# 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 x^{2}

+ y^{2} + z^{2} + 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 x^{2} + y^{2}

+ z^{2} – 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)**

x^{2} + y^{2} + z^{2} =

k^{2}

**b)**

x^{2} + y^{2} + z^{2} =

4 k^{2}

**c)**

9 ( x^{2} + y^{2} + z^{2} ) =

4 k^{2}

**d)**

9 ( x^{2} + y^{2} + z^{2} ) =

k^{2}

**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 ( x^{2} + y^{2} + z^{2}

) – 2 ( x + y + z ) – 1 = 0

**b)**

x^{2} + y^{2} + z^{2}

– x – y – z – 1 = 0

**c)**

3 ( x^{2} + y^{2} + z^{2}

) – 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)**

x^{2} + y^{2} + z^{2} = 4k^{2
}

**b)**

x^{2} + y^{2} + z^{2} =

k^{2}

**c)**

2 ( x^{2} + y^{2} + z^{2} )

= k^{2}

**d)** None of the given

Answers