The problems of a single semitransparent cone with longitudinal slots excitation are considered. The conical structure is excited by either electrical or magnetic radial dipole. The imperfect surface conductivity (semitransparency) is simulated by means of the average two-sided boundary conditions for the conical surface. It is shown that the original electromagnetic problem can be reduced to a boundary one of mathematical physics. The solution algorithm is based on a combination of the Kontorovich-Lebedev integral transforms, Fourier series, the method of Riemann-Hilbert problem and other rigorous approaches. Both numerical and analytical solutions are derived for specific cases of the investigated problem.