The first `(IE_(1))` and second `(IE_(2))` ionisation energies `(k J mol^(-1))` of a new elements designated by roman numerals are shown below:
`{:(,,IE_(1),IE_(2),),(I,,2370,5250,),(II,,520,7300,),(III,,900,1800,),(IV,,1700,3400,):}`
Which of the above elements is likely to be :
a. A reactive metal
b. A reactive non-metal
c. a noble gas
d. A metal that forms a stable binary halide of the formula `AX_(2)` (`X=` the halogen).

Explanation

To solve the problem, we need to analyze the given ionization energies (IE) for the four elements and determine which element corresponds to each of the categories: reactive metal, reactive non-metal, noble gas, and a metal that forms a stable binary halide of the formula AX₂.

### Step-by-Step Solution:

1. **Understanding Ionization Energies:**
- Ionization energy is the energy required to remove an electron from an atom. The first ionization energy (IE₁) is the energy needed to remove the first electron, and the second ionization energy (IE₂) is for removing the second electron.
- A large jump in ionization energy between IE₁ and IE₂ indicates that after the removal of the first electron, the atom achieves a stable electron configuration (like that of a noble gas).

2. **Analyzing Each Element:**
- **Element I:** IE₁ = 2370 kJ/mol, IE₂ = 5250 kJ/mol
- There is a significant increase from IE₁ to IE₂, indicating that after losing the first electron, the atom reaches a stable configuration. This suggests it is a reactive metal (likely from Group 1).
- **Element II:** IE₁ = 520 kJ/mol, IE₂ = 7300 kJ/mol
- There is a very large jump in ionization energy, suggesting that this element is a non-metal, likely a halogen, as it requires a lot of energy to remove the second electron after achieving a stable configuration.
- **Element III:** IE₁ = 900 kJ/mol, IE₂ = 1800 kJ/mol
- The increase from IE₁ to IE₂ is moderate, indicating that this element is likely a metal that can lose two electrons but does not have a drastic jump, suggesting it may form stable compounds.
- **Element IV:** IE₁ = 1700 kJ/mol, IE₂ = 3400 kJ/mol
- Similar to Element III, the increase is moderate, but both values are relatively high, indicating this could also be a noble gas.

3. **Matching Elements to Categories:**
- **a. Reactive Metal:** Element I (2370, 5250)
- **b. Reactive Non-Metal:** Element II (520, 7300)
- **c. Noble Gas:** Element IV (1700, 3400)
- **d. Metal that forms stable binary halide AX₂:** Element III (900, 1800)

### Final Answers:
- a. Element I is a reactive metal.
- b. Element II is a reactive non-metal.
- c. Element IV is a noble gas.
- d. Element III is a metal that forms a stable binary halide of the formula AX₂.