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Let $x(t)$ be the input and $y(t)$ be the output of a continuous-time system. Match the system properties P1, P2, and P3 with the system relations R1, R2, R3, and R4. Properties P1: Linear but not time-invariant P2: Time-invariant but not linear P3: Linear and time-invariant Relations R1: $y(t)=t^2x(t)$ R2: $y(t)=t|x(t)|$ R3: $y(t)=|x(t)|$ R4: $y(t)=x(t-5)$

1. (P1, R1), (P2, R3), (P3, R4)
2. (P1, R2), (P2, R3), (P3, R4)
3. (P1, R3), (P2, R1), (P3, R2)
4. (P1, R1), (P2, R2), (P3, R3)

Correct answer: (P1, R1), (P2, R3), (P3, R4)

Solution

For $y(t)=t^2x(t)$, the system is linear because scaling and addition pass through multiplication by $t^2$, but it is not time-invariant since the factor depends explicitly on time. For $y(t)=|x(t)|$, the system is time-invariant but not linear. The delay $y(t)=x(t-5)$ is both linear and time-invariant.

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