If , `0 < m < n < 1`. Then the expression km < kn is true if :

Options

A. `k lt 0`
B. `k gt 0`
C. `k = 1` only
D. all of a,b,c

Answer

To solve the problem, we need to analyze the expression \( km < kn \) under the given conditions \( 0 < m < n < 1 \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \( km < kn \). This can be rewritten as:
\[ k \cdot m < k \cdot n \]
Here, \( k \) is a constant that we need to analyze in relation to \( m \) and \( n \).

2. **Dividing Both Sides by k**:
To isolate \( m \) and \( n \), we can divide both sides of the inequality by \( k \). However, we need to consider the sign of \( k \):
- If \( k > 0 \), the inequality remains the same:
\[ m < n \]
- If \( k < 0 \), the inequality reverses:
\[ m > n \]

3. **Analyzing the Conditions**:
Since we know from the problem statement that \( 0 < m < n < 1 \), it follows that:
\[ m < n \]
This means that for the inequality \( km < kn \) to hold true, \( k \) must be positive.

4. **Conclusion**:
Therefore, the expression \( km < kn \) is true if \( k > 0 \).

### Final Answer:
The expression \( km < kn \) is true if \( k > 0 \).