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Solution


Q.

For the system of linear equations 2x-y+3z=5; 3x+2y-z=7; 4x+5y+αz=β, which of the following is NOT correct       [2023]
1 The system has infinitely many solutions for α=-6 and β=9  
2 The system has infinitely many solutions for α=-5 and β=9  
3 The system has a unique solution for α≠-5 and β=8  
4 The system is inconsistent for α=-5 and β=8  

Ans.

(1)

We have,

2x-y+3z=5

3x+2y-z=7

4x+5y+αz=β

By Cramer's rule,

Δ=|2-1332-145α|=7α+35

Δ1=|5-1372-1β5α|=17α-5β+130

Δ2=|25337-14βα|=11β-α-104

Δ3=|2-1532745β|=7(β-9)

For infinite many solutions, Δ=Δ1=Δ2=Δ3=0

⇒α=-5 and β=9

So, option (a) is incorrect and option (b) is correct.

For unique solution, Δ≠0⇒α≠-5 and β can be any value.

∴Option (c) is correct.

At α=-5 and β=8

Δ=0 and Δ1=5≠0                ∴ Solution is inconsistent.

 So, option (4) is correct.