If x be a rational number and y be an irrational number, then :

Options

A. both `x + y` and `xy` are necessarily irrational
B. both `x + y` and `xy` are necessarily rational
C. `xy` is necessarily irrational, but x + y can be either rational or irrational
D. `x + y` necessarity irrational , but xy can be either rational or irrational

Answer

To solve the question, we need to analyze the properties of rational and irrational numbers when combined through addition and multiplication.

### Step 1: Understanding Rational and Irrational Numbers
- A rational number is defined as a number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An irrational number cannot be expressed in this form. Examples include numbers like \( \sqrt{2} \), \( \pi \), and \( e \).

**Hint:** Remember that rational numbers can be written as fractions, while irrational numbers cannot.

### Step 2: Adding a Rational Number and an Irrational Number
- Let \( x \) be a rational number (e.g., \( x = 2 \)) and \( y \) be an irrational number (e.g., \( y = \sqrt{3} \)).
- When we add \( x \) and \( y \):
\[ x + y = 2 + \sqrt{3} \]
- The result \( 2 + \sqrt{3} \) cannot be expressed as a fraction \( \frac{p}{q} \) because the sum of a rational number and an irrational number is always irrational.

**Hint:** The sum of a rational number and an irrational number is always irrational.

### Step 3: Multiplying a Rational Number and an Irrational Number
- Now, consider the multiplication of \( x \) and \( y \):
\[ x \cdot y = 2 \cdot \sqrt{3} \]
- The product \( 2 \cdot \sqrt{3} \) is also irrational. This is because multiplying a non-zero rational number by an irrational number results in an irrational number.

**Hint:** The product of a non-zero rational number and an irrational number is always irrational.

### Conclusion
- From the analysis:
- \( x + y \) is always irrational.
- \( x \cdot y \) is always irrational.

Thus, the correct conclusion is that both \( x + y \) and \( x \cdot y \) are irrational.

**Final Answer:** Both \( x + y \) and \( x \cdot y \) are irrational.