Let S be the set of all column matrices 
such that b 1 , b 2 , b 3 
R and the system of equations (in real variables)
-x + 2y + 5z = b 1
2x-4y + 3z = b 2
x-2y + 2z = b 3
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each 
?
Text Solution
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(a, d)
given equation represent family of lines
2x - 4y + 3z - b 2 + 
(-x + 2y + 5z – b 1 ) = 0 is same as x - 2y + 2z - b 3 = 0 for same 
and 13b 3 =b 1 +7b 2
as planes are non parallel is must represent family of planes for solution to exist
(5 
+ l)x + (2 
+ l)y + (6 
+ 3)z = b 1 + 
b 2
Must be 2x + b + 3z + b 3 = 0 which gives
= 1, b 1 + b 2 + 3b 3 = 0 which is not true always
C 
x - 2y + 5z = -b 1 , x - 2y + 5z = 
, x - 2y + 5z = b 3 as these are parallel plane for specific case when 
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