Key Concepts and Formulas

• Location of Roots: For a quadratic $f(x) = ax^2 + bx + c$, if $f(x_1)f(x_2) < 0$, there's at least one root in the interval $(x_1, x_2)$.
• Solving Quadratic Inequalities: Find the roots of the corresponding quadratic equation. The sign of the quadratic expression changes at the roots.
• Intersection of Intervals: To satisfy multiple conditions, find the intersection of the solution intervals.

Step-by-Step Solution

Step 1: Define the Quadratic Function

We are given the quadratic equation $(c – 5)x^2 – 2cx + (c – 4) = 0$, with $c \neq 5$. Let's define the corresponding quadratic function: $f(x) = (c – 5)x^2 – 2cx + (c – 4)$

Step 2: Apply the Root Location Conditions

The problem states that one root lies in $(0, 2)$ and the other in $(2, 3)$. This implies:

• $f(0)$ and $f(2)$ have opposite signs: $f(0)f(2) < 0$
• $f(2)$ and $f(3)$ have opposite signs: $f(2)f(3) < 0$

Step 3: Calculate Function Values at x = 0, 2, and 3

We need to find $f(0)$, $f(2)$, and $f(3)$ in terms of $c$:

• $f(0) = (c – 5)(0)^2 – 2c(0) + (c – 4) = c – 4$
• $f(2) = (c – 5)(2)^2 – 2c(2) + (c – 4) = 4(c – 5) – 4c + c – 4 = 4c - 20 - 4c + c - 4 = c – 24$
• $f(3) = (c – 5)(3)^2 – 2c(3) + (c – 4) = 9(c – 5) – 6c + c – 4 = 9c - 45 - 6c + c - 4 = 4c – 49$

Step 4: Solve the Inequality f(0)f(2) < 0

Substitute the expressions for $f(0)$ and $f(2)$: $(c – 4)(c – 24) < 0$ The roots of $(c – 4)(c – 24) = 0$ are $c = 4$ and $c = 24$. Since the parabola opens upwards, the inequality is satisfied when $4 < c < 24$.

Step 5: Solve the Inequality f(2)f(3) < 0

Substitute the expressions for $f(2)$ and $f(3)$: $(c – 24)(4c – 49) < 0$ The roots of $(c – 24)(4c – 49) = 0$ are $c = 24$ and $c = \frac{49}{4} = 12.25$. Since the parabola opens upwards, the inequality is satisfied when $12.25 < c < 24$.

Step 6: Find the Intersection of the Solution Intervals

We need to find the values of $c$ that satisfy both $4 < c < 24$ and $12.25 < c < 24$. The intersection of these intervals is $12.25 < c < 24$.

Step 7: Determine the Integral Values of c

We need to find the integers $c$ such that $12.25 < c < 24$. These are: $c \in \{13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\}$

Step 8: Count the Number of Integral Values

The number of integers in the set is $23 - 13 + 1 = 11$.

Common Mistakes & Tips

• Sign Errors: Be careful with signs when expanding and simplifying expressions for $f(0)$, $f(2)$, and $f(3)$.
• Endpoints: Remember that the roots lie strictly within the intervals, so use strict inequalities ($<$ and $>$) rather than non-strict inequalities ($\le$ and $\ge$).
• Quadratic Inequality Direction: Always determine whether the parabola opens upwards or downwards to correctly interpret the solution to quadratic inequalities.

Summary

We used the location of roots concept to establish two inequalities based on the given intervals. Solving these inequalities gave us a range for $c$, and we then identified the integers within that range. The number of these integers represents the number of elements in the set $S$. The final count is 11.

Final Answer The final answer is \boxed{11}, which corresponds to option (D).