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JEE Main202331 Jan 2023Morning ShiftMathematicsDeterminantsActual

For the system of linear equations x + y + z = 6 α x + β y + 7 z = 3 x + 2 y + 3 z = 14 which of the following is NOT true ?

Options

  1. AIf α = β = 7 , then the system has no solution
  2. BIf α = β and α ≠ 7 then the system has a unique solution.
  3. CThere is a unique point ( α , β ) on the line x + 2 y + 18 = 0 for which the system has infinitely m
  4. DFor every point ( α , β ) ≠ ( 7 , 7 ) on the line x - 2 y + 7 = 0 , the system has infinitely

Correct answer

D. For every point ( α , β ) ≠ ( 7 , 7 ) on the line x - 2 y + 7 = 0 , the system has infinitely

Step-by-step solution

Given system of linear equations: x + y + z = 6 α x + β y + 7 z = 3 x + 2 y + 3 z = 14 On taking determinant, we get A = 1 1 1 α β 7 1 2 3 = ( 3 β - 14 ) - ( 3 α - 7 ) + ( 2 α - β ) = 2 β - α - 7 Here, if α = β = 7 then A = 0 and the system has no solution. So, option A is correct. If α = β ,   α ≠ 7 then the system has a unique solution. So, option B is correct. If ( α ,   β ) ≠ ( 7 ,   7 ) then A &#8800

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