# 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 **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)

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)

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
*

**The centre of the required circle is same as the centre of the given circle**

Ans Desc 5)

Ans Desc 5)

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)

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)

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.*

**Let the required ratio be k:1 internal**

Ans Desc 10)

Ans Desc 10)

(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.*

**The equations of the first line are 2x+3y+z-4=0 (!)**

Ans Desc 11)

Ans Desc 11)

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)*

**Any point on the first line is .**

Ans Desc 12)

Ans Desc 12)

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.
*

**The given lines are**

Ans Desc 13)

Ans Desc 13)

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,
*

**Let the given points be A(+2,6,-6), B(+3,10,-9) and**

Ans Desc 14)

Ans Desc 14)

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)
*

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

Ans Desc 16)

Ans Desc 16)

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*

**The given plane is 2x-3y+5z=2 (1)**

Ans Desc 17)

Ans Desc 17)

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
*

**The equation to the sphere on the line joining**

Ans Desc 20)

Ans Desc 20)

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.*

**x =0 and y = 0 represent z-axis, therefore x = a and y = b, represent**

Ans Desc 21)

Ans Desc 21)

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.*

**Let xy-plane divide the line joining A (-1, 3,4) and B (2,-5,6) in**

Ans Desc 22)

Ans Desc 22)

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
Squaring and adding
*

Ans Desc 24)

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

Squaring and adding

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

**The equation of the line**

Ans Desc 25)

Ans Desc 25)

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*

**Let l, m,n be the direction**

Ans Desc 27)

Ans Desc 27)

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.*

**Let O (0,0,0), A (3,4,7)**

Ans Desc 28)

Ans Desc 28)

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
*

**Let the position vectors**

Ans Desc 30)

Ans Desc 30)

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
*

**Equation of the line through**

Ans Desc 31)

Ans Desc 31)

(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.
*

**Let A,B,C be the points**

Ans Desc 33)

Ans Desc 33)

(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
*

**Equation of a plane through**

Ans Desc 34)

Ans Desc 34)

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
*

**Let A be the image of the**

Ans Desc 36)

Ans Desc 36)

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.
*

**We have, z = 0 for the point**

Ans Desc 37)

Ans Desc 37)

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)
*

**Let the foot of the perpendicular**

Ans Desc 38)

Ans Desc 38)

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)

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.*

**For rectangular parallelopiped,**

Ans Desc 41)

Ans Desc 41)

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.

**The mid point of AC**

Ans Desc 43)

Ans Desc 43)

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.*

**Given equation can be**

Ans Desc 44)

Ans Desc 44)

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)

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)
*

**The direction ratios of x – axis are 1, 0, 0. The line passes through**

Ans Desc 51)

Ans Desc 51)

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
*

**The direction ratios of the normals to the two planes are**

Ans Desc 52)

Ans Desc 52)

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*

**The direction ratios of the line are 3 , 4, 5. If a line is parallel**

Ans Desc 55)

Ans Desc 55)

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
*

**The direction ratios of the line joining (2,3,4) and (6,7,8) are 6- 2, 7-3,**

Ans Desc 57)

Ans Desc 57)

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*

**Any plane through the intersection of two planes can be taken as x + 2y +3z**

Ans Desc 58)

Ans Desc 58)

– 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)
*

**Writing the equation to the sphere in the standard form (with coefficients**

Ans Desc 59)

Ans Desc 59)

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
*

**The direction ratio of**

Ans Desc 60)

Ans Desc 60)

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)

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
*

**The required plane can be taken as**

Ans Desc 64)

Ans Desc 64)

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