JEE Main202329 Jan 2023Morning ShiftMathematicsDeterminantsActual
Consider the following system of questions α x + 2 y + z = 1 2 α x + 3 y + z = 1 3 x + α y + 2 z = β For some α , β ∈ ℝ . Then which of the following is NOT correct.
Options
- AIt has no solution if α = - 1 and β ≠ 2
- BIt has no solution for α = - 1 and for all β ∈ ℝ
- CIt has no solution for α = 3 and for all β ≠ 2
- DIt has a solution for all α ≠ - 1 and β = 2
Correct answer
B. It has no solution for α = - 1 and for all β ∈ ℝ
Step-by-step solution
Given, System of questions α x + 2 y + z = 1 2 α x + 3 y + z = 1 3 x + α y + 2 z = β Now finding, D = α 2 1 2 α 3 1 3 α 2 = 0 ⇒ α ( 6 - α ) - 2 ( 4 α - 3 ) + 1 ( 2 α 2 - 9 ) = 0 ⇒ α - 3 ( α + 1 ) = 0 ⇒ α = - 1 , 3 D x = 2 1 1 3 1 1 α 2 β = 0 ⇒ β = 2 ⇒ 2 β - 2 - 1 ( 3 β - α ) + ( 6 - α ) = 0 ⇒ β = 2 D y = α 1 1 2 α 1 1 3 2 β = 0 ⇒ α ( β - 2 )