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First sentence - we need a base form verb to fit in the infinitive construction TO + VERB. RIVAL (provide competition to) will fit here as the sentence tells us how he planned to set up a bank to compete with the Bank of New York. Second sentence - we need a noun here to be modified by the adjective ONE. RIVAL (foe; competitor) will fit here, as the sentence tells us how Charles executed his enemy to claim the throne. Thus, (a) is the right answer. B and D will fit only in 1, while C and E will only fit in 2.
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