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JEE Main202126 Feb 2021Evening ShiftMathematicsDeterminantsActual

Consider the following system of equations: x + 2 y - 3 z = a 2 x + 6 y - 11 z = b x - 2 y + 7 z = c where a , b and c are real constants. Then the system of equations :

Options

  1. Ahas a unique solution when 5 a = 2 b + c
  2. Bhas no solution for all a , b and c
  3. Chas infinite number of solutions when 5 a = 2 b + c
  4. Dhas a unique solution for all a , b and c

Correct answer

C. has infinite number of solutions when 5 a = 2 b + c

Step-by-step solution

D = 1 2 - 3 2 6 - 11 1 - 2 7 = 20 - 2 ( 25 ) - 3 ( - 10 ) = 20 - 50 + 30 = 0 D 1 = a 2 - 3 b 6 - 11 c - 2 7 = 20 a - 2 ( 7 b + 11 c ) - 3 ( - 2 b - 6 c ) = 20 a - 14 b - 22 c + 6 b + 18 c = 20 a - 8 b - 4 c = 4 ( 5 a - 2 b - c ) D 2 = 1 a - 3 2 b - 11 1 c 7 = 7 b + 11 c - a ( 25 ) - 3 ( 2 c - b ) = 7 b + 11 c - 25 a - 6 c + 3 b = - 25 a + 10 b + 5 c = - 5 ( 5 a - 2 b - c ) D 3 = 1 2 a 2 6 b 1 - 2 c = 6 c + 2 b - 2 ( 2 c - b ) - 10 a = - 10 a + 4 b + 2 c = - 2 ( 5 a - 2 b - c )   for infinite solution D = D 1 =

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