Which of the following is true?

Options

A. `(2)/(sqrt2),(-2)/(sqrt3),(-1)/(sqrt3)` are the direction cosines of a directed line.
B. `(-2)/(sqrt3),(-2)/(sqrt3),(-1)/(sqrt3)` are the direction cosines of a directed line.
C. `(2)/(sqrt3),(-2)/(sqrt3),(-1)/(sqrt3)` are not the direction cosines of a directed line.
D. `(1)/(sqrt3),(-1)/(sqrt3),(-1)/(sqrt3)` are not the direction cosines of a directed line.

Answer

To determine which of the given options regarding direction cosines is true, we will follow these steps:

### Step 1: Understand the Concept of Direction Cosines
Direction cosines are defined as the cosines of the angles that a line makes with the coordinate axes. If a line makes angles α, β, and γ with the x-axis, y-axis, and z-axis respectively, then the direction cosines are given by:
- \( l = \cos \alpha \)
- \( m = \cos \beta \)
- \( n = \cos \gamma \)

### Step 2: Use the Property of Direction Cosines
One important property of direction cosines is that:
\[ l^2 + m^2 + n^2 = 1 \]
This means that the sum of the squares of the direction cosines must equal 1.

### Step 3: Analyze Each Option
We will analyze each option to check if they satisfy the property \( l^2 + m^2 + n^2 = 1 \).

#### Option 1:
Assume \( l = \frac{2}{\sqrt{2}}, m = \frac{4}{\sqrt{2}}, n = \frac{\sqrt{3}}{1} \).
- Calculate \( l^2 + m^2 + n^2 \):
\[ l^2 = \left(\frac{2}{\sqrt{2}}\right)^2 = \frac{4}{2} = 2 \]
\[ m^2 = \left(\frac{4}{\sqrt{2}}\right)^2 = \frac{16}{2} = 8 \]
\[ n^2 = \left(\sqrt{3}\right)^2 = 3 \]
\[ l^2 + m^2 + n^2 = 2 + 8 + 3 = 13 \quad (\text{not equal to } 1) \]
Thus, this option cannot be direction cosines.

#### Option 2:
Assume \( l = \frac{4}{3}, m = \frac{1}{3}, n = \text{some value} \).
- Calculate \( l^2 + m^2 + n^2 \):
\[ l^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ m^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
\[ l^2 + m^2 = \frac{16}{9} + \frac{1}{9} = \frac{17}{9} \quad (\text{not equal to } 1) \]
Thus, this option cannot be direction cosines.

#### Option 3:
Assume \( l = \frac{4}{3}, m = \frac{4}{3}, n = \frac{4}{3} \).
- Calculate \( l^2 + m^2 + n^2 \):
\[ l^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ m^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ n^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ l^2 + m^2 + n^2 = \frac{16}{9} + \frac{16}{9} + \frac{16}{9} = \frac{48}{9} \quad (\text{not equal to } 1) \]
Thus, this option cannot be direction cosines.

#### Option 4:
Assume \( l = \frac{1}{3}, m = \frac{1}{3}, n = \frac{1}{3} \).
- Calculate \( l^2 + m^2 + n^2 \):
\[ l^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
\[ m^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
\[ n^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
\[ l^2 + m^2 + n^2 = \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{3}{9} = \frac{1}{3} \quad (\text{not equal to } 1) \]
Thus, this option cannot be direction cosines.

### Conclusion
After analyzing all the options, we find that the only true statement is the one that states that the direction cosines do not satisfy the property \( l^2 + m^2 + n^2 = 1 \).

### Final Answer
The true statement is that the direction cosines of the directed line are not valid for the options provided.