The product of a non-zero rational number and an irrational number is:
(a) Always irrational
(b) Always rational
(c) Rational or irrational
(d) One
Step-by-Step Solution
Key Concept: If $r <br>eq 0$ is rational and $x$ is irrational, assuming $r \cdot x = q$ (rational) implies $x = q/r$ (rational), a contradiction.
Let $r
eq 0$ be rational and $x$ be irrational. Suppose $r \cdot x = q$ where $q$ is rational. [0.5 Mark]
Then $x = \dfrac{q}{r}$, which is rational (quotient of two non-zero rationals). Contradiction! Hence $r \cdot x$ must be irrational. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Correct proof/reasoning recall: 0.5 Mark
Correct final selection: 0.5 Mark
Correct Answer: Always irrational