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Question 10

The ordered pair of numbers $$(x,y)$$ satisfy both the equations $$x+y=3$$ and $$x^5+y^5+162=0$$. Then

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Let $$p=xy$$. Using $$x+y=3$$ and the identity for $$x^5+y^5$$ reduces the second equation to $$5p^2-45p+135=0$$. Thus $$p=\frac{9\pm3\sqrt{3}i}{2}$$, which is non-real. Since $$x+y$$ is real but $$xy$$ is non-real, all resulting pairs are non-real, so option d is correct.

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