## Task B-3.1.

Solve the equation $x^{2}-6 x+\frac{17}{2}=\left|\sin \frac{(2021+18 k) \pi}{6}\right|$, for each $k \in \mathbb{Z}$.
## Solution.

Let's determine the value of the expression on the right side of the given equation.

$$
\begin{aligned}
\left|\sin \frac{(2021+18 k) \pi}{6}\right| & =\left|\sin \left(\frac{2021 \pi}{6}+3 k \pi\right)\right|= \\
& =\left|\sin \left(\frac{5 \pi}{6}+3 k \pi\right)\right|
\end{aligned}
$$

If $k$ is an even integer, $\sin \left(\frac{5 \pi}{6}+3 k \pi\right)$ has the same value as $\sin \frac{5 \pi}{6}=\frac{1}{2}$, and if $k$ is an odd integer, it has the same value as $\sin \frac{11 \pi}{6}=-\frac{1}{2}$.

Therefore, $\left|\sin \left(\frac{5 \pi}{6}+3 k \pi\right)\right|=\frac{1}{2}$, so the given equation reduces to the quadratic

$x^{2}-6 x+\frac{17}{2}=\frac{1}{2}$, i.e.,

1 point

$x^{2}-6 x+8=0$.

1 point

This gives $x_{1}=2, x_{2}=4$.