If $s^{3} + 3 s^{2} + 4 s + A = 0$ , then all the roots of this equation are in the left half plane provided that

A > 12

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- 3 < A < 4

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0 < A < 12

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5 < A < 12

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Solution

The correct option is C $0 < A < 12$
$s^{3} + 3 s^{2} + 4 s + A = 0$
Using R- H criteria
$\begin{matrix} s^{3} & 1 & 4 s^{2} & 3 & A s^{1} & \frac{12 - A}{3} & 0 s^{0} & A \end{matrix}$
For stable system
$A > 0$
$a n d \frac{12 - A}{3} > 0$
$12 - A > 0$
$12 > A$
$0 < A < 12$

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