Across the lanthanoid series, the basic character of the lanthanoid hydroxide

decreases

The correct answer is option 2. decreases.

The basic character of the lanthanoid hydroxides decreases across the lanthanoid series.

The basicity of a hydroxide is determined by the ability of the metal ion to donate electrons to the hydroxide ion. As you go down the lanthanoid series, the nuclear charge increases. This pulls the electrons in the 4f orbitals closer to the nucleus, making it more difficult for the metal ion to donate electrons to the hydroxide ion.

As a result, the basicity of the lanthanoid hydroxides decreases across the lanthanoid series.

The other options are incorrect.

Option (1): The basicity of the lanthanoid hydroxides does not increase across the lanthanoid series.

Option (3): The basicity of the lanthanoid hydroxides does not first increase and then decrease across the lanthanoid series.

Option (4): The basicity of the lanthanoid hydroxides does not first decrease and then increase across the lanthanoid series.