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GATE 2025 EE – Question 29

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 1 mark · Multiple select

The open-loop transfer function of the system shown in the figure, is

$$G(s)=\frac{Ks(s+2)}{(s+5)(s+7)}$$

For $K\ge0$, which of the following real axis point(s) is/are on the root locus?

Diagram for GATE 2025 EE question 29
  1. −1
  2. −4
  3. −6
  4. −10

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Correct answer: (A) −1; (C) −6

Explanation

The real-axis root locus lies to the left of an odd number of open-loop poles and zeros. The critical points are the zeros at 0 and −2 and the poles at −5 and −7. Between 0 and −2 there is 1 point to the right (odd): on the locus, so −1 qualifies. Between −2 and −5 there are 2 (even): −4 is not on it. Between −5 and −7 there are 3 (odd): −6 is on it. Left of −7 there are 4 (even): −10 is not on it.