1. V a r i a n t A. According to the condition, the function $y=\sin x+a-b x$ turns into zero at exactly two points $x_{1}$ and $x_{2}, x_{1}<x_{2}$.

These points divide the number line into 3 intervals $\left(-\infty, x_{1}\right]$, $\left(x_{1}, x_{2}\right],\left(x_{2},+\infty\right)$. Since $b \neq 0$, and $|\sin x| \leqslant 1$, the function has different signs on the intervals $\left(-\infty, x_{1}\right)$ and $\left(x_{2},+\infty\right)$. Therefore, on some two adjacent intervals $\left(-\infty, x_{1}\right),\left(x_{1}, x_{2}\right)$ or $\left(x_{1}, x_{2}\right),\left(x_{2},+\infty\right)$, the function has the same sign, and then either the point $x_{1}$ or the point $x_{2}$ is an extremum point, and the derivative at this point $y^{\prime}=(\sin x+a-b x)^{\prime}=\cos x-b$ turns into zero.
The solution to problem B is analogous.

Note. Prove independently that the system has exactly one solution.

Comment. This problem is based on the following physical problem proposed by T. Golenishcheva-Kutuzova and I. Yashchenko.

Vasya is descending with a parachute at a constant speed of $3 \mathrm{~m} / \mathrm{c}$, and his friend Misha is riding a Ferris wheel, which rotates at a constant speed (the wheel is below the starting point of the parachutist's descent). It is known that while Vasya was descending, they were at the same height exactly 4 times, including the moment when Vasya landed and the Ferris wheel cabin was at the bottom. Considering Vasya and Misha as points, find the rate of change of Misha's height at the moment they first reached the same height.

This problem allows for an intuitive solution based on the consideration of the graphs of the height changes of Misha and Vasya.

The graph of the height change of Vasya is a straight line $L$, and that of Misha is a sine wave (see Fig. 14). At the moment of Vasya's landing, the sine wave touches the $t$-axis, so "to the left" of the landing moment, the line $L$ is "above" the sine wave. Moving along $L$ to the left, if the next common point with the sine wave is a point of tangency, there will be no more common points (see Fig. 15). If not, then $L$ will be below the sine wave, and after the next common point, it will be above again. There is one common point left, so it is a point of tangency, and the rate of change of Misha's height at this moment is equal to the rate of change of Vasya's height, which is $3 \mathrm{~m} / \mathrm{c}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_562c0940e9bf5ec7656bg-23.jpg?height=316&width=1026&top_left_y=822&top_left_x=212)

Fig. 14

![](https://cdn.mathpix.com/cropped/2024_05_06_562c0940e9bf5ec7656bg-23.jpg?height=308&width=583&top_left_y=1196&top_left_x=442)

Fig. 15