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
Answer : Option B
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 


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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
Answer : Option C
Explaination / Solution:

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

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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
Answer : Option B
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

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


Hence lines are perpendicular

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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
Answer : Option A
Explaination / Solution:
No Explaination.


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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
Answer : Option D
Explaination / Solution:
No Explaination.


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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
Answer : Option B
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 )


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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
Answer : Option D
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

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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 )
Answer : Option D
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 )

 


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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
Answer : Option B
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


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