TANCET 2025 Solved Papers

Concept:A rational number plus an irrational number is always irrational, but the sum of two irrational numbers is not necessarily irrational.Explanation:Statement (1): Both $a$ and $b$ are irrational.This does not guarantee that $a+b$ is always irrational.For example, take $a = \sqrt{2}$ and $b = -\sqrt{2}$.Both are distinct irrational numbers, but$$a + b = \sqrt{2} + (-\sqrt{2}) = 0,$$which is rational.So statement (1) alone is not sufficient.Statement (2): $a$ is rational and $b$ is irrational.Let $a = r \in \mathbb{Q}$ and let $b$ be irrational.Assume $a+b$ were rational, say $a+b = q \in \mathbb{Q}$.Then $b = q - a$.Since rational numbers are closed under subtraction, $q-a$ would be rational.This contradicts the fact that $b$ is irrational.Therefore, $a+b$ is always irrational.So statement (2) alone is sufficient.Answer:Option B: Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.