Multiple choice

Let $\Delta =\begin{vmatrix} -bc & { b }^{ 2 }+bc & { c }^{ 2 }+bc \ { a }^{ 2 }+ac & -ac & { c }^{ 2 }+ac \ { a }^{ 2 }+ab & { b }^{ 2 }+ab & -ab \end{vmatrix}$ and the equation $p{ x }^{ 3 }+q{ x }^{ 2 }+rx+s=0$ has roots $a,b,c$, where $a,b,c\in { R }^{ +\quad }$. The value of $\Delta$ is

  1. $\le 9{ r }^{ 2 }/{ p }^{ 2 }$
  2. $\ge 27{ s }^{ 2 }/{ p }^{ 2 }$
  3. $\le 27{ s }^{ 3 }/{ p }^{ 3 }$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The determinant Delta evaluates to (abc)^2 * (a+b+c) * (1/a + 1/b + 1/c). Given roots a, b, c of px^3 + qx^2 + rx + s = 0, we have a+b+c = -q/p, ab+bc+ca = r/p, and abc = -s/p. The expression simplifies using these relations to show Delta >= 27s^2/p^2 based on AM-GM inequality.

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