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BITSAT2019MathematicsDeterminantsActual

If system of equation a x + y + z = a , x + b y + z = b and x + y + c z = c is inconsistent, then which of the following is correct?

Options

  1. Aa b c - a - b - c + 2 = 0
  2. Ba b c - a - b - c + 3 = 0 ,   a = 1
  3. Ca b c - a - b - c + 3 = 0
  4. Da b c - a - b - c + 2 = 0 ,   a ≠ 1 , b ≠ 1 , c ≠ 1

Correct answer

D. a b c - a - b - c + 2 = 0 ,   a ≠ 1 , b ≠ 1 , c ≠ 1

Step-by-step solution

We have, a   x + y + z = a . . . ( i ) x + b y + z = b . . . ( ii ) x + y + c z = c . . . ( iii ) are inconsistent. ∴ a 1 1 1 b 1 1 1 c = 0   and   a 1 1 b b 1 c 1 c ≠ 0 ⇒   a 1 1 1 b 1 1 1 c = 0 ⇒   a ( b c - 1 ) - 1 ( c - 1 ) + 1 ( 1 - b ) = 0 a b c - a - c + 1 + 1 - b = 0 a b c - a - b - c + 2 = 0 and a 1 1 b b 1 c 1 c ≠ 0 It is possible only when a ≠ 1 , b ≠ 1 and c ≠ 1

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