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If the system of equations 2 x - 3 y + 5 z = 12 , 3 x + y + p z = q and x - 7 y + 8 z = 17 is consistent, then which of the following is not true?

Options

  1. Ap = 2 , q = 7
  2. Bp ≠ 2 , q = 7
  3. Cp ≠ 2 , q ≠ 7
  4. Dp = 2 , q ≠ 7

Correct answer

A. p = 2 , q = 7

Step-by-step solution

Using Cramer’s rule: ∆ = 2 - 3 5 3 1 p 1 - 7 8 = 2 8 + 7 p + 3 24 - p + 5 - 21 - 1 = - 22 + 11 p = - 11 p - 2 ∆ 1 = 12 - 3 5 q 1 p 17 - 7 8 = 12 8 + 7 p + 3 8 q - 17 p + 5 - 7 q - 17 = 33 p - 11 q + 11 = 11 3 p - q + 1 ∆ 2 = 2 12 5 3 q p 1 17 8 = 2 8 q + 17 p - 12 24 - p + 5 51 - q = 11 a - 2 p - 3 ∆ 3 = 2 - 3 12 3 1 q 1 - 7 17 = 2 17 + 7 q + 3 51 - q + 12 - 21 - 1 = 11 q - 77 = 11 q - 7 For unique solution → p ≠ 2 , q ∈ R For infinite solutions → p = 2 , q = 7

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