GATE EC 2017 Set 1 — Question 5

Let's analyze each statement:Statement I: There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane.
For a continuous-time LTI system to be BIBO stable, its region of convergence (ROC) must include the $j\omega$-axis. If there is a pole in the right half of the complex plane (RHP), the ROC cannot include the $j\omega$-axis if the system is causal. However, a non-causal system can be BIBO stable even with RHP poles, provided its ROC is to the left of the RHP pole and includes the $j\omega$-axis. For example, a system with a pole at $s=1$ can be stable if its ROC is $\text{Re}(s) < 1$. Such a system would be non-causal. Therefore, statement I is false.Statement II: There is no causal and BIBO stable system with a pole in the right half of the complex plane.
For a causal LTI system, the ROC is always to the right of the rightmost pole. For the system to be BIBO stable, the ROC must include the $j\omega$-axis. If a causal system has a pole in the RHP, its ROC will be to the right of this RHP pole, and thus the $j\omega$-axis will not be included in the ROC. Therefore, a causal system with a pole in the RHP cannot be BIBO stable. This statement is true.Based on the analysis:
Statement I is false.
Statement II is true.Therefore, only statement II is true.The final answer is $\boxed{\text{D}}$