If a radioactive element having half-life of $$30$$ min is undergoing beta decay, the fraction of radioactive element remains undecayed after $$90$$ min will be:

For radioactive decay, the fraction of undecayed material after time $$t$$ is given by $$N/N_0 = \left(\dfrac{1}{2}\right)^{t/t_{1/2}}$$, where $$t_{1/2}$$ is the half-life. The type of decay (beta decay) does not affect this formula.

Here $$t = 90$$ min and $$t_{1/2} = 30$$ min, so the number of half-lives elapsed is $$n = 90/30 = 3$$. The fraction remaining is $$\left(\dfrac{1}{2}\right)^3 = \boxed{\dfrac{1}{8}}$$, which is option (A).