For the system of equations x + y + z = 6 x + 2 y + α z = 10 x + 3 y + 5 z = β , which one of the…

For the system of equations

x+y+z=6

x+2y+αz=10

x+3y+5z=β, which one of the following is NOT true?

  1. System has no solution for α=3, β=24
  2. System has a unique solution for α=-3, β=14
  3. System has infinitely many solutions for α=3, β=14
  4. System has a unique solution for α=3, β≠14

Solution

Given, the system of equations

x+y+z=6

x+2y+αz=10

x+3y+5z=β

Now finding,

△=11112α135

⇒△=110-3α-5-α+3-2

⇒△=6-2α

Now for unique solution, △≠0⇒α≠3

So, for unique solution ⇒α≠3

But in option D, system has unique solution for α=3 which is completely wrong.

Asked in: JEE Main 2023 (06 Apr Shift 2)

Practice more Determinants questions on Aicharya