A two-band model with repulsive interband coupling and interband {it transport} (potential) scattering is considered to elucidate their effects on material properties. In agreement with previous work, we find that the bands order parameters $Delta_{1,2}$ differ and the large is at the band with a smaller normal density of states (DOS), $N_{n2}<N_{n1}$. However, the bands energy gaps, as determined by the energy dependence of the DOS, are equal due to scattering. For each temperature, the gaps turn zero at a certain critical interband scattering rate, i.e. for strong enough scattering the model material becomes gappless. In the gapless state, the DOS at the band 2 is close to the normal state value, whereas at the band 1 it has a V-shape with non-zero minimum. When the normal bands DOS are mismatched, $N_{n1} e N_{n2}$, the critical temperature $T_c$ is suppressed even in the absence of interband scattering, $T_c(N_{n1})$ has a dome-like shape. With increasing interband scattering, the London penetration depth at low temperatures evolves from being exponentially flat to the power-law and even to near linear behavior in the gapless state, the latter being easily misinterpreted as caused by order parameter nodes.
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