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AP EAMCET202420 May 2024Evening ShiftMathematicsSequences and SeriesActual

If the roots of the equation x^3+a x^2+b x+c=0 are in arithmetic progression, then

Options

  1. Aa^3-3 a b+c=0
  2. B9 a b=2 a^3+27 c
  3. Ca^2-2 b c+c=0
  4. D3 a b-3 c-a^3=0

Correct answer

B. 9 a b=2 a^3+27 c

Step-by-step solution

Let the terms of AP be A -d, ~A , ~A +d aligned & 3 ~A =-a ~A = -a 3 & ~A ( ~A -d)+ A ( A +d)+ A ^2-d^2=b aligned 3 ~A ^2-d^2=b ... (i) A ( A ^2-d^2 )=c -a 3 ( ~A ^2+d^2 )=-c ~A ^2-d^2= 3 c a ... (ii) From equation (i) - (ii), aligned & 2 ~A ^2=b- 3 c a & 2 a^2 9 =b- 3 c a 2 a^3=9 a b-27 c & 9 a b=2 a^3+27 c aligned

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