If ` alpha, beta` be the roots of `4x^2-17x+lambda=0, lambda in R` such that `1 lt alpha lt 2` and `2 lt beta lt 3` then the number of integral values of `lambda` is `

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the quadratic equation \(4x^2 - 17x + \lambda = 0\) given the conditions on its roots \( \alpha \) and \( \beta \).

### Step 1: Understand the roots and their conditions
We know that:
- \(1 < \alpha < 2\)
- \(2 < \beta < 3\)

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = \frac{17}{4} \)
- The product of the roots \( \alpha \beta = \frac{\lambda}{4} \)

### Step 3: Find the range for \( \alpha + \beta \)
Calculating the range for \( \alpha + \beta \):
- The minimum value occurs when \( \alpha = 1 \) and \( \beta = 2 \):
\[ \alpha + \beta > 1 + 2 = 3 \]
- The maximum value occurs when \( \alpha = 2 \) and \( \beta = 3 \):
\[ \alpha + \beta < 2 + 3 = 5 \]
Thus, we have:
\[ 3 < \alpha + \beta < 5 \]

### Step 4: Relate \( \alpha + \beta \) to \( \lambda \)
Using Vieta's:
\[ \frac{17}{4} \in (3, 5) \]
Multiplying through by 4 gives:
\[ 12 < 17 < 20 \]
This is always true, so we can proceed to the product of the roots.

### Step 5: Find the range for \( \alpha \beta \)
Calculating the range for \( \alpha \beta \):
- The minimum value occurs when \( \alpha \) is close to 1 and \( \beta \) is close to 2:
\[ \alpha \beta > 1 \times 2 = 2 \]
- The maximum value occurs when \( \alpha \) is close to 2 and \( \beta \) is close to 3:
\[ \alpha \beta < 2 \times 3 = 6 \]
Thus, we have:
\[ 2 < \alpha \beta < 6 \]

### Step 6: Relate \( \alpha \beta \) to \( \lambda \)
Using Vieta's:
\[ \frac{\lambda}{4} \in (2, 6) \]
Multiplying through by 4 gives:
\[ 8 < \lambda < 24 \]

### Step 7: Combine conditions
Now we have two conditions:
1. From \( \alpha + \beta \): \( 3 < \frac{17}{4} < 5 \) (always true)
2. From \( \alpha \beta \): \( 8 < \lambda < 24 \)

### Step 8: Find integral values of \( \lambda \)
The integral values of \( \lambda \) that satisfy \( 8 < \lambda < 24 \) are:
\[ 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 \]
Counting these gives us:
- Total integral values = 15

### Conclusion
Thus, the number of integral values of \( \lambda \) is **15**.