1. Check statement A: "The process of adding an electron to a neutral gaseous atom is always exothermic."

   This refers to electron gain enthalpy / electron affinity.

   It is not always exothermic. For some atoms, especially noble gases and some alkaline earth metals / nitrogen-like stable configurations, adding an electron can require energy, i.e. it can be endothermic.

   Hence, A is false.

2. Check statement B: "The process of removing an electron from an isolated gaseous atom is always endothermic."

   This is the definition of ionization enthalpy.

   Removing an electron from an isolated gaseous atom always requires energy input: $X(g) \rightarrow X^+(g) + e^-$ so the process is always endothermic.

   Hence, B is true.

3. Check statement C: "The $1^{\text{st}}$ ionization energy of boron is less than that of beryllium."

   Electronic configurations: $\text{Be}: 1s^2 2s^2$ $\text{B}: 1s^2 2s^2 2p^1$

   The electron removed from boron is a $2p$ electron, which is higher in energy and less tightly held than the $2s$ electron in Be. Therefore, $IE_1(\text{B}) < IE_1(\text{Be})$

   Hence, C is true.

4. Check statement D: "The electronegativity of C is 2.5 in $\mathrm{CH_4}$ and $\mathrm{CCl_4}$."

   Electronegativity is not a fixed constant in all compounds; it depends on the chemical environment, hybridization, and bonded atoms. Thus saying carbon has exactly the same electronegativity value $2.5$ in both $\mathrm{CH_4}$ and $\mathrm{CCl_4}$ is not correct in this context.

   Hence, D is false.

5. Check statement E: "Li is the most electropositive among elements of group I."

   Electropositivity increases down the group. Among alkali metals, the lower elements are more electropositive than Li.

   Therefore, Li is not the most electropositive.

   Hence, E is false.

6. Collect correct statements

   • A: False
   • B: True
   • C: True
   • D: False
   • E: False

   So the correct set is: $\boxed{\text{B and C only}}$

7. Match with options

   Option B = B and C Only.

Therefore, the correct answer is Option B.