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We propose a variable-rate variable-power M-level quadrature amplitude modulation (MQAM) scheme employed under delay quality-of-service (QoS) constraints over spectrum-sharing channels. An underlay cognitive radio system, with one primary user and a secondary user, with constraints on interference leakage imposed by the primary receiver is considered. The transmission parameters of the secondary user are set optimally such that the successful communications for the primary user in terms of a minimum rate to be supported is satisfied, irrespective of co-existence with the secondary user. We obtain interference constraints that, when satisfied by the secondary users, is sufficient to satisfy the service outage requirement of the primary user. We further study the performance of the secondary user's link employing an adaptive MQAM scheme when, on top of the aforementioned interference constraint, the secondary user is also required to satisfy a statistical delay QoS constraint. Considering two modulation schemes, i.e., continuous MQAM and discrete MQAM with restricted constellations, we obtain the effective capacity of the secondary user's link and derive the optimum power allocation that maximizes the effective capacity of the secondary user.