If the equation `x^(3) - 3x + a = 0` has distinct roots between 0 and 1, then the value of a is

Options

A. 2
B. `1//2`
C. 3
D. none of these

Answer

To solve the problem, we need to analyze the cubic equation given by:

\[ x^3 - 3x + a = 0 \]

We are tasked with finding the value of \( a \) such that the equation has distinct roots between 0 and 1.

### Step 1: Understanding the Roots
The roots of the cubic equation are denoted as \( \alpha, \beta, \gamma \). Since the problem states that these roots are distinct and lie between 0 and 1, we can infer that:

\[ 0 < \alpha, \beta, \gamma < 1 \]

### Step 2: Using Vieta's Formulas
According to Vieta's formulas, for a cubic equation of the form \( x^3 + bx^2 + cx + d = 0 \):

1. The sum of the roots \( \alpha + \beta + \gamma = -b \)
2. The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = c \)
3. The product of the roots \( \alpha\beta\gamma = -d \)

In our case, the equation is \( x^3 + 0x^2 - 3x + a = 0 \), so:

- The sum of the roots \( \alpha + \beta + \gamma = 0 \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = -3 \)
- The product of the roots \( \alpha\beta\gamma = -a \)

### Step 3: Analyzing the Product of the Roots
Since \( \alpha, \beta, \gamma \) are all positive and distinct, their product \( \alpha\beta\gamma \) must also be positive. Thus, we have:

\[ \alpha\beta\gamma > 0 \]

From Vieta's, we know:

\[ \alpha\beta\gamma = -a \]

Since \( \alpha\beta\gamma > 0 \), it follows that:

\[ -a > 0 \]
\[ a < 0 \]

### Step 4: Checking the Options
The options provided are:
- Option a: \( 2 \) (positive)
- Option b: \( \frac{1}{2} \) (positive)
- Option c: \( 3 \) (positive)
- Option d: None of these

Since we have determined that \( a < 0 \), none of the options a, b, or c can be correct because they are all positive. Therefore, the only valid choice is:

### Conclusion
The value of \( a \) must be less than 0, which leads us to conclude that the answer is:

**Option d: None of these**

---