If a, p are the roots of the quadratic equation `x^2 - 2p (x - 4) - 15 = 0`, then the set of values of p for which one root is less than 1 and the other root is greater than 2 is:

Answer

To solve the problem, we need to analyze the quadratic equation given and determine the values of \( p \) for which one root is less than 1 and the other root is greater than 2.

### Step 1: Write the quadratic equation in standard form
The given equation is:
\[ x^2 - 2p(x - 4) - 15 = 0 \]
We can simplify this equation:
\[ x^2 - 2px + 8p - 15 = 0 \]
This is now in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = -2p \), and \( c = 8p - 15 \).

### Step 2: Identify the roots of the quadratic equation
Let the roots of the equation be \( a \) and \( p \). By Vieta's formulas, we know:
- The sum of the roots \( a + p = -\frac{b}{a} = 2p \)
- The product of the roots \( ap = \frac{c}{a} = 8p - 15 \)

### Step 3: Set conditions for the roots
We need one root to be less than 1 and the other root to be greater than 2. This can be expressed as:
- \( a < 1 \)
- \( p > 2 \)

### Step 4: Find the conditions on \( p \)
Using the conditions on the roots, we can substitute \( x = 1 \) and \( x = 2 \) into the quadratic equation to find the conditions on \( p \).

#### Condition 1: Evaluate at \( x = 1 \)
Substituting \( x = 1 \):
\[ 1^2 - 2p(1 - 4) - 15 < 0 \]
This simplifies to:
\[ 1 + 6p - 15 < 0 \implies 6p < 14 \implies p < \frac{14}{6} = \frac{7}{3} \]

#### Condition 2: Evaluate at \( x = 2 \)
Substituting \( x = 2 \):
\[ 2^2 - 2p(2 - 4) - 15 < 0 \]
This simplifies to:
\[ 4 + 4p - 15 < 0 \implies 4p < 11 \implies p < \frac{11}{4} \]

### Step 5: Combine the conditions
Now we have two conditions:
1. \( p < \frac{7}{3} \)
2. \( p < \frac{11}{4} \)

Since \( \frac{7}{3} \approx 2.33 \) and \( \frac{11}{4} = 2.75 \), the more restrictive condition is \( p < \frac{7}{3} \).

### Step 6: Determine the range of \( p \)
We also need to ensure that \( p > 2 \) (since one root must be greater than 2). Therefore, we combine the conditions:
\[ 2 < p < \frac{7}{3} \]

### Final Answer
The set of values of \( p \) for which one root is less than 1 and the other root is greater than 2 is:
\[ p \in (2, \frac{7}{3}) \]