GATE 2025 EC – Question 37
Consider the polynomial $p(s)=s^5+7s^4+3s^3-33s^2+2s-40$. Let $(L,I,R)$ be defined as follows:
$L$ is the number of roots of $p(s)$ with negative real parts.
$I$ is the number of roots of $p(s)$ that are purely imaginary.
$R$ is the number of roots of $p(s)$ with positive real parts.
Which one of the following options is correct?
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Correct answer: (A) $L=2,\ I=2,$ and $R=1$
Explanation
Routh array: $s^5$: 1, 3, 2; $s^4$: 7, −33, −40; $s^3$: 54/7, 54/7; $s^2$: −40, −40; $s^1$: 0 (row of zeros). The auxiliary polynomial from the $s^2$ row is $-40s^2-40=0$, i.e. $s^2+1=0$, giving two purely imaginary roots $\pm j$ ($I=2$). Continuing with its derivative ($-80s$) the first column is $1, 7, 54/7, -40, -80, -40$, which has one sign change, so $R=1$. Then $L=5-2-1=2$.