Given x2 − sqrt(3)x + 1 = 0, find the values of x + (1)/(x) and x2 + (1)/(x2).

Here are the solutions for parts (ক) and (খ).

a) Step 1: Rearrange the given equation to find $x + \frac{1}{x}$.
Given the equation:

$$x^2 - \sqrt{3}x + 1 = 0$$

Since $x=0$ would lead to $1=0$, which is false, we know $x \neq 0$. We can divide the entire equation by $x$:

$$\frac{x^2}{x} - \frac{\sqrt{3}x}{x} + \frac{1}{x} = \frac{0}{x}$$ $$x - \sqrt{3} + \frac{1}{x} = 0$$

Add $\sqrt{3}$ to both sides:

$$x + \frac{1}{x} = \sqrt{3}$$

The value of $x + \frac{1}{x}$ is $\boxed{\sqrt{3}}$.

b) Step 1: Use the result from part (a) to find $x^2 + \frac{1}{x^2}$.
From part (a), we have $x + \frac{1}{x} = \sqrt{3}$.
Square both sides of this equation:

$$\left(x + \frac{1}{x}\right)^2 = (\sqrt{3})^2$$

Expand the left side using the formula $(a+b)^2 = a^2 + 2ab + b^2$:

$$x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 3$$ $$x^2 + 2 + \frac{1}{x^2} = 3$$

Subtract 2 from both sides:

$$x^2 + \frac{1}{x^2} = 3 - 2$$ $$x^2 + \frac{1}{x^2} = 1$$

The value of $x^2 + \frac{1}{x^2}$ is $\boxed{1}$.