If `x= (sqrt3 - sqrt2)/(sqrt3+sqrt2) and y = (sqrt3+sqrt2)/(sqrt3-sqrt2) ` then `x^2 +xy +y^2` is a multiple of

Options

A. 11
B. 3
C. 9
D. All (a) , (b) and (c)

Answer

To solve the problem, we need to find the value of \( x^2 + xy + y^2 \) given \( x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \) and \( y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \).

### Step 1: Calculate \( x \)
We start with the expression for \( x \):
\[ x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \]
To simplify \( x \), we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{3} - \sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]
Calculating the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Calculating the numerator:
\[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \]
Thus, we have:
\[ x = 5 - 2\sqrt{6} \]

### Step 2: Calculate \( y \)
Now, we calculate \( y \):
\[ y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \]
Rationalizing the denominator:
\[ y = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]
Calculating the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Calculating the numerator:
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]
Thus, we have:
\[ y = 5 + 2\sqrt{6} \]

### Step 3: Calculate \( x^2 \) and \( y^2 \)
Now we calculate \( x^2 \):
\[ x^2 = (5 - 2\sqrt{6})^2 = 25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6} \]

Next, we calculate \( y^2 \):
\[ y^2 = (5 + 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6} \]

### Step 4: Calculate \( xy \)
Now we calculate \( xy \):
\[ xy = (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 25 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

### Step 5: Calculate \( x^2 + xy + y^2 \)
Now we can find \( x^2 + xy + y^2 \):
\[ x^2 + y^2 + xy = (49 - 20\sqrt{6}) + (49 + 20\sqrt{6}) + 1 \]
Simplifying this:
\[ = 49 + 49 + 1 + (-20\sqrt{6} + 20\sqrt{6}) = 98 + 1 = 99 \]

### Conclusion
Thus, \( x^2 + xy + y^2 = 99 \).

### Final Answer
99 is a multiple of 3, 9, and 11.