Assertion (A): If `alpha, beta` are the roots of `x^(2)-x+1=0`, then `alpha^(5)+beta^(5)=1`
Reason (R) : The roots of `x^(2)-x+1=0` are `omega,omega^(2)`.

Options

A. Both A, R are true and R explain Assertion
B. Both A, R are true but R does't explain A
C. A is true R is false
D. A is false R is true

Answer

To solve the problem, we need to analyze the assertion (A) and the reason (R) given in the question.

### Step-by-Step Solution:

1. **Identify the roots of the quadratic equation:**
The given quadratic equation is:
\[ x^2 - x + 1 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = 1 \):
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm i\sqrt{3}}{2} \]
Let \( \alpha = \frac{1 + i\sqrt{3}}{2} \) and \( \beta = \frac{1 - i\sqrt{3}}{2} \).

2. **Calculate \( \alpha^5 + \beta^5 \):**
To find \( \alpha^5 + \beta^5 \), we can use the property of roots of unity. The roots \( \alpha \) and \( \beta \) can be expressed in terms of \( \omega \) (the cube roots of unity):
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
The roots of the equation can be represented as \( \omega \) and \( \omega^2 \).

3. **Use the property of roots of unity:**
We know that:
\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]
Therefore, we can express \( \alpha^5 \) and \( \beta^5 \) as:
\[ \alpha^5 = (\omega)^5 = \omega^2 \quad \text{and} \quad \beta^5 = (\omega^2)^5 = \omega \]
Thus,
\[ \alpha^5 + \beta^5 = \omega^2 + \omega \]

4. **Simplify \( \alpha^5 + \beta^5 \):**
From the property of roots of unity:
\[ \omega + \omega^2 = -1 \]
Therefore,
\[ \alpha^5 + \beta^5 = -1 \]

5. **Conclusion on Assertion (A):**
The assertion states that \( \alpha^5 + \beta^5 = 1 \), which is incorrect since we found that \( \alpha^5 + \beta^5 = -1 \). Thus, assertion (A) is false.

6. **Conclusion on Reason (R):**
The reason states that the roots of the equation are \( \omega \) and \( \omega^2 \). This is true, as we have identified the roots correctly as cube roots of unity.

### Final Answer:
- Assertion (A) is false.
- Reason (R) is true.