The correct answer is (P1, R1), (P2, R3), (P3, R4).
A linear system is one that satisfies the superposition principle, which states that the output of a linear system is the sum of the outputs of the system due to each individual input. A time-invariant system is one that does not change its behavior over time.
Property P1 is linear but not time-invariant. This means that the output of the system is the sum of the outputs of the system due to each individual input, but the output depends on the time at which the input is applied. For example, if the input is $x(t)$, then the output is $y(t) = t^2x(t)$.
Property P2 is time-invariant but not linear. This means that the output of the system does not depend on the time at which the input is applied, but the output is not the sum of the outputs of the system due to each individual input. For example, if the input is $x(t)$, then the output is $y(t) = t|x(t)|$.
Property P3 is linear and time-invariant. This means that the output of the system is the sum of the outputs of the system due to each individual input, and the output does not depend on the time at which the input is applied. For example, if the input is $x(t)$, then the output is $y(t) = |x(t)|$.
Relation R1 is $y(t) = t^2x(t)$. This is a linear relation, because the output is the sum of the outputs of the system due to each individual input. However, it is not time-invariant, because the output depends on the time at which the input is applied. Therefore, R1 matches with P1.
Relation R2 is $y(t) = t|x(t)|$. This is not a linear relation, because the output is not the sum of the outputs of the system due to each individual input. However, it is time-invariant, because the output does not depend on the time at which the input is applied. Therefore, R2 matches with P2.
Relation R3 is $y(t) = |x(t)|$. This is a linear and time-invariant relation. Therefore, R3 matches with P3.