The element that reacts vigorously with cold water is (b) sodium.
Explanation: Sodium is a highly reactive alkali metal that readily reacts with cold water, producing hydrogen gas and sodium hydroxide, often with a noticeable release of heat.
Why other options are incorrect:
(a) carbon:
Carbon is not reactive with cold water. It is a non-metal and forms covalent bonds, which are not easily broken by water.
(c) magnesium:
While magnesium can react with water, it does not react as vigorously as sodium. Magnesium reacts more readily with hot water.
(d) sulphur:
Sulphur is a non-metal and is not reactive with cold water. It is more likely to react with other non-metals or with metals under specific conditions.