a) A system X is in thermal equilibrium with Y but not with Z. System Y and Z may be in thermal equilibrium with each other

b) A system X is in thermal equilibrium with Y but not with Z. Systems Y and Z are not in thermal equilibrium with each other

c) A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z must be in thermal equilibrium with each other

d) A system X is neither in thermal equilibrium with Y nor with Z. The system Y and Z may be in thermal equilibrium with each other

Answer:

The correct options are

b) A system X is in thermal equilibrium with Y but not with Z. Systems Y and Z are not in thermal equilibrium with each other

d) A system X is neither in thermal equilibrium with Y nor with Z. The system Y and Z may be in thermal equilibrium with each other

Explanation: In this question, we’ll use the Zeroth Law of Thermodynamics, which states that if two thermodynamic systems are both in thermal equilibrium with a third, they’re both in thermal equilibrium.

In option (a) here Tx=Ty but Tx ≠ Tz hence Ty ≠ Tz which means Y and Z are never in thermal equilibrium so option (a) is wrong.
In option (b) Tx=Ty but Tx ≠ Tz hence Ty ≠ Tz so this option is correct.
In option (c) Tx≠ Ty also Tx ≠ Tz hence Ty ≠ Tz so this option is also wrong.
In option (d) Tx≠ Ty and Tx ≠ Tz hence Ty may be equal Tz so option (d) is correct.