+3y+4z-4=
0

a) 1
b) 2
c) 3
d) none of these

40) The smallest radius of
the sphere passing through (1,0,0), (0,1,0) and (0,0,1) is

a)
b)
c)
d)

41) A parallelopiped
is formed by planes drawn through the points( 5,7,9) and (2,3,7) parallel
to co-ordinate planes. The length of an edge of this rectangular parallelopiped
is

a) 2
b) 3
c) 4
d) all the above

42) The equation to the
plane through the points (2,-1,0) (3,-4,5) and parallel to the line 2x= 3y =
4z is

a) 29x + 27y – 22z = 85
b) 29x – 27y -22z = 85
c) 29 x-27 y+ 22z = 85
d) none of these

43) The points A( 5,-1,1)
, B ( (7,-4,7) C ( 1,-6,10) and D( -1,-3,4) are the vertices of a

a) parallelogram
b) rectangle
c) rhombus
d) Both (a) and (c)

44) The equation ,
represents

a) A pair of straight
lines
b) A pair of planes
c) A pair of planes
passing through the origin
d) Both (b) and
(c)

45) If centroid of a
tetrahedron OABC , where A, B and C are (a,2,3),(1,b,2) and (2,1,c) respectively,
be (1,2,3) , then distance of P(a,b,c ) from origin is equal to

a)
b)
c)
d) None of these

46) The coplanar points A,B,C,D are (2-x,2,2), (2,2-y,2), (2,2,2-z)
and (1,1,1) respectively. Then

a)
b) x+y+z=1
c)
d) none of these

47)
a)
b)
c)
d) none of these

48) Let a, b, c be positive real numbers. The following system
of equations in x, y, and z

a) no solution
b) unique solution
c) infinitely many solution
d) finitely many solutions

49) The distance of the point (1,1,1) from the plane passing
through the points (2,1,1), (1,2,1) and (1,1,2) is

a)
b) 1
c)
d) none of these

50) The direction cosines of a line are (1/a, 1/a, 1/a) then
a) 0 < a < 1
b)
c)
d)

51) The line through (a, b,c) and parallel to the x axis is
a)
b)
c)
d)

52) The angle between the two planes 3x-4y + 5z = 0 and 2x-y-2z= 5 is
a)
b)
c)
d) none of these

53) The plane passing through the point (a,b,c) and parallel to the plane
x + y +z = 0 is

a) x +y +z = a + b + c
b) x + y + z + (a + b + c ) = 0
c) x + y + z + abc = 0
d) ax + by + cz = 0

54) The equation of the plane through the intersection of the planes x
+ 2y + 3z – 4 = 0, 2x + 3y + 4z – 5 = 0 and perpendicular to the plane x
+ y + z -1 = 0 is

a) x – y + 2 = 0
b) x – z + 2 = 0
c) y – z + 2 = 0
d) z – x + 2 = 0

55) The line
is parallel to the plane

a) 2x + y – 2z = 0
b) 3x + 4y + 5z = 5
c) x +y+z = 2
d) 2x +3y +4z=0

56) The equation of the sphere which circumscribes the tetrahedron with vertices (0,0,0), (1, 0,
0), (0,1,0) and (0,0,1) is

a)
b)
c)
d)

57) The equation of the palne which bisects the line joining (2,3,4) and (6,7,8)
at right angles is

a) x + y +z= 15
b) x + y + z+15 = 0
c) x + y -z = 15
d) x – y + z + 15 = 0

58) The equation of the plane through the intersection of planes x + 2y +3z
– 4= 0 and 4x+ 3y + 2z + 1 = 0 and passing through origin is

a) 17x + 14 y + 11 Z = 0
b) 17x + y + Z = 0
c) 7x + 4y + z = 0
d) x + 14 y + 11Z = 0

59) The radius of the sphere
a)
b)
c)
d)

60) A vector r has length 15 and direction ratio are 3, -4, 5. The components
of r=

a) 3i- 4j+5k
b) 15(3i-4j+5k)
c)
d) None of these

61) The ratio in which the line joining the points ( 2,3,4) and (-1,4,5)
is divided by the plane 3x+2y-z+2=0 is

a) 5:2 external
b) 5:2 internal
c) 5:1 internal
d) 5:1 external

62) The image of the point (1,3,4) in the line
is

a)
b)
c)
d)

63) The angle between the lines whose direction cosines are
given by

3l+ m+5n=0, 6mn-2nl+5lm=0 is



a)
b)
c)
d) None of these

64) The equation to the plane through the line 3x-4y+5z=10, 2x+2y-3z=4
and parallel to the line x=2y=3z is

a) x-20y+27z=14
b) x+4y+27z=14
c) x-20y+3z=14
d) x-4y+27z=14

65) The ratio in which the line segment joining the points P ( 2, 3, 4 ) and Q ( -3, 5, -4 ) is divided by yz- plane is
a) 1:2
b) 2:3
c) 3:2
d) 2:1

66) The angle between any two diagonals of a cube is
a)

Cos-1 (1/2)

b)

Cos-1 (1/3)

c)

Cos-1 (1/4)

d) None of the given

67) The point ( 1, 2, 3 ) , ( 4, 0, 4 ) , ( – 2, 4, 2 ) , ( 7, – 2, 5 ) are the
a) Vertices of a square
b) Vertices of a parallelogram
c) Vertices of a rhombus
d) Collinear

68)

The straight lines whose direction Cosines are given by
al + bm + cn = 0. f m n + g n l + h l m = 0 are perpendicular
if


a) ( f / a ) + ( g / b ) + ( h / c ) = 0
b)
c)
d)

69) The equation of plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x – y + z + 3 = 0 and passing through the origin is
a) x – 10y + 5z = 0
b) x – 10y – 5z = 0
c) – x + 10y + 5z = 0
d) None of the given

70) The distance of the point ( – 2, 3, – 4 ) from the line ( x + 2 ) / 3 = ( 2y + 3 ) / 4 = ( 3y + 4 ) / 5 measured parallel to the plane 4x + 12y + 3z + 1 = 0 is
a) 17/2
b) 13/2
c) 27/2
d) 17

71)

The centre of the circle in which the sphere x2
+  y2 +  z2 + 2x – 2y – 14 = 0 is cut by the
plane x + 2y + z = 0 is


a) ( – 1 / 6, ( – 1/ 3 ) – ( 1 / 6 ) )
b) ( – 7 / 6, 2 / 3, – 1/ 6 )
c) ( 1 / 6, 2 / 3, – 1 / 6 )
d) None of the given

72) The image of the point P ( 1, 3, 4 ) in the plane 2x – y + z + 3 = 0 is
a) ( 3, 5, – 2 )
b) ( – 3, 5, 2 )
c) 3, – 5, 2 )
d) ( 3, 5, 2 )

73)

A ( 3, 2, 0 ) B ( 5, 3, 2 ) and C ( – 9, 6, – 3 ) are
the vertices of a triangle ABC.  If the bisector of 
ABC meets BC at D, then Co-ordinates of D are


a)
b)
c)
d) None of the given

74) 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 the given

75)
a) 3x + 4y – 5z = 9
b) 3x + 4y – 5z = 9
c) 3x + 4y – 5z + 9 = 0
d) None of the given

76)

If one end of a diameter of the sphere x2 + y2
+ z2 – 2x – 2y – 2z + 2 =  0 is


a)
b)
c)
d) None of the given

77) If a sphere of constant radius k passes the origin and meets the axis in A, B, C then the centroid of the triangle ABC lies on
a)

x2 + y2 + z2  =
 k2

b)

x2 + y2 + z2  =
 4 k2

c)

9 ( x2 + y2 + z2 ) =
 4 k2

d)

9 ( x2 + y2 + z2 ) =
 k2

78) The equations of a sphere which passes through the points ( 1, 0 , 0 ) ( 0, 1 , 0 ) ( 0, 0, 1 ) and having radius as small as possible is
a)

3 ( x2 + y2 + z2
) – 2 ( x + y + z ) – 1  =  0

b)

x2 + y2 + z2
– x – y – z – 1  =  0

c)

3 ( x2 + y2 + z2
) – 2 ( x + y + z ) + 1 =  0

d) None of the given

79) A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C . The locus of the centroid of the tetrahedron OABC is
a)

x2 + y2 + z2 =  4k2

b)

x2 + y2 + z2  =
 k2

c)

2 ( x2 + y2 + z2 )
 =  k2

d) None of the given

Answers

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