Lanthanides are a series of 15 metallic elements that are located in the f-block of the periodic table. They are also known as rare earth elements.

a) Lanthanide metal ions are mostly coloured:
This statement is correct. Lanthanide ions are known for their characteristic colors. The presence of f-electrons in their electronic configuration leads to the absorption of certain wavelengths of light, resulting in the appearance of different colors. For example, cerium ions are yellow, praseodymium ions are green, and europium ions are red.

b) Lanthanides have high melting and boiling points:
This statement is correct. Lanthanides generally have high melting and boiling points compared to other metallic elements. This is due to the strong metallic bonding resulting from the presence of partially filled 4f orbitals. The strong metallic bonds require a large amount of energy to break, resulting in high melting and boiling points.

c) Lanthanides have low densities:
This statement is incorrect. Lanthanides actually have relatively high densities compared to other metallic elements. The high density is attributed to the combination of their high atomic masses and compact atomic radii. For example, the density of neodymium is about 7 g/cm3, which is higher than most common metals such as iron and copper.

d) Lanthanides have high ionization energy:
This statement is correct. Lanthanides have high ionization energies, which means that it requires a large amount of energy to remove an electron from a lanthanide atom or ion. This is because the f-electrons are relatively well shielded by the outer electrons, making them more tightly held by the nucleus. The high ionization energies of lanthanides contribute to their stability and chemical inertness.

In conclusion, the incorrect statement is option C, which states that lanthanides have low densities. Lanthanides actually have high densities due to their high atomic masses and compact atomic radii.