Let `alpha, beta " the roots of " x^(2) -4x + A =0 and gamma, delta " be the roots of " x^(2) -36x +B =0. " If " alpha, beta , gamma, delta ` forms an increasing G.P. Then

Options

A. B = 81A
B. A=3
C. B =243
D. A + B = 251

Answer

To solve the problem, we will follow these steps systematically:

### Step 1: Identify the roots of the quadratic equations
Given the equations:
1. \( x^2 - 4x + A = 0 \)
2. \( x^2 - 36x + B = 0 \)

Let \( \alpha \) and \( \beta \) be the roots of the first equation, and \( \gamma \) and \( \delta \) be the roots of the second equation.

From Vieta's formulas:
- For the first equation:
- \( \alpha + \beta = 4 \)
- \( \alpha \beta = A \)

- For the second equation:
- \( \gamma + \delta = 36 \)
- \( \gamma \delta = B \)

### Step 2: Express the roots in terms of a geometric progression (G.P.)
Since \( \alpha, \beta, \gamma, \delta \) form an increasing G.P., we can express them as:
- \( \alpha = A \)
- \( \beta = AR \)
- \( \gamma = AR^2 \)
- \( \delta = AR^3 \)

### Step 3: Set up equations based on the sums of the roots
From the first equation:
\[ \alpha + \beta = A + AR = 4 \quad \text{(1)} \]

From the second equation:
\[ \gamma + \delta = AR^2 + AR^3 = 36 \quad \text{(2)} \]

### Step 4: Substitute and simplify
From equation (1):
\[ A(1 + R) = 4 \implies 1 + R = \frac{4}{A} \quad \text{(3)} \]

From equation (2):
\[ AR^2(1 + R) = 36 \quad \text{(4)} \]

### Step 5: Substitute equation (3) into equation (4)
Substituting \( 1 + R = \frac{4}{A} \) into equation (4):
\[ AR^2 \cdot \frac{4}{A} = 36 \]
This simplifies to:
\[ 4R^2 = 36 \implies R^2 = 9 \implies R = 3 \quad \text{(since R is positive for increasing G.P.)} \]

### Step 6: Find the value of A
Substituting \( R = 3 \) back into equation (3):
\[ 1 + 3 = \frac{4}{A} \implies 4 = \frac{4}{A} \implies A = 1 \]

### Step 7: Find the value of B
Now we can find \( B \) using the product of the roots:
\[ \gamma \delta = AR^2 \cdot AR^3 = A^2 R^5 \]
Substituting \( A = 1 \) and \( R = 3 \):
\[ B = 1^2 \cdot 3^5 = 243 \]

### Conclusion
The values we found are:
- \( A = 1 \)
- \( B = 243 \)

### Final Answer
Thus, the values of \( A \) and \( B \) are \( A = 1 \) and \( B = 243 \).