maths 3d geometry
TRANSCRIPT
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LEVEL-1
Distance between two points in space
1. If L, M are the feet of the perpendiculars from
(2, 4, 5) to the xy-plane, yz-plane respectively,
then the distance LM is
1) 41 2) 20 3) 29 4) 53
2. If A(0, 4, 1), B(a, b, c), C(4, 5, 0), D(2, 6, 2) are
the consecutive vertices of a square then the
distance BD is
1) 34 2) 6 3) 18 4) 23
3. If the extremities of a diagonal of a square are
(1, -2, 3) and (2, -3, 5) then the length of its
side is
1) 6 2) 3 3) 5 4) 7
3-D GEOMETRY
section formula
4. A = (2, 4, 5) and B = (3, 5, -4) are two points. If the
xy-plane, yz-plane divide AB in the ratios
a : b, p : q respectively then q
p
b
a
1)12
232)
12
73)
12
74)
15
22
5. In the ABC if A = (-2, 3, 4) and mid points of
BC, CA, AB are (1, -4, 2), (-5, 2, -3), F
respectively then zx FF
1) 1 2) 13 3) 6 4) 7
collinear points
6. If A = (1, 2, 3), B = (2, 10, 1), Q are collinear
points and 1Qx then zQ
1) -3 2) 7 3) -14 4) -7
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centroid of triangle
7. If (1, 1, a) is the centroid of the triangle
formed by the points (1, 2, -3), (b, 0, 1) and (-1,
1, -4) then a - b =
1) -5 2) -7 3) 5 4) 1
centroid of tetrahedron
8. If (4, 2, p) is the centroid of the tetrahedron
formed by the points (k, 2, -1), (4, 1, 1), (6,2, 5)
and (3, 3, 3) then k + p =
1)3
172) 1 3)
3
54) 5
locus
9. If the sum of the squares of the perpendicular
distances of P from the coordinate axes is 12
then the locus of P is
1) 6zyx 222 2) 6zyx
3) 12zyx 222 4) 12zyx
10. The locus of a point which is equidistant from
yz-plane and zx-plane is
1) x + y = 0 2) 0yx 22
3) 0zyx 222 4) 0yx 33
11. The locus of the point
( cos cosr , cos sinr , sinr )
where , are variables and r is a constant is
1) r zyx 2) 2222 r zyx
3) r zyx 222 4) 2222 r zyx
plane divides line segment
12. If the zx-plane divides the line segment joining
(1, -1, 5) and (2, 3, 4) in the ratio p : 1 then
p + 1 =
1)3
12) 1 : 3 3)
4
34)
3
4
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KEY
1)3 2)4 3)2 4)3 5)2 6)2
7) 1 8)4 9) 1 10) 2 11) 2 12) 4
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LEVEL-1
distance between two points in space
PRACTICE QUESTIONS
1. The name of the figure formed by the points
(3, -5, 1), (-1, 0, 8) and (7, -10, -6) is
1) a triangle 2) a straight line
3) an isosceles triangle 4) an equilateral triangle
2. The point P is on the y-axis. If P is equidistant
from (1,2, 3) and (2,3, 4) then yP
1)2
152) 15 3) 30 4)
2
3
3. Two opposite vertices of a square are (2, -3, 4)
and (4, 1, -2). The length of the side of the square
is
1) 58 2) 72 3) 14 4) 7
section formula
4. A = (1, -1, 2) and B = (2, 3, 7) are two points. If
P, Q divide AB in the ratios 2 : 3, -2 : 3
respectively then yx QP
1)5
382)
5
383)
5
24)
6
47
5. If A = (1, 2, 3), B = (2, 3, 4) and C is a point of
trisection of AB such that 3
13CC yx
then zC
1)3
102)
3
113)
2
114) 11
collinear points
6. If the points (h, 3, -4), (0, -7, 10) and (1, k, 3) are
collinear then h + k =
1) 4 2) 0 3) -4 4) 14
7. The circum centre of the triangle formed by the
points (2, 5, 1), (1, 4, -3) and (-2, 7 , -3) is
1) (6,0,1) 2) (0,6,-1) 3) (-1,6,2) 4) (6,1,-2
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centroid of tetrahedron
8. A = (1, -2, 3), B = (2, 1, 3), C = (4, 2, 1) and
G = (-1, 3, 5) is the centroid of the tetrahedron
ABCD. If yD p and zDq then q 11 p13
1) 0 2) 1 3) -1 4) 2
locus
9. If the locus of a point which is equidistant from
(2, 3, -1) and (-3, 4, -3) is ax + by + cz = d and
a > 0 then a + b + c + d =
1) -4 2) 4 3) 16 4) -2
10. If the ends of the hypotenuse of a right-angled
triangle are (0, 1, 2), (1, 2, 0) and the locus of
the third vertex of the triangle is
0d cz byaxzyx 222 then
d c ba
1) 4 2) -4 3) -8 4) 8
plane divides line segment
11. The ratio in which yz-plane divides the line
segment joining (-3, 4, 2), (2, 1, 3) is
1) -4 : 1 2) 3 : 2 3) -2 : 3 4) 1 : 4
KEY
123 2)1 3)2 4)1 5)2 6)2
7) 2 8)1 9) 1 10) 2 11) 3
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123 2)1 3)2 4)1 5)2 6)2
7) 2 8)1 9) 1 10) 2 11) 3
2- DIRECTION COSINES AND DIRECTION
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RATIOS
LEVEL - I
direction cosines of a directed line
1. The product of the d.cs of the line which makes
equal angles with ox, oy, oz is
1) 1 2) 3 3) 331 4)
3
1
3
1
3
1 , ,
2. The direction cosines of the line passing through
p 2,3, 1 and the origin are
1)2
14
3 1, ,
14 142)
2 3,
14 14
,
1
14
3)2 3 1
, ,14 14 14
4)
2 3 1, ,
14` 14 14
3. The number of lines which make equal angles
with ox, oy, oz where O = (0, 0, 0) is
1) 8 2) 4 3) 1 4) 2
4. If A (2, 4, 5), B(-7, -2, 8), C are collinear points
then C =
1) (1, 2, 6) 2) (2, -1,6) 3) (-1, 2, 6) 4) (2, 6, -1)
5. If a line makes angles 43
,
with the x-axis, y-
axis respectively then the angle made by that line
with the z-axis is
1)2
2)
3
3)
4
4)
12
5
6. ox, oy are posit ive x-axis , posi tive y-axis
respectively where O = (0, 0, 0). The d.cs of
the line which bisects xoy are
1) 1, 1, 0 2) 02
1
2
1 , , 3)
2
10
2
1 , , 4) 0, 0, 1
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direction cosines with direction ratios
7. If O = (0, 0, 0), OP = 5 and the d.rs of OP are
1, 2, 2 then z y x PPP
1) 25 2)9
253)
3
25 4)
3
10
3
10
3
5 , ,
8. The coordinates of a point P are(3,12,4) w.r.t
origin O, then the direction cosines of OP are
1) 3,12,4 2)1 1 1
, ,4 3 2
3)3 1 2
, ,13 13 13
4)3 12 4
, ,13 13 13
9. The direction ratios of the diagonal of the cube
joining the origin to the opposite corner are
(when the 3 concurrent edges of the cube are
coordinate axes)
1)2 2 2
, ,3 33
2) 1,1,1
3) 2,-2,1 3) 1,2,3
angle between two lines
10. If the angle between two lines whose d.rs are
2, 3, -1 and 0, k, -2 is o90 then k =
1)23 2)
32 3)
32 4)
31
11. If
is the angle between two lines whose d.rs
are 1, -2, 1 and 4, 3, 2 then
22
eccossec =
1) 2 2) 3) 22 4) 22
1
d.r’s of normal to the plane
12. If the d.rs of OA and OB are 1, -1, -1 and 2, -1,
1 then the d.cs of the line perpendicular to both
OA and OB are
1) 0,1, -1 2) -2, -3,1
3)14
1
14
3
14
2 , ,
4)
41
1
41
3
41
2 , ,
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13. The co-ordinate of the foot of the perpendicu-
lar from (1,8,4) to the line joining the points
(0,1,3), (2,-3,-1)
1)5 2 19
, ,3 3 3
2)1 2 19
, ,3 3 3
3) 5 2 19, ,3 3 3
4) 1 2, ,13 3
14. If P = (3, 4, 5), Q = (4, 6, 3), R = (-1, 2, 4) and
S = (1, 0, 5) are four points then the projection
of RS on PQ is
1)3
82)
3
43) 4 4) 0
15. If the projections of the line segment AB on the
coordinate planes are 2, k, 6 and2
7 AB then
12 k k
1) 7 2) -7 3) 0 4) 1
16. The projections of a line segment on x,y and
z axes are respectively 2,3,5 . The length
of the line segment is
1) 6 2) 11 3) 8 4) 5
17. The name of the figure formed by the points
(-2, 4, 1), (-1, 5, 5), (2, 2, 5) and (1, 1, 1) is a
1) rectangle 2) rhombus
3) square 4) parallelogram
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KEY
1)1 2)3 3)2 4)3 5)2 6)2
7) 3 8)4 9)2 10) 2 11)3 12) 3
13)3 14)2 15)4 16)1 17) 3
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LEVEL - I
direction cosines with direction ratios
1. If the d.rs of a line are 3,1, 2 3 then the sum
of the d.cs of that line is...
1) 132 2)3 3 1
4
3) 232 4)3 3 2
4
angle between two lines
2. The angle between the lines whose direction
cosines are3 1 3 3 1 3
, , , ,4 4 2 4 4 2
and
,
is
1) 2)2
3)
3
4)
4
3. If A = (1, 2, 3), B = (2, 3, 4), C = (1, -2, 3) and
the lines AB, CD are perpendicular then D =
1) (2, 4, -4) 2) (2, 2, -4) 3) (4, 2, -1) 4) (-4, 2, 1)
4. AB and CD are two perpendicular lines. If the
d.rs of AB are 2,3,4 then the d.rs of CD are
1) -1, 6, -5 2) 1, -6, -5 3) 1, 6, -54) -1, -6, -5
5. T he d. cs of t wo lines a re2
3
4
1
4
3 , , and
k , ,4
1
4
3. If the angle between the lines is o120
then k =
1) 3 2) 3
23)
32
14)
2
3
PROJECTION of a point ON a line
6. If the projections of the line segment AB on the
coordinate axes are 12, 3, k and AB = 13 then
322 k k
1) 0 2) 1 3) 11 4) 27
7. If the projections of the line segment AB on the
yz-plane, zx-plane, xy-plane are 5153160 , ,
respectively then the projection of AB on the
z-axis is
1) 12 2) 13 3) 12 4) 144
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8. If the projections of the line segment AB on the
coordinate axes are 2, 3, 6 then the sum of the
d.cs of the line AB is
1) 11 2) 1 3)
49
114)
7
11
MISCELLANEOUS
9. The name of the figure formed by the points
(-1, -3, 4), (5, -1,1), (7, -4, 7) and (1, -6, 10) is a
1) square 2) rhombus
3) parallelogram 4) rectangle
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KEY
1) 2 2)3 3)1 4)3 5)4 6) 3
7)3 8) 4 9) 2
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PLANES
GENERAL EQUATION OF A PLANE:
1. The product of the d.r's of a line perpendicular
to the plane passing through the points (4,0,0),
(0,2,0) and ( 1,0,1) is
1) 6 2) 2 3) 0 4) 1
2. If the plane 2 3 5 2 0 x y z divides the line
segment joining (1, 2, 3) and (2, 1, k) in the ratio
9 : 11 then k =
1) 1 2) -2 3) -10 4)1
2
3. Equation of the plane through the mid-point of the
join of A(4,5,-10) and B(-1,2,1) and perpen-
dicular to AB is
1) 135
. 5 3 11 02
r i j k
2) 135
. 5 3 112
r i j k
3)3 7 9 ˆ ˆˆ ˆ ˆ ˆ. 5 3 112 2 2
r i j k i j k
4) 185
. 5 3 11 02
r i j k
Normal Form of a Plane:
4. The distance between the planes
2x - 3y + 6z + 12 = 0 and 2x - 3y + 6z - 2 =0 is
1)10
7 2)
2
7 3) 2 4)
24
7
5. The d.c' s of the normal to the plane
2 2 5 0 x y z are
1) (3, -2, 6) 2)2 3 6
, ,
7 7 7
3)
3 2 6, ,
7 7 7
4)2 1 2
, ,3 3 3
Intercept Form of a Plane:
6. (2, 4, 6) is a point on the plane making equals
intercepts on the co-ordinate axes. The square
of the per pendicular distances from the origin
to that plane is
1) 4 3 2) 4 3) 48 4) 16
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7. The equation of a plane which cuts equal inter-
cepts of unit length on the axes is
1) 0 x y z 2) 1 x y z
3) 1 x y z 4) 1 x y z
a a a
Angle between Two Planes:
8. If the planes2 3 7 0
x y z and
4 2 5 9 0 x y kz are parallel then
25 6k
1)6
5
2)
5
6
3)
36
54)
66
5
9. The acute angle between the plane 5x-4y+7z-
13=0 and the y-axis is given by
1)1 5sin
90
2)1 4sin
90
3)1 7
sin 90
4)1 4
sin 90
10. Equation of the plane bisecting the acute
angle between the planes
2 2 3 0 x y x ,3 6 2 2 0 x y z
is
1) 2 8 16 4 27 0 x y z
2) 8 16 4 27 0 x y z
3) 16 32 8 27 0 x y z
4) 16 32 8 27 0 x y z
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KEY
1) 1 2)2 3)2 4)3 5) 4 6) 3
7)2 8) 4 9) 4 10) 1
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PRACTICE QUESTIONS
GENERAL EQUATION OF A PLANE:
1. The ratio of the distances (-1, 1, 3) and (3, 2, 1)
to the plane 2 5 7 9 0 x y z is
1) 2 : 1 2) 1 : 3 3) 1 : 1 4) 1 : 2
2. The equation of the parallel plane lying midway
between the parallel planes 2x-3y+6z-7=0 and
2x-3y+6z+7=0.
1) 2x-3y+6z+1=0 2) 2x-3y+6z-1=0
3) 2x-3y+6z=0 4) 2x-3y+6z = 11
3 Equation of a plane through the line of
intersection of planes 2 3 4 1 x y z and
3 2 0 x y z and parallel to
12 0 x y is
2 3 4 1 3 2 0 x y z x y z
then is
1)1
2 2) 29 3) 4 4)
1
2
4. Distance between parallel planes
2 2 3 0 x y z and 4 4 2 5 0 x y z is
1) 6 2)1
6 3) 3 4)
1
3
5. The normal form of 2 2 5 x y z is
1) 12 4 3 39 x y z
2)6 2 3
17 7 7
x y z
3)12 4 3
313 13 13
x y z
4) 2 2 1 53 3 3 3
x y z
Intercept Form of a Plane:
6. A plane intersects the co ordinate axes at A, B,
C If 0 = (0, 0, 0) and (1, 1, 1) is the centroid of
the tetrahedron O ABC then the sum of the
reciprocals of the intercepts of the plane
1) 12 2)4
3 3) 1 4)
3
4
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7. The area o f the t riangle formed by p lane
2 3 6 9 0 x y z with y - axis, z- axis is
(in Sq.units)
1) 9 2)9
2 3)
9
4 4)
9
8
8. The plane 12 3 4
x y z cuts the axes in A, B,
C then the area of the ABC is (squ)
1) 29 2) 41 3) 61 4) 2 61
KEY
1) 4 2)3 3)3 4)2 5) 4 6) 4
7)3 8) 3