The given question asks which synthetically produced radioactive noble gas is formed by the collision of \(^{249}_{98}\text{Cf}\) with \(^{48}_{20}\text{Ca}\). Let's analyze the process and find the correct answer.
The collision of californium (\(^{249}_{98}\text{Cf}\)) with calcium (\(^{48}_{20}\text{Ca}\)) nuclei is a nuclear reaction used to synthesize superheavy elements. In these reactions, two nuclei combine to form a heavier nucleus.
Oganesson (Og) is one of the superheavy elements that belong to the noble gas group. It was first synthesized in 2002, with the atomic number 118 and symbol Og.
The reaction can be represented as follows:
\(^{249}_{98}\text{Cf} + ^{48}_{20}\text{Ca} \rightarrow ^{294}_{118}\text{Og} + 3 \ ^{1}_{0}\text{n}\)
Thus, during the reaction, three neutrons are released, and Oganesson is produced.
Therefore, the synthetically produced radioactive noble gas is Oganesson.
By understanding the nuclear reaction between \(^{249}_{98}\text{Cf}\) and \(^{48}_{20}\text{Ca}\), we conclude that the correct answer is Oganesson. This matches with option 3.