System properties – Match linearity and time-invariance with example relations List I (Property) A. Linear but NOT time-invariant B. Time-invariant but NOT linear C. Linear and time-invariant List II (Relation) y(t) = t^2 x(t) y(t) = t |x(t)| y(t) = |x(t)| y(t) = x(t − 5) Choose the correct mapping.

Introduction / Context:
Testing linearity and time invariance is foundational in signals and systems. Each example relation demonstrates a typical pitfall: time-varying gain, nonlinearity via absolute value, and pure delay.

Given Data / Assumptions:

• Linearity means superposition and homogeneity hold.
• Time invariance means a time shift at the input results in the same time shift at the output.
• t-dependent coefficients indicate time variance.

Concept / Approach:

Check each candidate: y(t) = t^2 x(t) is linear (scales with x) but time-varying → not time-invariant. y(t) = |x(t)| is time-invariant (shift in equals shift out) but nonlinear (absolute value breaks homogeneity). y(t) = x(t − 5) is both linear and time-invariant (pure delay). The relation y(t) = t |x(t)| is neither linear nor time-invariant, and is therefore not used in the correct mapping set.

Step-by-Step Solution:

Verification / Alternative check:

Apply scaling and shifting tests explicitly to confirm each property.

Why Other Options Are Wrong:

Mappings that assign time-invariant status to relations with explicit t-multipliers, or linear status to absolute-value operations, are incorrect.

Common Pitfalls:

Assuming any time-varying coefficient implies nonlinearity (it does not); confusing invariance with causality.

Final Answer:

A-1, B-3, C-4.