Multiple choice

Let $a,b,c\in R$ such that o two of them are equal and satisfy $\left| \begin{matrix} 2a & b & c \ b & c & 2a \ c & 2a & b \end{matrix} \right| =0$, then equation ${ 24ax }^{ 2 }+4bx+c=0$ has

  1. attest one root in [0, 1/2]
  2. attest one root in $\left[ -\dfrac { 1 }{ 2 } ,0 \right] $

  3. attest one root in [1, 2]

  4. None of these
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A Correct answer
Explanation

Interpreting the garbled condition as no two of a, b, and c being equal, the determinant condition reduces to 2a + b + c = 0. Substituting this relation into the quadratic shows that at least one root lies between 0 and 1/2.

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