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Statement 1 - Given any two real numbers $$p$$, $$q$$ there always exist two real numbers $$a$$ and $$b$$ such that $$a + b = p$$ and $$ab = q$$.<br>Statement 2 - There exist two real numbers $$a$$, $$b$$ such that $$a + b = 3$$ and $$ab = -400$$.
$$a$$ and $$b$$ are the roots of $$t^2 - pt + q = 0$$, which are real only when $$p^2 \ge 4q$$; for example $$p = 0$$, $$q = 1$$ fails, so Statement 1 is false. For Statement 2 the equation $$t^2 - 3t - 400 = 0$$ has discriminant $$9 + 1600 > 0$$, so real $$a$$, $$b$$ exist and Statement 2 is true. Hence options b and c are both correct.
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