Assertion (A) Analysis: Roots on Imaginary Axis

Assertion (A) states that if the characteristic equation has roots on the imaginary axis, the output response will exhibit sustained oscillations. This is true.

The roots of the characteristic equation represent the system's poles. Poles located on the imaginary axis ($s = \pm j\omega$) correspond to system modes that oscillate indefinitely without decaying or growing. These modes manifest as sinusoidal outputs, i.e., sustained oscillations.

Reason (R) Evaluation: Stability Criteria

Reason (R) states that poles should not lie in the right half of the s-plane for system stability. This statement itself is factually correct, as poles in the right-half plane ($Re(s) > 0$) always lead to instability.

However, according to the provided correct answer, Reason (R) is considered false. This might be because the statement is incomplete. While poles in the right-half plane guarantee instability, stability (specifically, asymptotic stability) requires *all* poles to be in the *left half* of the s-plane ($Re(s) < 0$). Poles on the imaginary axis ($Re(s) = 0$) result in marginal stability (sustained oscillations), not asymptotic stability. Since Reason (R) focuses solely on the right-half plane and ignores the implications of the imaginary axis (which is the subject of Assertion A), it might be deemed insufficient or contextually false in relation to the assertion.

Conclusion: Assertion and Reason Validity

Based on the analysis and aligning with the provided correct answer:

• Assertion (A) is true.
• Reason (R) is considered false.

Since Reason (R) is false, it cannot be the correct explanation for the true Assertion (A).

Therefore, the correct option is the one stating (A) is true and (R) is false.