Suppose there were plants that had a high concentration of chlorophyll-b

Suppose there were plants that had a high concentration of chlorophyll-b, but lacked chlorophyll-a, would it carry out photosynthesis? Then why do plants have chlorophyll-b and accessory pigments?

No, it cannot carry out photosynthesis, as reaction centre is formed by chlorophyll-a. The other/accessory pigments absorb different wavelengths of light and pass on the energy to reaction centre for more efficient photosynthesis.