Identify the equation that DOES NOT represent oscillatory motion....

Let us analyze each option to determine if it represents oscillatory motion.

For the equation $y = A\sin\omega t + A\cos 2\omega t$, the motion is a superposition of two harmonic functions with commensurable frequencies. This represents a periodic and oscillatory motion.

For the equation x = A\cos \left(\omegat - \dfrac{\pi}{4}\right), the motion is simple harmonic, which is inherently oscillatory.

For the equationy = A\sin$\omega$t + A\cos($\omega$t + $\pi$/2)$, we can simplify it using the trigonometric identity$\cos($\theta + \pi$/2) = -\sin$\theta:<br>$y = A\sin$\omega$t - A\sin$\omega$t = 0$<br>Since the displacement$y$is zero for all time$t$, the particle remains at rest at the origin. Therefore, this does not represent oscillatory motion.<br><br>For the equation$y = A\sin \left(\omega t - \dfrac{\pi}{4}\right)$, the motion is simple harmonic, which is oscillatory.<br><br>Thus, the equation that does not represent oscillatory motion is$y = A\sin\omega t + A\cos(\omega t + \pi/2)$.<br><br>Answer:$y = A\sin\omega t + A\cos(\omega t + \pi/2)