Mathematics 3D Geometry MCQ
Mathematics 3D Geometry MCQ
Mathematics 3D Geometry MCQ
Page 1
(1)
(a)
(b) -
3 5
(c)
3 4
(d) -
4 3
(2)
If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining 2 2 2 the centres of the spheres x + y + z + 6x - 8y - 2z = 13 2 2 2 and x + y + z - 10x + 4y - 2z = 8, then a equals (a) -1 (b) 1 (c) -2 (d) 2 [ AIEEE 2005 ]
ra
^ ^
ce .c
^ ^ ^
(3)
2 i - 2 j + 3 k + ( i - j + 4 k ) and the
plane (a) 10 9
.(
i + 5 j + k ) = 5 s (b)
xa
10
3 3
3 10
(d)
10 3
(4)
w w
.e
(5)
+ y + z - x + z - 2 = 0 in a circle
(a) 3
( 6 ) A line makes the same angle with each of the X- and Z- axis. If the angle , which 2 2 2 it makes with the y-axis, is such that sin = 3 sin , then cos equals (a) 2 3 (b) 1 5 (c) 3 5 (d) 2 5 [ AIEEE 2004 ]
om
^
[ AIEEE 2005 ]
[ AIEEE 2005 ]
[ AIEEE 2005 ]
Page 2
(7)
Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is (a) 3 2 (b) 5 2 (c) 7 2 (d) 9 2 [ AIEEE 2004 ]
(8)
ce .c
2
A line with direction cosines proportional to 2, 1, 2 meets each o the nes x = y + a = z and x + a = 2y = 2z. The coordinates of each of the p ints of intersection are given by
(9)
xa
(a) -2
(b) -1
(c)
1 2
ra
(d) 0
2 2
t , y = 1 + t, 2 z = 2 - t, with parameters s and t respectively, are co-planar, then equals If the straight lines x = 1 + s, y = - 3 - s, z = 1 + s and x = [ AIEEE 2004 ]
( 10 ) The intersection of the sphe es x + y + z + 7x - 2y - z = 13 and 2 2 2 x + y + z - 3x + 3y + 4z = 8 is the same as the intersection of one of the spheres and the plane (a) x - y - z = ( c ) x - y 2z = 1
.e
( b ) x - 2y - z = 1 ( d ) 2x - y - z = 1
w w
( 12 ) The lines
x -2 1
y -3 1
z -4 -k
and
x -1 = k
y -4 2
( a ) k = 0 or - 1 ( c ) k = 0 or - 3
( b ) k = 1 or - 1 ( d ) k = 3 or - 3
om
= z -5 1
[ AIEEE 2004 ]
[ AIEEE 2004 ]
[ AIEEE 2003 ]
are coplanar, if
[ AIEEE 2003 ]
Page 3
( 13 )
Two systems of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and a, b c from the origin, then (a) (c) 1 a2 1 + 1 + 1 + 1 + 1 + 1 = 0 (b) (d) 1 a2 1 a
2
( 14 ) The direction cosines of the normal to the plane x + 2y - 3z (a) (c) 1 14 1 14 , , 2 14 2 14 , 3 14 , 3 14 (d) (b) 1 14 1 14 , 2 , 3
ce .c
14 2 1 , 14 3 14
2 2 2
ra
(d) 4 ( d ) 11 ( d ) 13
( 15 ) The radius of a circle in which the sphe e x + y + z the plane x + 2y + 2z + 7 = 0 is (a) 1 (b) 2 (c) 3
+ 2x - 2y - 4z = 19 is cut by
( 16 ) The shortest distance f om the plane 12x + 4y + 3z = 327 to the sphere 2 2 2 x + y + z + 4x - 2y - 6z = 155 is ( a ) 13
.e
( b ) 26
xa
( c ) 39
w w
( 17 ) The dist nce of a point ( 1, - 2, 3 ) from the plane x - y + z = 5 and parallel to the y z x line = is = 2 3 -6 (a) 1 (b) 7 (c) 3 [ AIEEE 2002 ]
( 18 )
The co-ordinates of the point in which the line joining the points ( - 2, 1, 8 ) and intersected by the YZ-plane are 13 , 2 ( a ) 0, 5 13 2 , ( c ) 0, 5 5 13 ( b ) 0, , -2 5 13 2 ( d ) 0, , 5 5
om
a'
2
1 b2 1
1 c2 1
1 a' 2 1
+ -
1 b' 2 1
2
1 c' 1
b'
= 0 c2 [ AIEEE 2003 ]
4 = 0 are
[ AIEEE 2003 ]
[ AIEEE 2003 ]
[ AIEEE 2003 ]
( 3, 5, - 7 )
and
[ AIEEE 2002 ]
Page 4
( 20 ) If the lines
and
y - 5 x - 1 z - 6 = = 3k 1 -5
(b) -
( c ) - 10
ce .c
(d) -7
7 10
om
are at right
[ AIEEE 2002 ]
( 21 )
vector
perpendicular
to
the
plane
of
2 i - 6 j - 3k
and
b = 4 i + 3 j - k 4 i + 3 j - k 26
is
(a)
(b)
2 i - 6 j - 3k 7
(c)
3 i - 2 j + 6k 7
ra
(d)
2 i - 3 j - 6k 7
[ AIEEE 2002 ]
xa
i, 2 j
and 3 k
is
.e
(b)
i + 2 j + 3k
(c)
6 i + 3 j + 2k 7
(d)
6 i + 3 j + 2k 7
[ AIEEE 2002 ]
w w
(a) 1
( 23 ) A plane at a unit distance from the origin intersects the coordinate axes at P, Q and 1 1 1 = k, R If the locus of the centroid of PQR satisfies the equation + + 2 2 x y z2 then the value of k is (b) 3 (c) 6 (d) 9 [ IIT 2005 ]
y +1 x -1 z -1 = = 2 3 4 (b) 9 2 (c) 2 9
and
y -k x -3 z = = 1 2 1
(d) 2
[ IIT 2004 ]
Page 5
( 25 ) If the line
( 26 )
( a ) 12
( b ) 23
( c ) 67
( d ) 47
(a) 3
(b) 4
(c) 5
(a) 1
(b) 2
xa
( 28 ) The ratio of magnitudes of tota surface area to volume of a right circular cone with vertex at origin, having sem - ertical angle equal to 30 and the circular base on the plane x + y + z = 6 s c) 3 (d) 4
( 29 )
The direct on of normal to the plane passing through origin and the line of intersection of the planes x + 2y + 3z = 4 and 4x + 3y + 2z = 1 is ( a ) ( 1, 2, 3 ) ( b ) ( 3, 2, 1 ) ( c ) ( 2, 3, 1 ) ( d ) ( 3, 1, 2 )
w w
( 30 ) T e volume of the double cone having vertices at the centres of the spheres 2 2 2 2 2 2 x + y + z = 25 and x + y + z - 4x - 8y - 8z + 11 = 0 and the common circle of the spheres as the circular base of the double cone is ( a ) 24 ( b ) 32 ( c ) 28 ( d ) 36
.e
ra
(d) 6 ( d ) 64
( 27 ) The mid-points of the chords cut off by th lines through the point ( 3, 6, 7 ) 2 2 2 intersecting the sphere x + y + z - 2x 4y - 6z = 11 lie on a sphere whose radius =
ce .c
2
There are infinite planes passing through the points ( 3, 6, 7 ) touching the sphere 2 2 2 x + y + z - 2x - 4y - 6z = 11. If the plane passing through th circle of contact cuts intercepts a, b, c on the co-ordinate axes, then a + b + c =
( 31 )
om
2 2
(a) 7
( d ) no real value
[ IIT 2003 ]
= 36 in points
Page 6
( 32 )
+ y
+ z
= 144 from th
( 34 )
The equation of the plane containing the line x + y through the point ( 1, 1, 1 ) is ( a ) 3x + 4y - 5z = 2 (c) x + y + z = 3 ( b ) 4x + 5y - 6z 3 ( d ) 3x + 6y - 5z = 4
ra
Answers
9 a 29 b 10 d 30 b 11 a 31 d
ce .c
( d ) 36 12 c 32 c 13 d 33 d 14 d 34 d 15 c 35 b
( c ) ( 4, 8, 8 )
z = 0 = 2x - y + 4 and passing
om
(d)
2 2 2 2
point ( 2, 4, 4 )
- 4, - 8, - 8 )
( 35 )
A plane passes through the points of intersection of the spheres x + y + z = 36 2 2 2 and x + y + z - 4x - 4y - 8z - 12 = 0. A line joining the centres of the spheres intersects this plane at ( a ) ( 1, 1, 1 )
xa
(b
( 1, 1 2 )
( c ) ( 1, 2, 1 )
( d ) ( 2, 1, 1 )
.e
( 36 )
The area of the circle formed by the intersection of the spheres x + y + z 2 2 2 x + y + z - 4x - 4y - 8z - 12 = 0 is (a) 9 ( b ) 18 ( c ) 27
= 36 and
w w
( 37 )
A line joining the points ( 1, 1, 1 ) and ( 2, 2, 2 ) intersects the plane x + y + z = 9 at the point ( a ) ( 3, 4, 2 ) ( b ) ( 2, 3, 4 ) ( c ) ( 3, 2, 4 ) ( d ) ( 3, 3, 3 )
1 a 21 c
2 c 22 c
3 b 23 d
4 b 24 b
5 b 25 a
6 c 26 d
7 c 27 a
8 b 28 c
16 a 36 c
17 a 37 d
18 a 38
19 d 39
20 a 40