A shake is an informal metric unit of time equal to 10 nanoseconds, or 10−8 seconds. It was originally coined for use in nuclear physics, helping to conveniently express the timing of various events in a nuclear reaction, especially neutron reactions.

Light-year is the distance light travels in one year. Light zips through interstellar space at 186,000 miles (300,000 kilometers) per second and 5.88 trillion miles (9.46 trillion kilometers) per year.

So shake is unlit of time and light year is unit of distance hen statement I is wrong.

Velocity gradient   $\frac{\mathrm{dV}}{\mathrm{dx}}=\frac{\text{ Dimension of Velocity }}{\text{ Dimension of Distance }}=\frac{\left[{\mathrm{LT}}^{-1}\right]}{\left[\mathrm{L}\right]}=\left[{\mathrm{T}}^{-1}\right]$. Velocity has dimensional formula ${\mathrm{LT}}^{-1}$. So they do not have same dimensional formula.