Correct Option: C

Given discrete time system
y(n) = x(n²)
The order to solve such type of problems, check causality, linearity and time invariant one by one The given system
● y(n) = x(n²) is non-causal
Because for any negative value of n the output goes to future. For example:
y(– 2) = x(– 2)² = x(4)
i.e., non-causal system
● y(n) = x(n²)
To check linearity let a1 and a2 are the weights added to the input and H[x(n²)] = y (n) is the response of a discrete-time system.
Since, H[a₁x₁(n²) + a₂x₂(n²)]
= a₁H[x₁(n²)] + a₂H[x₂(n²)]
Hence, the given system is linear.
● For a system to be time invariant time delay of the input signal leads to an identical time shifts in the input signal.
Let y₁(n) = x₁(n²)
Consider input obtained by shifting x1(n) by time, n₀
i.e., x₁(n) = x₁(n – n₀)²
or y₂(n) = x₂(n²) = x₁(n – n₀)² …(A)
Similarly, when output changes by same amount n₀.
y₁(n – n₀) = x₁(n – n₀)² …(B)
From equations (A) and (B)
y₁(n – n₀) = y₂(n)
Therefore, this system is time invariant.
Hence, alternative (C) is the correct answer.