In a sine wave,what is the minimum distance between two particles that always have the same speed?

Write the equation of the relation between wave speed $(v)$,angular frequency $(\omega)$,and angular wave number $(k)$.

The displacement of a particle in a medium is $y = 10^{-4} \sin(100t + 20x + \frac{\pi}{3}) \ m$,where $t$ is in seconds and $x$ is in metres. The speed of the wave is (in $m/s$)

The rope shown at an instant is carrying a wave travelling towards the right,created by a source vibrating at a frequency $n$. Consider the following statements:
$I.$ The speed of the wave is $4n \times ab$
$II.$ The medium at $a$ will be in the same phase as $d$ after $\frac{4}{3n} \text{ s}$
$III.$ The phase difference between $b$ and $e$ is $\frac{3\pi}{2}$
Which of these statements are correct?

The phase difference between two waves represented by $y_1 = 10^{-6} \sin [100t + (x/50) + 0.5] \, m$ and $y_2 = 10^{-6} \cos [100t + (x/50)] \, m$,where $x$ is expressed in meters and $t$ is expressed in seconds,is approximately .... $rad$.

For a wave represented by the equation $y = 3 \cos \left( \frac{x}{4} - 10t - \frac{\pi}{2} \right)$,what is the maximum velocity of the particle of the medium?