The number of values of k for which the equation `x ^(2) - 3x + k =0` has two distinct roots lying in the interval `(0,1)` is

Options

A. three
B. two
C. infinite
D. no value of k will satisfy

Answer

To determine the number of values of \( k \) for which the equation \( x^2 - 3x + k = 0 \) has two distinct roots lying in the interval \( (0, 1) \), we can follow these steps:

### Step 1: Understand the conditions for distinct roots
For the quadratic equation \( ax^2 + bx + c = 0 \) to have two distinct roots, the discriminant must be positive:
\[ D = b^2 - 4ac > 0 \]
In our case, \( a = 1 \), \( b = -3 \), and \( c = k \). Thus, the discriminant becomes:
\[ D = (-3)^2 - 4 \cdot 1 \cdot k = 9 - 4k \]
We require:
\[ 9 - 4k > 0 \implies 4k < 9 \implies k < \frac{9}{4} \]

### Step 2: Roots must lie in the interval (0, 1)
Next, we need to ensure that both roots of the equation lie within the interval \( (0, 1) \). The roots of the quadratic can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{9 - 4k}}{2} \]
Let the roots be \( r_1 \) and \( r_2 \):
\[ r_1 = \frac{3 + \sqrt{9 - 4k}}{2}, \quad r_2 = \frac{3 - \sqrt{9 - 4k}}{2} \]

For both roots to be in the interval \( (0, 1) \), we need:
1. \( r_1 < 1 \)
2. \( r_2 > 0 \)

### Step 3: Solve the inequalities
**Inequality 1: \( r_1 < 1 \)**
\[ \frac{3 + \sqrt{9 - 4k}}{2} < 1 \]
Multiplying both sides by 2:
\[ 3 + \sqrt{9 - 4k} < 2 \implies \sqrt{9 - 4k} < -1 \]
This inequality is not possible since the square root cannot be negative. Thus, we need to check the second inequality.

**Inequality 2: \( r_2 > 0 \)**
\[ \frac{3 - \sqrt{9 - 4k}}{2} > 0 \]
Multiplying both sides by 2:
\[ 3 - \sqrt{9 - 4k} > 0 \implies \sqrt{9 - 4k} < 3 \]
Squaring both sides:
\[ 9 - 4k < 9 \implies -4k < 0 \implies k > 0 \]

### Step 4: Combine the results
From the inequalities, we have:
1. \( k < \frac{9}{4} \)
2. \( k > 0 \)

Thus, the values of \( k \) must satisfy:
\[ 0 < k < \frac{9}{4} \]

### Step 5: Determine the number of values of \( k \)
The interval \( (0, \frac{9}{4}) \) contains infinitely many values. Therefore, the number of values of \( k \) for which the equation has two distinct roots lying in the interval \( (0, 1) \) is infinite.

### Final Answer
The number of values of \( k \) is infinite.