# 3D Geometry MCQs Part I

**1) The co-ordinates of the foot of the perpendicular from the point
(1, 3, 4) on the plane 2x – y +z + 3 = 0 are**

**a)**(1, 1, -4)

**b)**(4, -3, 1)

**c)**(1, -3, 4)

**d)**(-1, 4, 3)

**2) The shortest distance between the lines ****a)** 1/6**b)** **c)** 1/3**d)**

**3) The image of the point (1, 3, 4) by the plane 2x – y +z +3 =
0 is**

**a)**(-1, 4, 3)

**b)**(1, 3, 4)

**c)**(-3, 5, 2)

**d)**(3, 5,-2)

**4) G is the centroid of the triangle whose vertices are (1, 2,
0), (2, 1, 1) and (0, 0, 2). The equation of the line OG, where
O is the origin, is**

**a)**x = y = z

**b)**

**c)**

**d)**

**5) The equation of the sphere passing through the point (0,0,0),
(a, 0, 0),(0,b,0) and (0,0,c)
is**

**a)**

**b)**

**c)**

**d)**

**6) The equation of the sphere concentric withÂ
and passing through the origin is **

**a)**

**b)**

**c)**

**d)**

**7) The radius of the sphere which passes through the point (3,
0, 2), (-1, 1, 1) and (2,
-5, 4) and whose centre lies on the plane 2x+3y+4z- 6= 0
is**

**a)**

**b)**

**c)**

**d)**

**8) The equation of the sphere with centre (-1, 1, 1) and radius
equal to that of the sphere **

**a)**

**b)**

**c)**

**d)**

**9) The centre of the sphere ****a)** **b)** **c)** **d)**

**10) The centre of the sphere that passes through (0,0,0), (2,0,0),
(0,2,0) and (0,0,2) is**

**a)**(1,0,0)

**b)**(0, 1,0)

**c)**(0,0,1)

**d)**(1,1,1)

**11) The radius of the circle given by
is**

**a)**3

**b)**4

**c)**

**d)**

**12) If (2,3,5) is one end of a diameter of the sphere
then the co-ordinates of the other end of the diameter are**

**a)**(4,3,5)

**b)**(4,3,-3)

**c)**(4,9,-3)

**d)**(4,3,-9)

**13) A sphere of constant radius k passes through the origin and meets the axis
in A,B,C. Then the centroid of triangle ABC lies on**

**a)**

**b)**

**c)**

**d)**

**14) The centre of the circle ****a)** (0,1,2)**b)** (1,3,-4)**c)** (1,-3,4)**d)** (1, 3,4)

**15) If A (0,0,0), B (2,-3,3), C (-2,3,-3) and 3 collinear points, then C divides
BA in the ratio**

**a)**-1:2

**b)**-2:1

**c)**1:1

**d)**1:2

**16) The points (5,2,4), (6,-1,2) and (8,-7,K) are collinear if K is equal to****a)** 1**b)** -1**c)** 2**d)** -2

**17) A straight line which makes an angle of ,
with each of the y and z axes is inclined with the x axis at an angle**

**a)**

**b)**

**c)**

**d)**

**18) The equation of the plane through P (2,3,-1) at right angles to O P is****a)** 2x + y -z = 14**b)** 2x + y -z+ 14 = 0**c)** 2x +3y -z – 14 = 0**d)** 2x – 3y + z +14 = 0

**19) The equation of the plane through (2,-3,1) and perpendicular to the line
joining the points
(3, 4, 1) and (2, -1,5) is**

**a)**x + 5y -6z = 19

**b)**x + 5y – 6z + 19 = 0

**c)**x – 5y +6z = 19

**d)**x -5y +6z + 19 = 0

**20) If q is the angle between the planes 2x
– y +2z = 3, 6x-2y +3z = 5, the cos q is equal
to**

**a)**

**b)**

**c)**

**d)**

**21) The shortest distance between the lines ****a)** 4**b)** **c)** **d)** 0

**22) The shortest distance between ****a)** **b)** **c)** **d)**

**23) The equation of the straight line 3x+ 2y -z- 4 = 0, 4x+y -2z+3 = 0 in the
symmetrical form is**

**a)**

**b)**

**c)**

**d)**

**24) The equation to the line x- 2y + 3z -4 = 0 = 2x – 3y + 4z -5 in the symmetrical
form is**

**a)**

**b)**

**c)**

**d)**

**25) If the points (1,0,3), (-1, 3,4) , (1,2,1) and (K, 2,5) are coplanar then
K = **

**a)**1

**b)**2

**c)**0

**d)**-1

**26) The ratio in which x y plane divides the line joining (1,2,3) and (4,2,1)
is**

**a)**3 : 1

**b)**-3 : 1

**c)**2 : 1

**d)**-2 : 1

**27) For a point in the x,y plane****a)** x = 0**b)** y = 0**c)** z = 0**d)** x = 0, y = 0

**28) The equation of the plane containing the line is
,Where**

**a)**

**b)**

**c)**

**d)**

**29) For every point P (x, y, z) on the xy-plane****a)** **b)** **c)** **d)** none of these

**30) For every point P (x, y, z) on the x-axis (except the origin),
(i)
(ii)
(iii)
(iv)
**

**a)**i and ii

**b)**ii and iii

**c)**iii and iv

**d)**none of these

**31) The locus of a point P (x, y, z) which moves in such a way that
(constant), is a**

**a)**line parallel to z-axis

**b)**plane parallel to xy-plane

**c)**line parallel to y-axis

**d)**line parallel to x-axis

**32) The locus of a point P (x, y, z) which moves in such a way that
and ,
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

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

- 2
- 3
- 4
- 9

**a)**i ,ii and iii

**b)**ii and iii

**c)**iii and iv

**d)**none of these

**34) A parallelopiped is formed by planes drawn through the points (2, 3, 5)
and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal
of the parallelopiped is**

**a)**7

**b)**

**c)**

**d)**none of these

**35) 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

**36) If the equation of a plane is
is in the normal form, then
**

- l, m and n are the direction cosines of the normal to the plane
- p is the length of the perpendicular from the origin to the plane
- the plane passes through the origin for all values of p

**a)**i ,ii and iv

**b)**ii and iii

**c)**iii and iv

**d)**none of these

**37) The points A (5, -1, 1), B (7, -4, 7), C (1, -6, 10) and D (-1, -3,
4) are the vertices of a
(i) Parallelogram (ii) rectangle (iii) rhombus (iv) square
**

**a)**i and iii

**b)**ii and iii

**c)**iii and iv

**d)**none of these

**38) The equation
represents a plane perpendicular to the**

**a)**xy-plane

**b)**yz-plane

**c)**zx-plane

**d)**none of these

**39) The plane
passes through the intersection of the planes**

**a)**

**b)**

**c)**

**d)**none of these

**40) If a plane meets the coordinate axes at A, B and C, in such
a way that the centroid of
is at the point (1, 2, 3), the equation of the plane is**

**a)**

**b)**

**c)**

**d)**none of these

**41) 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

**42) The image of the point P (1,3,4) in the plane
is**

**a)**(3,5,-2)

**b)**(-3,5,2)

**c)**(3,-5,2)

**d)**(3,5,2)

**43) The line intersects the curve
if c =**

**a)**

**b)**

**c)**

**d)**none of these

**44) If the x-coordinate of a point P on the join of Q(2,2,1) and R(5,
1, -2) is 4, then its z-coordinate is**

**a)**2

**b)**1

**c)**-1

**d)**– 2

**45) The distance of the point P (a, b, c) from the x-axis is****a)** **b)** **c)** **d)** none of these

**46) Ratio in which the xy-plane divides the join of (1, 2, 3) and (4,
2, 1) is**

**a)**3 : 1 internally

**b)**3 : 1 externally

**c)**1 : 2 internally

**d)**2 : 1 externally

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

**a)**

**b)**

**c)**

**d)**none of these

**48) A line passes through the points (6, -7, -1) and (2, -3, 1). The
direction cosines of the line so directed that the angle made by it
with the positive direction of x-axis is acute, are**

**a)**

**b)**

**c)**

**d)**

**49) If
are the angles which a directed line makes with the positive directions
of the coordinate axes, then
is equal to**

**a)**1

**b)**2

**c)**3

**d)**none of these

**50) The coordinates of the foot of the perpendicular drawn from the point A
(1,0,3) to the join of the points B (4,7,1) and C (3,5,3) are**

**a)**

**b)**(5,7,17)

**c)**

**d)**

**51) The projections of a directed line segment on the coordinate axes are 12,
4, 3. The DCs of the line are**

**a)**

**b)**

**c)**

**d)**none of these

**52) 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
on the axes are 13/5, 19/5, 26/5 respectively, then P divides QR in the ratio**

**a)**1 : 2

**b)**3 : 2

**c)**2 : 3

**d)**1 : 3

**53) A mirror and a source of light are situated at the origin O and at a point
on OX respectively. A ray of light from the source strikes the mirror and
is reflected. If the DRs of the normal to the plane are 1, -1, 1, then DCs
of the reflected ray are**

**a)**

**b)**

**c)**

**d)**

**54) The equation of the plane perpendicular to the line
and passing through the point (2, 3,1) is**

**a)**

**b)**

**c)**

**d)**none of these

**55) 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

**56) The values of
for which the lines
are perpendicular to each other is**

**a)**0

**b)**1

**c)**-1

**d)**none of these

**57) A line passes through two points
The coordinates of a point on this line at a distance of 14
units from A are**

**a)**

**b)**

**c)**

**d)**both (a) and (b)

**58) Equation of a line passing through (-1, 2, -3) and
perpendicular to the plane **

**a)**

**b)**

**c)**

**d)**none of these

**59) The equation of the plane through the points (2, 2,
1) and (9, 3, 6) and
to the plane **

**a)**

**b)**

**c)**

**d)**none of these

**60) The equation of the plane containing the line
and the point (0,7, -7) is**

**a)**

**b)**

**c)**

**d)**none of these

**61) If one end of a diameter of the sphere then the other end is****a)** **b)** **c)** **d)** none of these

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

**a)**

**b)**

**c)**

**d)**none of these

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

**a)**

**b)**

**c)**

**d)**

**64) The equation 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)**

**b)**

**c)**

**d)**none of these

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

**a)**

**b)**

**c)**

**d)**none of these

**66) The equation of the z axis is****a)** x = 0**b)** y = 0**c)** x = 0, y = 0**d)** y = 0, z = 0

**67) The co-ordinates of the point equidistant from (0,0,0), (a, 0,0), (0,b,0)
and (0,0,c) is**

**a)**(a,b,c)

**b)**(2a, 2b,2c)

**c)**(a/2, b/2, c/2)

**d)**(a/3, b/3, c/3)

**68) The direction cosines of the x axis are****a)** (1,1,1)**b)** (1,0,0)**c)** (0,1,0)**d)** (0,0,1)

**69) Which of the following represents the
direction cosines of a line**

**a)**(1,1,1)

**b)**

**c)**

**d)**

**70) The direction cosines of the normal to the x, y plane is****a)** (0,0,1)**b)** (0,1,0)**c)** (1,0,0)**d)** (1,1,0)

**71) The distance of the point (x, y, z)
from the x y plane is**

**a)**x

**b)**y

**c)**z

**d)**None

**72) The graph of the equation
in the 3 – dimensional space is **

**a)**x axis

**b)**y axis

**c)**z axis

**d)**xy plane

**73) The plane ax + by + cz = 1 meets the co-ordinate
axes in A,B,C . The centroid of the triangle
ABC is**

**a)**(3a, 3b, 3c)

**b)**(a/3, b/3, c/3)

**c)**(a, b,c)

**d)**

**74) If the XOZ plane divides the join of (1,-1,5) and
(2,3,4) in the ratio l
: 1, then the value of l is**

**a)**3

**b)**1/3

**c)**-3

**d)**-1/3

**75) The co-ordinates of the foot of the perpendicular drawn from the point
A (1,0,3) to the join of the points B (4,7,1) and
C(3,5,3) are**

**a)**(5,7,17)

**b)**(5,-5,17)

**c)**

**d)**

**76) If l, m, n and are the direction
ratios of two perpendicular lines, then **

**a)**

**b)**

**c)**

**d)**

**77) The numbers (3,4,5) can be****a)** the direction cosines of a line**b)** the direction ratios of a line**c)** the co-ordinates of a point in the plane z = 5**d)** Answer is b or c

**78) The direction cosines of a line equally inclined to the co-ordinate axes
are**

**a)**(1,1,1)

**b)**

**c)**(1,0,0)

**d)**(0, 1, 0)

**79) The projection of the line joining (3,4,5) and (4, 6,3) on the line joining
(-1,2,4) and (1,0,5) is**

**a)**-4/3

**b)**3/4

**c)**2/3

**d)**3/2

**80) The angle between the lines whose direction ratios are
is
**

**a)**

**b)**

**c)**

**d)**none of these

**81) The points ( 4, 7, 8) (2,3,4) (-1,-2,1) and (1,2,5) are the vertices
of a **

**a)**rectangle

**b)**square

**c)**parallelogram

**d)**rhombus

**82) The distance of the plane 2x-3y+6z+14 = 0 from the origin is****a)** 14**b)** 2**c)** -2**d)** 11

**83) A plane meets the co-ordinate axes in A,B, C such that the centroid
of
is ( 3, 3, 3). The equation of the plane is**

**a)**x + y+ z = 3

**b)**3x+ 3y + 3z = 1

**c)**x + y+ z = 9

**d)**9x + 9y + 9z = 1

**84) A line makes angles a , b
, g , d
with the four diagonals of a cube. Then **

**a)**0

**b)**2

**c)**4/3

**d)**5/3

**85) The distance of the point (3, -4, 5) from the y axis is****a)** 3**b)** 5**c)** **d)**

**86) The angle between the planes 2x – y +z = 6 and x+ y + 2z = 7 is****a)** **b)** **c)** **d)**

**87) The intercepts of the plane 2x – 3y + 4z = 12 on the co-ordinate axes
are given by **

**a)**2, -3, 4

**b)**6, -4, -3

**c)**6, -4, 3

**d)**3, -2, 3/2

**88) A plane meets the co-ordinate axes in P, Q, R such that the centroid
of the triangle PQR is (a,b,c) The equation of the plane is**

**a)**

**b)**

**c)**

**d)**

**89) The direction cosines of the normal to the plane 2x – 3y +6z = 7 are****a)** 2, -3, 6**b)** 2/7, -3/7, 6/7**c)** 1/3, 2/3, 7/3**d)** 2/3, -1, -2

**90) The equation of the plane through the points (2, 1, -1) and (-1, 3,
4) and perpendicular to the plane x – 2y
+4z = 0 is given by**

**a)**18x + 17y + 4z= 49

**b)**18x – 17y + 4z = 49

**c)**18x +17 y -4z + 49 = 0

**d)**18x – 17y – 4z +49 = 0

**91) The equation of the line joining the points (-2, 4, 2 ) and (7, -2,
5) are**

**a)**

**b)**

**c)**

**d)**

**92) The equation of the line through the point (1,2,3) and parallel to
the line
are**

**a)**

**b)**

**c)**

**d)**

**93) The line ****a)** parallel**b)** perpendicular**c)** skew**d)** coincident

**94) The line ****a)** parallel to the x axis**b)** parallel to the y axis**c)** parallel to the z axis**d)** lies in a plane parallel to the xy plane

**95) The lines ****a)** parallel**b)** intersecting**c)** skew**d)** perpendicular

**96) The equation of the line passing through (1,2,3) and parallel to the
plane 2x + 3y + z + 15 = 0 are**

**a)**

**b)**

**c)**

**d)**

**97) The equation of the line passing through (-1, 2, -3) and perpendicular
to the plane 2x+3y +z +5 = 0 are**

**a)**

**b)**

**c)**

**d)**

**98) The value of k so that the lines
are perpendicular to each other is **

**a)**10/7

**b)**7/10

**c)**-8/7

**d)**None of these

**99) The distance of the point (-1,-5,10) from the point of intersection
of the line
and the x – y + z = 5 is **

**a)**14/5

**b)**11

**c)**13

**d)**15

**100) The equation of the line through the point (2,3,-5) and equally
inclined to the axes are**

**a)**x – 2 = y – 3 = z + 5

**b)**

**c)**

**d)**

**Answers**