If `x < 0 < y`, then :

Options

A. `1/(x^2) lt 1/(xy) lt 1/(y^2)`
B. `1/(x^2) gt 1/(xy) gt 1/(y^2)`
C. `1/x lt 1/y`
D. `1/x gt 1`

Answer

To solve the problem, we need to analyze the given conditions and evaluate the options one by one. We know that \( x < 0 < y \), meaning \( x \) is negative and \( y \) is positive.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- Since \( x < 0 \), we can conclude that \( x \) is a negative number.
- Since \( 0 < y \), we can conclude that \( y \) is a positive number.

2. **Evaluating Option A**:
- Option A states: \( \frac{1}{x^2} < xy < \frac{1}{y^2} \)
- Calculate \( \frac{1}{x^2} \):
- Since \( x \) is negative, \( x^2 \) is positive. Therefore, \( \frac{1}{x^2} \) is positive.
- Calculate \( xy \):
- Since \( x \) is negative and \( y \) is positive, \( xy \) is negative.
- Calculate \( \frac{1}{y^2} \):
- Since \( y \) is positive, \( y^2 \) is positive. Therefore, \( \frac{1}{y^2} \) is positive.
- So we have: \( \frac{1}{x^2} > 0 \), \( xy < 0 \), and \( \frac{1}{y^2} > 0 \).
- This means \( \frac{1}{x^2} > xy \) is true, but \( xy < \frac{1}{y^2} \) is false since \( xy \) is negative and \( \frac{1}{y^2} \) is positive. Thus, Option A is incorrect.

3. **Evaluating Option B**:
- Option B states: \( \frac{1}{x^2} > xy > y^2 \)
- We already know \( \frac{1}{x^2} > 0 \) and \( xy < 0 \).
- Since \( y^2 > 0 \), we have \( xy < y^2 \) is also false. Thus, Option B is incorrect.

4. **Evaluating Option C**:
- Option C states: \( \frac{1}{x} < \frac{1}{y} \)
- Calculate \( \frac{1}{x} \):
- Since \( x \) is negative, \( \frac{1}{x} \) is negative.
- Calculate \( \frac{1}{y} \):
- Since \( y \) is positive, \( \frac{1}{y} \) is positive.
- Therefore, \( \frac{1}{x} < \frac{1}{y} \) is true since a negative number is less than a positive number. Thus, Option C is correct.

5. **Evaluating Option D**:
- Option D states: \( \frac{1}{x} > 1 \)
- Since \( x \) is negative, \( \frac{1}{x} \) is negative. Therefore, \( \frac{1}{x} > 1 \) is false. Thus, Option D is incorrect.

### Conclusion:
The correct answer is **Option C**: \( \frac{1}{x} < \frac{1}{y} \).