Topic: Introduction to Three Dimensional Geometry (Test 1)

Topic: Introduction to Three Dimensional Geometry
Q.1
The distance of the point ( 3, 4, 5) from X- axis is
A. 5
B. √41
C. 3
D. √34
Explaination / Solution:

The distance of the point ( 3, 4, 5) from X- axis is

let L be the foot of perpendicular from the point ( 3, 4, 5) to X axis ,then coordinate of L will be (3,0,0)       [ because on X axis y and z coordinate are zero]

then distance of the point ( 3, 4, 5) from X- axis i.e. from L (3,0,0) is given by

Workspace
Report
Q.2
the numbers 3, 4 , 5 can be
A. coordinates of a point on the line y = 4 , z = 0
B. direction cosines of a line in space
C. direction numbers of a line in space
D. coordinates of a point in the plane x + y – z = 0
Explaination / Solution:

the numbers 3, 4 , 5 can be direction ratio of any line these not satisfying any other option

Workspace
Report
Q.3
The equation xy = 0 in three dimensional space represents
A. a plane
B. a pair of planes at right angles
C. a pair of straight lines
D. a pair of parallel lines
Explaination / Solution:

since xy=0 implies x=0 or y=0.i.e YZ plane or XZ plane.Hence it represents a pair of planes at right angles

Workspace
Report
Q.4
The lines having drs as a1a2+b1b2+c1c2=0 are
A. coincident
B. skew
C. perpendicular
D. parallel
Explaination / Solution:

Hence lines are perpendicular

Workspace
Report
Q.5
The plane x = 0 divides the join of ( - 2 , 3 , 4 ) in the ratio
A. it is 2 : 1
B. it is 3 : 2
C. it is 1: 2
D. it is - 4 : 3
Explaination / Solution:
No Explaination.

Workspace
Report
Q.6
The locus of the equation xy + yz = 0 is
A. none of these.
B. a pair of parallel planes
C. a pair of straight lines
D. a pair of perpendicular planes
Explaination / Solution:
No Explaination.

Workspace
Report
Q.7
The point equidistant from the points ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 2 , 0 ) , and ( 0 , 0 , 3 ) is
A. ( 1 , 2 ,3 )
B. ( 1/2 , 1 , 3/2 )
C. ( - 1/2 ,- 1 , - 3/2 )
D. None of these
Explaination / Solution:

let the point equidistant from the points ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 2 , 0 ) , and ( 0 , 0 , 3 ) is (x ,y ,z)

then according to the given condition and  distance formula between two points we have

taking ist two expressions and solving them we get

Similarly by taking ist and 3rd we get y = 1 and by taking ist and 4th we get z = 3/2

So the required point is ( 1/2 , 1 , 3/2 )

Workspace
Report
Q.8
Volume of a tetrahedron is k X area of one face X length of perpendicular from the opposite vertex upon it, where k is
A. it is 1/3
B. it is 1/2
C. it is 1/4
D. it is 1/6
Explaination / Solution:

Volume of tetrahedron  where a,b,c are co-terminus edges of tetrahedron.a X b is area of one face and c is the perpendicular from the opposite vertex

Workspace
Report
Q.9
The centre of the sphere , which passes through ( a , 0 , 0 ) , ( 0 , b , 0 ) ( 0 , 0 , c ) and ( 0 , 0 ,0 ) is ? where abc ≠ 0
A. ( 0 , 0 , c/2 )
B. ( 0 , b/2 , 0 )
C. ( a/2 , 0 ,0 )
D. ( a/2 , b/2 , c/2 )
Explaination / Solution:

General equation of the sphere is ---------------------1)

Since 1) passes through the point (0,0,0) using this in 1) we get d=0

Similarly 1) passes through ( a , 0 , 0 ) , ( 0 , b , 0 ) ( 0 , 0 , c ) using these values in 1)

But as abc0  So , a 0 ,b 0 ,c 0

So from above equations , we have a =  - 2g , b= - 2f , c = -2h

centre is (-f ,-g , -h) = ( a/2 , b/2 , c/2 )

Workspace
Report
Q.10
Three planes x + y = 0 , y + z = 0 , and x + z = 0
A. meet taken two at a time in parallel lines
B. meet in a unique point
C. None of these
D. meet in a line
Explaination / Solution:

Explanation:

x + y = 0           (1)

y + z = 0            (2)

x + z = 0             (3)

Subtracting 1 and 2 we get x-z=0   (4)

adding 3 and 4 we get x-0,y=0 and z=0

Workspace
Report