If $x$ is a rational number and $y$ is an irrational number, then

### Concept & Rule This tests the definitive properties of Real Numbers. The sum of a rational and an irrational number is ALWAYS irrational. However, the product of a rational and an irrational number can sometimes be rational, specifically due to the Zero Property. ### Step-by-Step Solution * **Given:** $x$ is rational (can be written as a fraction $\frac{p}{q}$). $y$ is irrational (cannot be written as a simple fraction, e.g., $\sqrt{2}$). * Let's evaluate the sum: $x + y$. * If you add a non-terminating, non-repeating decimal ($y$) to any rational number ($x$), the decimal tail will never resolve or terminate. Therefore, the sum is **necessarily irrational**. * Let's evaluate the product: $xy$. * If $x$ is any non-zero rational number (e.g., $2$), then $2 \times \sqrt{2} = 2\sqrt{2}$ (Irrational). * However, the number $0$ is a rational number. If $x = 0$, then $0 \times \sqrt{2} = 0$. Since $0$ is rational, the product in this specific case is **rational**. * Therefore, the product $xy$ can be either rational or irrational depending entirely on whether $x = 0$. ### Exam Strategy & Shortcut Whenever a question asks about the absolute necessity of a condition involving multiplication and real numbers, immediately test the number zero. Zero destroys irrationality when multiplied. Recalling this edge-case instantly guides you to option (d). ### Common Pitfall The most common mistake is selecting option (b), assuming that "rational $\times$ irrational = irrational". Students frequently forget that $0$ is classified as a rational integer. ### Final Answer Therefore, the correct answer is **$x + y$ is necessarily irrational, but $xy$ can be either rational or irrational**.