JEE Main20231 Feb 2023Evening ShiftMathematicsDeterminantsActual
For the system of linear equations a x + y + z = 1 , x + a y + z = 1 , x + y + a z = β , which one of the following statements is NOT correct?
Options
- AIt has infinitely many solutions if α = 2 and β = - 1
- BIt has no solution if α = - 2 and β = 1
- Cx + y + z = 3 4 if α = 2 and β = 1
- DIt has infinitely many solutions if α = 1 and β = 1
Correct answer
A. It has infinitely many solutions if α = 2 and β = - 1
Step-by-step solution
For infinite solution ∆ = ∆ x = ∆ y = ∆ z = 0 ∆ = α 1 1 1 α 1 1 1 α = 0 α α 2 - 1 - 1 α - 1 + 1 1 - α = 0 ⇒ α 3 - 3 α + 2 = 0 ⇒ α 2 α - 1 + α α - 1 - 2 α - 1 = 0 ⇒ α - 1 α 2 + α - 2 = 0 ⇒ α = 1 , α = - 2 , 1 If β = 1 , then all planes are overlapping ∴ option (d) is correct. Put α = 2 ,   β = 1 the system equations as, 2 x + y + z = 1 x + 2 y