Correct : a,b,d

The correct answers are A, B, and D.
Option A — H(X) ≤ log₂K : TRUE
This is a fundamental property of entropy. Entropy is maximum when all K outcomes are equally likely, giving H(X) = log₂K. For any other distribution it is strictly less. So this always holds.

Option B and D — H(X) ≤ H(2X) : TRUE
Multiplying X by 2 is a one-to-one mapping, every distinct value of X maps to a unique value of 2X, with no merging of outcomes. Because of this, H(2X) = H(X) exactly. So H(X) ≤ H(2X) holds with equality, making it necessarily true.
Option C — H(X) ≤ H(X²) : NOT necessarily true
Squaring is not a one-to-one function. If X takes values like -1 and +1, both map to the same value 1 under X², reducing the number of distinct outcomes and therefore reducing entropy. So H(X²) can be less than H(X), and this inequality does not always hold.