The properties which are common to the elements belonging to group `1` and `17` of periodic tables are:

Options

A. Electropositive character increases down the group.
B. Reactivity decreases from a top to bottom
C. Atomic radii increases as atomic number increases
D. Electronegatively decreases on moving down a group.

Answer

To solve the question regarding the common properties of elements belonging to group 1 and group 17 of the periodic table, we will analyze each of the given statements one by one.

### Step 1: Analyze Electropositive Character
- **Statement**: Electropositive character increases down the group.
- **Group 1**: The elements in group 1 (alkali metals) are highly electropositive. As we move down the group from lithium to francium, the atomic radius increases, which leads to a decrease in ionization enthalpy. This means that it becomes easier for these elements to lose their outermost electron, thus increasing their electropositive character.
- **Group 17**: The elements in group 17 (halogens) are generally electronegative. However, as we move down the group, the size of the atoms increases, and their ability to gain electrons decreases. Thus, they can also exhibit some electropositive character as they can lose electrons in the presence of more electronegative elements.
- **Conclusion**: This property is common to both groups.

### Step 2: Analyze Reactivity
- **Statement**: Reactivity decreases from top to bottom.
- **Group 1**: In group 1, reactivity increases down the group. As the atomic number increases, the atomic radius increases, and ionization enthalpy decreases, making it easier for these metals to react.
- **Group 17**: In group 17, reactivity actually decreases down the group. For example, fluorine is more reactive than chlorine, which is more reactive than bromine, and so on.
- **Conclusion**: This property is not common to both groups.

### Step 3: Analyze Atomic Radii
- **Statement**: Atomic radii increase as atomic number increases.
- **Group 1**: As we move down group 1, the atomic radius increases due to the addition of electron shells.
- **Group 17**: Similarly, in group 17, the atomic radius also increases down the group for the same reason.
- **Conclusion**: This property is common to both groups.

### Step 4: Analyze Electronegativity
- **Statement**: Electronegativity decreases on moving down a group.
- **Group 1**: As we move down group 1, the electronegativity decreases because the atomic size increases, making it less effective in attracting electrons.
- **Group 17**: In group 17, electronegativity also decreases down the group for the same reason.
- **Conclusion**: This property is common to both groups.

### Final Conclusion
Based on the analysis:
- Common properties between group 1 and group 17 are:
- A: Electropositive character increases down the group.
- C: Atomic radii increase as atomic number increases.
- D: Electronegativity decreases on moving down a group.

Thus, the correct options are A, C, and D.