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JEE Main2003MathematicsDeterminantsActual

If the system of linear equations x+2 a y+a z=0x+3 b y+b z=0x+4 c y+c z=0 has a non-zero solution, then a , b , c

Options

  1. Asatisfy a+2 b+3 c=0
  2. Bare in A.P.
  3. Care in G.P.
  4. Dare in H.P.

Correct answer

D. are in H.P.

Step-by-step solution

| array lll 1 & 2 a & a 1 & 3 ~b & ~b 1 & 4 c & c array |=0 C ₂ C ₂-2 C ₃ | array ccc 1 & 0 & a 1 & ~b & ~b 1 & 2 c & c array |=0 R ₃ R ₃- R ₂, R ₂ R ₂- R ₁ | array ccc 1 & 0 & a 0 & b & b-a 0 & 2 c-b & c-b array |=0b(c-b)-(b-a)(2 c-b)=0 On simplification, 2 ~b = 1 a + 1 c a , b , c are in Harmonic Progression.

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