# 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

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

Ans 1) a

Ans Desc 1)

Ans 2) d

The direction ratios of the lines re 1,2,3 and 2,2,-2.

Ans Desc 2)

The direction ratios of the lines re 1,2,3 and 2,2,-2.

Ans 3) a

Let AB be the given line parallel to
and P be any general point on the line with position vector .
Let O be the origin of reference

Ans Desc 3)

Let AB be the given line parallel to
and P be any general point on the line with position vector .
Let O be the origin of reference

Ans 4) d

Ans Desc 4)

Ans 5) a
The centre of the required circle is same as the centre of the given circle
since they are concentric.
centre is (1,2,3) and

Ans Desc 5)
The centre of the required circle is same as the centre of the given circle
since they are concentric.
centre is (1,2,3) and

Ans 6) a

be
the root of the perpendicular from A to BC.

Ans Desc 6)

be
the root of the perpendicular from A to BC.

Ans 7) d
The direction cosines are given by
Ans Desc 7)
The direction cosines are given by

Ans 8) b

Ans Desc 8)

Ans 9) d

A ( 4,1,0), B( 3,-1,2) C( 5,0,4) and D(6,2,2) are given points

AB=BC=CD=DA ie all four sides and AC=BD ie the diagonals are equal.
Therefore, the figure is a square.

Ans Desc 9)

A ( 4,1,0), B( 3,-1,2) C( 5,0,4) and D(6,2,2) are given points

AB=BC=CD=DA ie all four sides and AC=BD ie the diagonals are equal.
Therefore, the figure is a square.

Ans 10) c
Let the required ratio be k:1 internal

(since any point on yz-plane, x=0, ie, 3k=-2. Therefore k= -2/3
Therefore, 2:3 external.

Ans Desc 10)
Let the required ratio be k:1 internal

(since any point on yz-plane, x=0, ie, 3k=-2. Therefore k= -2/3
Therefore, 2:3 external.

Ans 11) c
The equations of the first line are 2x+3y+z-4=0 (!)
X+y-2z-3=0 (2)
(1) – (2)2y+4z+2=0
(3)
(1) – (2) 3-x+7z+5=0
(4)
(3)
Similarly, the second line in symmetric form is

The angle between the lines

The lines are at right angles.

Ans Desc 11)
The equations of the first line are 2x+3y+z-4=0 (!)
X+y-2z-3=0 (2)
(1) – (2)2y+4z+2=0
(3)
(1) – (2) 3-x+7z+5=0
(4)
(3)
Similarly, the second line in symmetric form is

The angle between the lines

The lines are at right angles.

Ans 12) c
Any point on the first line is .
If the lines are intersecting at this point, then we must
have

satisfying the equation.
Therefore the point of intersection is given by (-1-1, -3(-1),
2(-1)+3. ie; (-3,2,1)

Ans Desc 12)
Any point on the first line is .
If the lines are intersecting at this point, then we must
have

satisfying the equation.
Therefore the point of intersection is given by (-1-1, -3(-1),
2(-1)+3. ie; (-3,2,1)

Ans 13) c
The given lines are
The line (1) passes through A(3,3,8) and has dr’s 1:3:-1.
The line (2) passes through B ( 6,-3,-7) and has dr,s
4: -3:2. Let PQ be the line of shortest distance between
the lines with P and Q on the lines (1) and (2) respectively.

Ans Desc 13)
The given lines are
The line (1) passes through A(3,3,8) and has dr’s 1:3:-1.
The line (2) passes through B ( 6,-3,-7) and has dr,s
4: -3:2. Let PQ be the line of shortest distance between
the lines with P and Q on the lines (1) and (2) respectively.

Ans 14) b
Let the given points be A(+2,6,-6), B(+3,10,-9) and
C(+5,0, +6). Let r be the position vector of any point
P (x,y,z) on the plane
Now AP = position vector of P- Position vector of A
= (xi+yj+zk)- (2i+6j-6k)
=(x-2) i+ (y-6) j+ (z-6)k
AB=PV of B- P.V of A= (3i+10j – 9k) – (2j+6j-6k)= (i+4j-3k)
AC =P.V of C – P.V of A = (5i+0j+6k) – (2i+6j-6k)
= 3i-6j+12k
Since P is on the plane passing through A,B,C the vectors
AP, AB and AC are coplanar.
Ie,

Ans Desc 14)
Let the given points be A(+2,6,-6), B(+3,10,-9) and
C(+5,0, +6). Let r be the position vector of any point
P (x,y,z) on the plane
Now AP = position vector of P- Position vector of A
= (xi+yj+zk)- (2i+6j-6k)
=(x-2) i+ (y-6) j+ (z-6)k
AB=PV of B- P.V of A= (3i+10j – 9k) – (2j+6j-6k)= (i+4j-3k)
AC =P.V of C – P.V of A = (5i+0j+6k) – (2i+6j-6k)
= 3i-6j+12k
Since P is on the plane passing through A,B,C the vectors
AP, AB and AC are coplanar.
Ie,

Ans 15) d

Ans Desc 15)

Ans 16) d
Let the equation of the required plane be a (x-1)+b(y-2)+c(z+3)=0

I is perpendicular to the planes 2x+y+z=5 and 3x+y+2x=6

2a+b+c=0 (2)
3a+b+c=0 (3)

Ans Desc 16)
Let the equation of the required plane be a (x-1)+b(y-2)+c(z+3)=0

I is perpendicular to the planes 2x+y+z=5 and 3x+y+2x=6

2a+b+c=0 (2)
3a+b+c=0 (3)

Ans 17) d
The given plane is 2x-3y+5z=2 (1)
Required plane is 2x-3y+5z= k(2)
passes through the point (1,2,3)
\ 2-3 (2)+5(3)=k ie,
2 – 6+15=k
ie k=11
\ The required plane
is 2x-3y+5z=11

Ans Desc 17)
The given plane is 2x-3y+5z=2 (1)
Required plane is 2x-3y+5z= k(2)
passes through the point (1,2,3)
\ 2-3 (2)+5(3)=k ie,
2 – 6+15=k
ie k=11
\ The required plane
is 2x-3y+5z=11

Ans 18) a

Ans Desc 18)

Ans 19) d

Ans Desc 19)

Ans 20) a
The equation to the sphere on the line joining
as diameter is

Ans Desc 20)
The equation to the sphere on the line joining
as diameter is

Ans 21) d
x =0 and y = 0 represent z-axis, therefore x = a and y = b, represent
a line parallel to z-axis.

Ans Desc 21)
x =0 and y = 0 represent z-axis, therefore x = a and y = b, represent
a line parallel to z-axis.

Ans 22) b
Let xy-plane divide the line joining A (-1, 3,4) and B (2,-5,6) in
the ratio l :1 at the point P, then,

Thus, the ratio is 2 :3 externally.

Ans Desc 22)
Let xy-plane divide the line joining A (-1, 3,4) and B (2,-5,6) in
the ratio l :1 at the point P, then,

Thus, the ratio is 2 :3 externally.

Ans 23) a

Since, the given line makes equal angles with the axes

Ans Desc 23)

Since, the given line makes equal angles with the axes

Ans 24) c

Let the given line AB has
d.c.`s l,m,n, then its projection on x-axis = AB.l = 12
Projection on y -axis = AB.m = 4
Projection on z – axis – AB.n = 3

Ans Desc 24)

Let the given line AB has
d.c.`s l,m,n, then its projection on x-axis = AB.l = 12
Projection on y -axis = AB.m = 4
Projection on z – axis – AB.n = 3

Ans 25) a
The equation of the line
joining B and C is

Ans Desc 25)
The equation of the line
joining B and C is

Ans 26) b

Ans Desc 26)

Ans 27) b
Let l, m,n be the direction
cosines of the line MN which is perpendicular to each of the given lines

It is obvious that the points P (-3,6,0) and Q (-2,0,7) are situated on the
given lines

Length of shortest distance
= Projection of PQ
on the common perpendicular MN

Ans Desc 27)
Let l, m,n be the direction
cosines of the line MN which is perpendicular to each of the given lines

It is obvious that the points P (-3,6,0) and Q (-2,0,7) are situated on the
given lines

Length of shortest distance
= Projection of PQ
on the common perpendicular MN

Ans 28) a
Let O (0,0,0), A (3,4,7)
and B (5,2,6) be the given points.

Ans Desc 28)
Let O (0,0,0), A (3,4,7)
and B (5,2,6) be the given points.

Ans 29) b

Ans Desc 29)

Ans 30) b
Let the position vectors
of given points A and B be

and that of the variable point P be

Ans Desc 30)
Let the position vectors
of given points A and B be

and that of the variable point P be

Ans 31) a
Equation of the line through
(1,-2,3) and parallel to the given line

Ans Desc 31)
Equation of the line through
(1,-2,3) and parallel to the given line

Ans 32) b

Ans Desc 32)

Ans 33) a
Let A,B,C be the points
(u,0,0), (0, v,0) , (0,0,w) respectively so that intercepts made by the plane
on co-ordinate axes are u, v and w.

Ans Desc 33)
Let A,B,C be the points
(u,0,0), (0, v,0) , (0,0,w) respectively so that intercepts made by the plane
on co-ordinate axes are u, v and w.

Ans 34) c
Equation of a plane through
the line of intersection of given planes is

Ans Desc 34)
Equation of a plane through
the line of intersection of given planes is

Ans 35) b

Ans Desc 35)

Ans 36) b
Let A be the image of the
point P (1,3,4) in the given plane. The equation of the line through P and normal
to the given plane is

Ans Desc 36)
Let A be the image of the
point P (1,3,4) in the given plane. The equation of the line through P and normal
to the given plane is

Ans 37) c
We have, z = 0 for the point
where the line intersects the curve.

Ans Desc 37)
We have, z = 0 for the point
where the line intersects the curve.

Ans 38) b
Let the foot of the perpendicular
from the origin on the given plane be P (a , b , c). Since the
plane passes through A(a,b,c)

Ans Desc 38)
Let the foot of the perpendicular
from the origin on the given plane be P (a , b , c). Since the
plane passes through A(a,b,c)

Ans 39) b

The equation of any plane
passing through given line is

If this plane parallel to z-axis whose direction cosines are 0,0,1;
then the normal to the plane will be perpendicular to z-axis

Ans Desc 39)

The equation of any plane
passing through given line is

If this plane parallel to z-axis whose direction cosines are 0,0,1;
then the normal to the plane will be perpendicular to z-axis

Ans 40) c

Ans 41) d
For rectangular parallelopiped,
the lengths of the edges are obtained by subtracting the corresponding co-ordinates.
Therefore lengths of the edges are 5-2=3, 7-3=4 and 9-7=2.

Ans Desc 41)
For rectangular parallelopiped,
the lengths of the edges are obtained by subtracting the corresponding co-ordinates.
Therefore lengths of the edges are 5-2=3, 7-3=4 and 9-7=2.

Ans 42) b

Ans Desc 42)

Ans 43) d
The mid point of AC
is
and so is that of BD. Thus ABCD is a parallelogram. It is easy to verify
that AB=BC =CD=DA=7,
So, ABCD is a rhombus. Now direction ratios of AB and BC are -2,3,-6
and 6,2,-3 respectively. The product of these two sets is
(-2)(6)+(3)(2)+(-6)(-3)=120.
This means that AB is not perpendicular to BC and hence ABCD is not a square.

Ans Desc 43)
The mid point of AC
is
and so is that of BD. Thus ABCD is a parallelogram. It is easy to verify
that AB=BC =CD=DA=7,
So, ABCD is a rhombus. Now direction ratios of AB and BC are -2,3,-6
and 6,2,-3 respectively. The product of these two sets is
(-2)(6)+(3)(2)+(-6)(-3)=120.
This means that AB is not perpendicular to BC and hence ABCD is not a square.

Ans 44) d
Given equation can be
factorized as (3x+y+3z)(4x-2y-2z)=0

Which clearly represent two planes passing through origin.

Ans Desc 44)
Given equation can be
factorized as (3x+y+3z)(4x-2y-2z)=0

Which clearly represent two planes passing through origin.

Ans 45) b

Co-ordinate of centroid
G of the tetrahedron OABC are

Ans Desc 45)

Co-ordinate of centroid
G of the tetrahedron OABC are

Ans 46) a

Ans Desc 46)

Ans 47) c

Ans Desc 47)

Ans 48) b

X+Y-Z=1, X-Y+Z=1, -X+Y+Z=1.
The coefficient matrix is

Clearly,
,so the given system of equations has a unique solution.

Ans Desc 48)

X+Y-Z=1, X-Y+Z=1, -X+Y+Z=1.
The coefficient matrix is

Clearly,
,so the given system of equations has a unique solution.

Ans 49) a

Ans Desc 49)

Ans 50) d

Ans Desc 50)

Ans 51) a
The direction ratios of x – axis are 1, 0, 0. The line passes through
the point (a, b, c)

Ans Desc 51)
The direction ratios of x – axis are 1, 0, 0. The line passes through
the point (a, b, c)

Ans 52) b
The direction ratios of the normals to the two planes are
3, -4, 5 and 2, -1, -2

Ans Desc 52)
The direction ratios of the normals to the two planes are
3, -4, 5 and 2, -1, -2

Ans 53) a

Ans 54) b

Ans 55) a
The direction ratios of the line are 3 , 4, 5. If a line is parallel
to the plane with direction ratios of the normal a,b,c then 3 a
+ 4 b +5 c = 0
Taking the plane 2x + y- 2z = 0, its direction ratios o f the normal
are 2, 1, -2

3 (2) + 4 (1) + 5 (-2) = 0

Ans Desc 55)
The direction ratios of the line are 3 , 4, 5. If a line is parallel
to the plane with direction ratios of the normal a,b,c then 3 a
+ 4 b +5 c = 0
Taking the plane 2x + y- 2z = 0, its direction ratios o f the normal
are 2, 1, -2

3 (2) + 4 (1) + 5 (-2) = 0

Ans 56) b

Ans Desc 56)

Ans 57) a
The direction ratios of the line joining (2,3,4) and (6,7,8) are 6- 2, 7-3,
8-4 = 4,4,4
Since the plane is at right angles to the given line its equation can be taken
as x+y+z=K (1)
It bisects the line joining (2,3,4) and (6,7,8) and hence it passes through
the mid point =
4,5,6) 4+5+6=k

Ans Desc 57)
The direction ratios of the line joining (2,3,4) and (6,7,8) are 6- 2, 7-3,
8-4 = 4,4,4
Since the plane is at right angles to the given line its equation can be taken
as x+y+z=K (1)
It bisects the line joining (2,3,4) and (6,7,8) and hence it passes through
the mid point =
4,5,6) 4+5+6=k

Ans 58) a
Any plane through the intersection of two planes can be taken as x + 2y +3z
– 4 + k (4x + 3y +2z +1) = 0
It passes through (0,0,0)
-4 + k (1) = 0 Þ
k = 4
I Þ
x + 2y + 3Z – 4 +4 (4 x+3y+2z +1) = 0
x + 2y +3z – 4 + 16x+ 12y +8z + 4 = 0
Þ 17 x + 14 y + 11 z = 0

Ans Desc 58)
Any plane through the intersection of two planes can be taken as x + 2y +3z
– 4 + k (4x + 3y +2z +1) = 0
It passes through (0,0,0)
-4 + k (1) = 0 Þ
k = 4
I Þ
x + 2y + 3Z – 4 +4 (4 x+3y+2z +1) = 0
x + 2y +3z – 4 + 16x+ 12y +8z + 4 = 0
Þ 17 x + 14 y + 11 z = 0

Ans 59) b
Writing the equation to the sphere in the standard form (with coefficients
of unity)

Ans Desc 59)
Writing the equation to the sphere in the standard form (with coefficients
of unity)

Ans 60) c
The direction ratio of
are 3,-4,5. The direction cosine of
are
If is
assumed to make acute angle with x axis, then the direction cosine of
are

Ans Desc 60)
The direction ratio of
are 3,-4,5. The direction cosine of
are
If is
assumed to make acute angle with x axis, then the direction cosine of
are

Ans 61) d

Let the ratio be k:1

Let P (x, y, z) be the point which divides the join of A (2,3,4) and
B ( -1,4,5) in the ratio k :1

P(x,y,z) lies in the plane 2x+2y-z+2=0
ie,

Therefore the required ratio is -5:1

Ans Desc 61)

Let the ratio be k:1

Let P (x, y, z) be the point which divides the join of A (2,3,4) and
B ( -1,4,5) in the ratio k :1

P(x,y,z) lies in the plane 2x+2y-z+2=0
ie,

Therefore the required ratio is -5:1

Ans 62) b

Ans 63) a

Ans Desc 63)

Ans 64) a
The required plane can be taken as
3x-4y-5z-10+l (2x+2y-3z-4)=0 _________(1)
Where l is to be determined.
Since the plane is parallel to the line x=2y=3z, its normal is ^
r to the line. The direction ratios of this line are 6,3,2 since
x=2y=3z

The direction ratios of the normal to the plane (1) are
Since the normal is perpendicular to the line

Ans Desc 64)
The required plane can be taken as
3x-4y-5z-10+l (2x+2y-3z-4)=0 _________(1)
Where l is to be determined.
Since the plane is parallel to the line x=2y=3z, its normal is ^
r to the line. The direction ratios of this line are 6,3,2 since
x=2y=3z

The direction ratios of the normal to the plane (1) are
Since the normal is perpendicular to the line

Ans 65) b

Ans 66) b

Ans 67) d

Ans 68) a

Ans 69) c

Ans 70) a

Ans 71) b

Ans 72) b

Ans 73) a

Ans 74) b

Ans 75) b

Ans 76) a

Ans 77) c

Ans 78) a

Ans 79) b

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