Let `alpha,beta in Rdot` If `alpha,beta^2` are the roots of quadratic equation `x^2-p x+1=0a n dalpha^2,beta` is the roots of quadratic equation `x^2-q x+8=0` , then the value of `rifr/8` is the arithmetic mean of `p a n d q ,` is

Options

A. `83/2`
B. 83
C. `83/8`
D. `83/4`

Explanation

To solve the problem step by step, we will analyze the given quadratic equations and use the properties of roots.

### Step 1: Identify the roots and their relationships
We have two quadratic equations:
1. \( x^2 - px + 1 = 0 \) with roots \( \alpha \) and \( \beta^2 \)
2. \( x^2 - qx + 8 = 0 \) with roots \( \alpha^2 \) and \( \beta \)

### Step 2: Use the product of roots
For the first equation:
- The product of the roots \( \alpha \cdot \beta^2 = 1 \) (Equation 1).

For the second equation:
- The product of the roots \( \alpha^2 \cdot \beta = 8 \) (Equation 2).

### Step 3: Multiply the two equations
From Equation 1 and Equation 2:
\[ (\alpha \cdot \beta^2) \cdot (\alpha^2 \cdot \beta) = 1 \cdot 8 \]
This simplifies to:
\[ \alpha^3 \cdot \beta^3 = 8 \]
Taking the cube root of both sides:
\[ \alpha \cdot \beta = 2 \quad \text{(Equation 3)} \]

### Step 4: Express beta in terms of alpha
From Equation 1:
\[ \alpha \cdot \beta^2 = 1 \implies \beta^2 = \frac{1}{\alpha} \implies \beta = \sqrt{\frac{1}{\alpha}} = \frac{1}{\sqrt{\alpha}} \quad \text{(Equation 4)} \]

### Step 5: Substitute beta in Equation 3
Substituting Equation 4 into Equation 3:
\[ \alpha \cdot \frac{1}{\sqrt{\alpha}} = 2 \]
This simplifies to:
\[ \sqrt{\alpha} = 2 \implies \alpha = 4 \]

### Step 6: Find beta
Using the value of \( \alpha \) in Equation 4:
\[ \beta = \frac{1}{\sqrt{4}} = \frac{1}{2} \]

### Step 7: Calculate p and q
Now we can find \( p \) and \( q \).

For \( p \):
\[ p = \alpha + \beta^2 = 4 + \left(\frac{1}{2}\right)^2 = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \quad \text{(Equation 5)} \]

For \( q \):
\[ q = \alpha^2 + \beta = 4^2 + \frac{1}{2} = 16 + \frac{1}{2} = 16 + \frac{1}{2} = \frac{32}{2} + \frac{1}{2} = \frac{33}{2} \quad \text{(Equation 6)} \]

### Step 8: Find R
We know that \( \frac{R}{8} \) is the arithmetic mean of \( p \) and \( q \):
\[ \frac{R}{8} = \frac{p + q}{2} \]
Substituting the values of \( p \) and \( q \):
\[ \frac{R}{8} = \frac{\frac{17}{4} + \frac{33}{2}}{2} \]
Finding a common denominator (which is 4):
\[ \frac{R}{8} = \frac{\frac{17}{4} + \frac{66}{4}}{2} = \frac{\frac{83}{4}}{2} = \frac{83}{8} \]
Thus:
\[ R = 83 \]

### Final Answer
The value of \( R \) is \( 83 \).

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