Let S be the set of all column matrices 
such that b 1 , b 2 , b 2 ∈ R and the system of equation (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 
∈ S ?
Text Solution
Verified by ExpertsB
(a,d)
Sol. Δ = 0 so for at least one solutions Δ 1 = Δ 2 = Δ 3 = 0 ⇒ b 1 + 7b 2 = 13b 3 ..........
option Δ ≠ 0 ⇒ unique solution ⇒ option is correct
option Δ ≠ 0 ⇒ unique solution ⇒ option is correct
option Δ = 0 ⇒ equations are x – 2y + 5z = –b 1
x – 2y + 5z = 
x – 2y + 5z = b 2
There planes are parallel so they must be coincident
⇒ –b 1 = 
= b 3
All b 1 , b 2 , b 3 obtained from equation may not satisfy this relation so option is wrong.
option Δ = 
= 0. Also Δ 1 = 0
For infinite solutions, Δ 2 and Δ 3 must be 0
⇒ 
= 0 ⇒ –b 1 – b 2 + 3b 3 = 0 which does not satisfy for all b 1 , b 2 , b 3 so option(b) wrong
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