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 system relations R1, R2, R3, R4.

Properties

P1 : Linear but NOT time-invariant
P2: Time-invariant but NOT linear
P3: Linear and time-invariant

Relations

R1: y(t) = $${t^2}$$ x(t)
R2: y(t) = t$$\left| {x(t)} \right|$$
R3: y(t) = $$\left| {x(t)} \right|$$
R4: y(t) = x(t-5)

In the following network (Fig .1), the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.
The expression and the region of convergence of the z-transform of the sampled signal are

{x(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point. Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence
y (n) = $${1 \over N}\,\sum\limits_{r = 0}^{N - 1} x \,\left( r \right)x\,(n + r\,)$$ is