maths 3d geometry

8/18/2019 Maths 3d Geometry

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1

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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9

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


(-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


coordinate axes are 12, 3, k and AB = 13 then

322 k k

1) 0 2) 1 3) 11 4) 27


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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12


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


(-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

maths 3d geometry

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