Consider the quadratic equation `(c - 5)x^(2) - 2cx + (c - 4) = 0, c ne 5`. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is

Options

A. 11
B. 18
C. 10
D. 12

Answer

To solve the quadratic equation \((c - 5)x^2 - 2cx + (c - 4) = 0\) for the values of \(c\) such that one root lies in the interval \((0, 2)\) and the other root lies in the interval \((2, 3)\), we will follow these steps:

### Step 1: Identify the function
The quadratic function can be expressed as:
\[ f(x) = (c - 5)x^2 - 2cx + (c - 4) \]

### Step 2: Evaluate \(f(0)\)
To find the value of \(f(0)\):
\[ f(0) = (c - 4) \]

### Step 3: Evaluate \(f(2)\)
Now, calculate \(f(2)\):
\[ f(2) = (c - 5)(2^2) - 2c(2) + (c - 4) \]
\[ = (c - 5)(4) - 4c + (c - 4) \]
\[ = 4c - 20 - 4c + c - 4 \]
\[ = c - 24 \]

### Step 4: Evaluate \(f(3)\)
Next, calculate \(f(3)\):
\[ f(3) = (c - 5)(3^2) - 2c(3) + (c - 4) \]
\[ = (c - 5)(9) - 6c + (c - 4) \]
\[ = 9c - 45 - 6c + c - 4 \]
\[ = 4c - 49 \]

### Step 5: Set up inequalities
For one root to be in \((0, 2)\) and the other in \((2, 3)\), we need:
1. \(f(0) \cdot f(2) < 0\)
2. \(f(2) \cdot f(3) < 0\)

#### Inequality 1: \(f(0) \cdot f(2) < 0\)
\[ (c - 4)(c - 24) < 0 \]
This inequality holds when \(c\) is in the interval \((4, 24)\).

#### Inequality 2: \(f(2) \cdot f(3) < 0\)
\[ (c - 24)(4c - 49) < 0 \]
To analyze this inequality, we find the roots:
1. \(c = 24\)
2. \(4c - 49 = 0 \Rightarrow c = \frac{49}{4} = 12.25\)

This inequality holds when \(c\) is in the intervals:
- \((-\infty, \frac{49}{4})\) or \((24, \infty)\)

### Step 6: Find the intersection of intervals
We need to find the intersection of the two intervals:
1. From the first inequality: \(c \in (4, 24)\)
2. From the second inequality: \(c \in (-\infty, \frac{49}{4})\) or \(c \in (24, \infty)\)

The intersection is:
\[ c \in (4, \frac{49}{4}) \]

### Step 7: Identify integral values of \(c\)
The integral values of \(c\) in the interval \((4, 12.25)\) are:
\[ 5, 6, 7, 8, 9, 10, 11, 12 \]
Counting these values gives us a total of \(8\) integral values.

### Final Answer
Thus, the number of elements in the set \(S\) is:
\[ \boxed{8} \]